Physics Book - User contributions [en]

Multi-particle analysis of Momentum

2018-04-18T20:28:34Z

Sdhal3: /* The Main Idea */

'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' '''Claimed by Simran Dhal (sdhal3) (Spring 2018)'''

==The Main Idea==

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

A multi-particle analysis of momentum can be applied to systems with multiple objects that may be interacting. Using the same principles of momentum applied on single objects, we apply these on the multiple objects that may make up a chosen system. However, one interesting characteristic of the multi-particle analysis of momentum is that it only factors in external forces applied on the object as opposed to internal forces which would be applied by objects on one another within a system even though objects within the system are interacting.

'''Multi-particle System Momentum Principle'''

<math> ∆\vec{p} = \vec{F}_{net} * {∆t}</math>

Where Fnet is the sum of all external forces on the system. Therefore:

<math>\vec{F}_{net} = \vec{F}_{1} + \vec{F}_{2} + \vec{F}_{3} + ...</math>

[[File:ExternalForces.png|300px|thumb|left|Figure 1]]

From Figure 1 we can see that there are internal forces within the system since each object contributes a force to its surrounding objects. However, when applying the multi-particle momentum principle, we only look at the external forces. So in this case, to sum the forces in order to find the change in momentum we would add F1,surr + F2,surr + F3,surr + F4,surr to sum up to an Ftotal that we would multiply by the time elapsed in order to determine the change in momentum.

The reason that the internal forces are not used within the momentum principle is because the internal forces would cancel due to the principle of reciprocity. Since each object exerts an equal and opposite force on each other, all of the internal forces would cancel, leaving only the external forces. This makes solving the momentum for the system, which would have been a complicated to solve for with a multitude of forces, much simpler.

Theory of reciprocity results in this:

<math>\vec{F}_{1,2} = \vec{-F}_{2,1}</math>

<math>\vec{F}_{1,3} = \vec{-F}_{3,1}</math>

<math>\vec{F}_{1,4} = \vec{-F}_{1,1}</math>

And this would be the case for every object in the system (objects 2,3 and 4).

'''More on Proving the Cancellation of Internal Forces:'''

Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as

<math>{\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net}</math> (or <math>∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

<math> P = p_1 + p_2 = m_1v_1 + m_2v_2</math>

and deriving from <math> F = m * a </math>, the rate of change is the sum of the vector forces, and therefore:

<math> \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 </math>.

Newton's third law gives that:

<math> \frac{d\vec{P}}{dt} = 0 </math>,

'''so,'''

<math> m_1\vec{a}\_1 + m_2\vec{a}_2 = 0</math>.

<math>\vec{F}_{12} = -\vec{F}_{21} </math>.

<math> \vec{F}_{12} = m_2\vec{a}_2 </math>.

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

<math> \vec{P} = \sum_{i=1}^N m_i\vec{v}_i </math>
'''and'''
<math> \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} </math>.

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.

See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]

===A Mathematical Model===

Mathematically, momentum is the product of a system's mass and the change in its velocity. It is also defined as the product of the net force on the system and the time in which the force was applied. This principle stays the same for a multi-particle system and for a multi-particle system we would define the momentum as this:

<math>\vec{p}_{sys}= \vec{p}_{1} + \vec{p}_{2} + \vec{p}_{3}</math>

And for a multi-particle system, since internal forces cancel, to find the total force on on the system, we add only the external forces applied to each object within the system:

And use the sum of the forces in the following equation:

<math>\vec{∆p}_{sys} = \vec{F}_{net,surr} * ∆t </math>

This equation can essentially be broken down further into this:

<math>\vec{∆p}_{1} + \vec{∆p}_{2} + \vec{∆p}_{3} = (\vec{F}_{1,surr} + \vec{F}_{2,surr} + \vec{F}_{3,surr}) * ∆t</math>

===Computational Models===

Click on the link below to understand the basics of the momentum principle in VPython,

[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

or check out this collision simulation to see a multi-particle analysis of it:

[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]

==Examples==

===Simple===

A loaded cart is travelling with momentum Mcart, and a brick is falling with momentum Mbrick. As the cart advances forward, the brick lands on it and the now heavier cart continues to move forward. The initial momentum of the system includes that of the cart and of the brick. Symbolically, what is the final momentum of the system?

'''Answer: Mcart + Mbrick'''

'''Explanation:'''

As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

===Middle===

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

'''Answer: 9.4e4 N'''

'''Explanation:'''

With the car as particle 1 and the truck as particle 2,
total initial momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>

Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.

===Difficult===

A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

'''Explanation:'''
Momentum before: 0,
Momentum after: <math> (2m(v - v_f) - m_{bin} * v_f) * i </math>
so,
<math> 0 = 2m_{people}(v - v_f) - m_{bin} * v</math>

Then solve for <math> v_f </math>.

==Connectedness==

Multi-particle analysis of momentum is just one way of analyzing a particular system, but multi-particle analysis is a way to examine other systems as well, and the momentum principle is very important for a variety of professional fields.

One application includes its use by automobile safety engineers, who must take into account the physics of momentum in order to safely design vehicles for their customers' use. However, this is by no means the only application--momentum is an applicable concept for literally anything that moves and has mass: figure skaters, roller coasters, pool balls...the list goes on.

For computer scientists, it's possible to create simulations like those listed in the "Computational Models" section of this article! In order to design those it is necessary for the creator to understand not only computer science concepts, but also the physics concepts that they are attempting to represent. In any field, it is possible that momentum comes into play.

==History==

The scientists most credited for the introduction of the concept of momentum to their field are Rene Descartes and Isaac Newton. Newton is more acknowledged for his Laws, from which the more specific concept of Conservation of Momentum can be derived, while Descartes holds the original "discovery" of momentum.

More information on the history of momentum can be found at:
[http://www.physicsbook.gatech.edu/Momentum_Principle#Connectedness] and
[http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html]

== See also ==

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit:
[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]

and to then explore the Conservation of Momentum page:
[http://www.physicsbook.gatech.edu/Conservation_of_Momentum].

Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[http://www.physicsbook.gatech.edu/Main_Page# Main Page].

Check out the links in the references below for more information on this subject as well as other physics concepts!

==References==
[1] Fitzpatrick, R. (2006, February 2). Angular momentum of a point particle. Retrieved November 27, 2016, from http://farside.ph.utexas.edu/teaching/301/lectures/node118.html

[2] Torre, Charles. "Newton's Third Law. Multi-Particle Systems." Newton’s Third Law. Multi-particle Systems. Conservation of Momentum. Physics 3550, Fall 2012 Newton’s Third Law. Multi-particle Systems. Relevant Sections in Text: §1.5, 3.1, 3.2, 3.3 (2012): n. pag. Intermediate Classical Mechanics. Utah State University, Sept. 2012. Web. 27 Nov. 2016.

[3] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 27 Nov. 2016. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[4] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[5] Fenton, Flavio. "Momentum and Second Newton's Law." 27 Nov. 2016. Lecture.

[6] Henderson, Tom. "Momentum and Its Conservation." Momentum and Its Conservation. The Physics Clasroom, 31 Dec. 2012. Web. 27 Nov. 2016.

[7] Dourmashkin, Peter. Lister, J. David. Pritchard, David E. Surrow, Bernd. "Momentum and Collisions." Massachusetts Institute of Technology. Sep. 2004. Lecture.

[[Category: Multiparticle Systems]]

Multi-particle analysis of Momentum

2018-04-18T20:24:58Z

Sdhal3: /* Middling */

'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' '''Claimed by Simran Dhal (sdhal3) (Spring 2018)'''

==The Main Idea==

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

A multi-particle analysis of momentum can be applied to systems with multiple objects that may be interacting. Using the same principles of momentum applied on single objects, we apply these on the multiple objects that may make up a chosen system. However, one interesting characteristic of the multi-particle analysis of momentum is that it only factors in external forces applied on the object as opposed to internal forces which would be applied by objects on one another within a system even though objects within the system are interacting.

'''Multi-particle System Momentum Principle'''

<math> ∆\vec{p} = \vec{F}_{net} * {∆t}</math>

Where Fnet is the sum of all external forces on the system. Therefore:

<math>\vec{F}_{net} = \vec{F}_{1} + \vec{F}_{2} + \vec{F}_{3} + ...</math>

[[File:ExternalForces.png|300px|thumb|left|Figure 1]]

From Figure 1 we can see that there are internal forces within the system since each object contributes a force to its surrounding objects. However, when applying the multi-particle momentum principle, we only look at the external forces. So in this case, to sum the forces in order to find the change in momentum we would add F1,surr + F2,surr + F3,surr + F4,surr to sum up to an Ftotal that we would multiply by the time elapsed in order to determine the change in momentum.

The reason that the internal forces are not used within the momentum principle is because the internal forces would cancel due to the principle of reciprocity. Since each object exerts an equal and opposite force on each other, all of the internal forces would cancel, leaving only the external forces. This makes solving the momentum for the system, which would have been a complicated to solve for with a multitude of forces, much simpler.

Theory of reciprocity results in this:

<math>\vec{F}_{1,2} = \vec{-F}_{2,1}</math>

<math>\vec{F}_{1,3} = \vec{-F}_{3,1}</math>

<math>\vec{F}_{1,4} = \vec{-F}_{1,1}</math>

And this would be the case for every object in the system (objects 2,3 and 4).

===A Mathematical Model===

Mathematically, momentum is the product of a system's mass and the change in its velocity. It is also defined as the product of the net force on the system and the time in which the force was applied. This principle stays the same for a multi-particle system and for a multi-particle system we would define the momentum as this:

<math>\vec{p}_{sys}= \vec{p}_{1} + \vec{p}_{2} + \vec{p}_{3}</math>

And for a multi-particle system, since internal forces cancel, to find the total force on on the system, we add only the external forces applied to each object within the system:

And use the sum of the forces in the following equation:

<math>\vec{∆p}_{sys} = \vec{F}_{net,surr} * ∆t </math>

This equation can essentially be broken down further into this:

<math>\vec{∆p}_{1} + \vec{∆p}_{2} + \vec{∆p}_{3} = (\vec{F}_{1,surr} + \vec{F}_{2,surr} + \vec{F}_{3,surr}) * ∆t</math>

Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as

<math>{\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net}</math> (or <math>∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

<math> P = p_1 + p_2 = m_1v_1 + m_2v_2</math>

and deriving from <math> F = m * a </math>, the rate of change is the sum of the vector forces, and therefore:

<math> \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 </math>.

Newton's third law gives that:

<math> \frac{d\vec{P}}{dt} = 0 </math>,

'''so,'''

<math> m_1\vec{a}\_1 + m_2\vec{a}_2 = 0</math>.

<math>\vec{F}_{12} = -\vec{F}_{21} </math>.

<math> \vec{F}_{12} = m_2\vec{a}_2 </math>.

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

<math> \vec{P} = \sum_{i=1}^N m_i\vec{v}_i </math>
'''and'''
<math> \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} </math>.

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.

See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]

===Computational Models===

Click on the link below to understand the basics of the momentum principle in VPython,

[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

or check out this collision simulation to see a multi-particle analysis of it:

[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]

==Examples==

===Simple===

A loaded cart is travelling with momentum Mcart, and a brick is falling with momentum Mbrick. As the cart advances forward, the brick lands on it and the now heavier cart continues to move forward. The initial momentum of the system includes that of the cart and of the brick. Symbolically, what is the final momentum of the system?

'''Answer: Mcart + Mbrick'''

'''Explanation:'''

As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

===Middle===

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

'''Answer: 9.4e4 N'''

'''Explanation:'''

With the car as particle 1 and the truck as particle 2,
total initial momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>

Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.

===Difficult===

A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

'''Explanation:'''
Momentum before: 0,
Momentum after: <math> (2m(v - v_f) - m_{bin} * v_f) * i </math>
so,
<math> 0 = 2m_{people}(v - v_f) - m_{bin} * v</math>

Then solve for <math> v_f </math>.

==Connectedness==

Multi-particle analysis of momentum is just one way of analyzing a particular system, but multi-particle analysis is a way to examine other systems as well, and the momentum principle is very important for a variety of professional fields.

One application includes its use by automobile safety engineers, who must take into account the physics of momentum in order to safely design vehicles for their customers' use. However, this is by no means the only application--momentum is an applicable concept for literally anything that moves and has mass: figure skaters, roller coasters, pool balls...the list goes on.

For computer scientists, it's possible to create simulations like those listed in the "Computational Models" section of this article! In order to design those it is necessary for the creator to understand not only computer science concepts, but also the physics concepts that they are attempting to represent. In any field, it is possible that momentum comes into play.

==History==

The scientists most credited for the introduction of the concept of momentum to their field are Rene Descartes and Isaac Newton. Newton is more acknowledged for his Laws, from which the more specific concept of Conservation of Momentum can be derived, while Descartes holds the original "discovery" of momentum.

More information on the history of momentum can be found at:
[http://www.physicsbook.gatech.edu/Momentum_Principle#Connectedness] and
[http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html]

== See also ==

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit:
[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]

and to then explore the Conservation of Momentum page:
[http://www.physicsbook.gatech.edu/Conservation_of_Momentum].

Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[http://www.physicsbook.gatech.edu/Main_Page# Main Page].

Check out the links in the references below for more information on this subject as well as other physics concepts!

==References==
[1] Fitzpatrick, R. (2006, February 2). Angular momentum of a point particle. Retrieved November 27, 2016, from http://farside.ph.utexas.edu/teaching/301/lectures/node118.html

[2] Torre, Charles. "Newton's Third Law. Multi-Particle Systems." Newton’s Third Law. Multi-particle Systems. Conservation of Momentum. Physics 3550, Fall 2012 Newton’s Third Law. Multi-particle Systems. Relevant Sections in Text: §1.5, 3.1, 3.2, 3.3 (2012): n. pag. Intermediate Classical Mechanics. Utah State University, Sept. 2012. Web. 27 Nov. 2016.

[3] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 27 Nov. 2016. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[4] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[5] Fenton, Flavio. "Momentum and Second Newton's Law." 27 Nov. 2016. Lecture.

[6] Henderson, Tom. "Momentum and Its Conservation." Momentum and Its Conservation. The Physics Clasroom, 31 Dec. 2012. Web. 27 Nov. 2016.

[7] Dourmashkin, Peter. Lister, J. David. Pritchard, David E. Surrow, Bernd. "Momentum and Collisions." Massachusetts Institute of Technology. Sep. 2004. Lecture.

[[Category: Multiparticle Systems]]

Multi-particle analysis of Momentum

2018-04-18T20:22:57Z

Sdhal3: /* A Mathematical Model */

'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' '''Claimed by Simran Dhal (sdhal3) (Spring 2018)'''

==The Main Idea==

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

A multi-particle analysis of momentum can be applied to systems with multiple objects that may be interacting. Using the same principles of momentum applied on single objects, we apply these on the multiple objects that may make up a chosen system. However, one interesting characteristic of the multi-particle analysis of momentum is that it only factors in external forces applied on the object as opposed to internal forces which would be applied by objects on one another within a system even though objects within the system are interacting.

'''Multi-particle System Momentum Principle'''

<math> ∆\vec{p} = \vec{F}_{net} * {∆t}</math>

Where Fnet is the sum of all external forces on the system. Therefore:

<math>\vec{F}_{net} = \vec{F}_{1} + \vec{F}_{2} + \vec{F}_{3} + ...</math>

[[File:ExternalForces.png|300px|thumb|left|Figure 1]]

From Figure 1 we can see that there are internal forces within the system since each object contributes a force to its surrounding objects. However, when applying the multi-particle momentum principle, we only look at the external forces. So in this case, to sum the forces in order to find the change in momentum we would add F1,surr + F2,surr + F3,surr + F4,surr to sum up to an Ftotal that we would multiply by the time elapsed in order to determine the change in momentum.

The reason that the internal forces are not used within the momentum principle is because the internal forces would cancel due to the principle of reciprocity. Since each object exerts an equal and opposite force on each other, all of the internal forces would cancel, leaving only the external forces. This makes solving the momentum for the system, which would have been a complicated to solve for with a multitude of forces, much simpler.

Theory of reciprocity results in this:

<math>\vec{F}_{1,2} = \vec{-F}_{2,1}</math>

<math>\vec{F}_{1,3} = \vec{-F}_{3,1}</math>

<math>\vec{F}_{1,4} = \vec{-F}_{1,1}</math>

And this would be the case for every object in the system (objects 2,3 and 4).

===A Mathematical Model===

Mathematically, momentum is the product of a system's mass and the change in its velocity. It is also defined as the product of the net force on the system and the time in which the force was applied. This principle stays the same for a multi-particle system and for a multi-particle system we would define the momentum as this:

<math>\vec{p}_{sys}= \vec{p}_{1} + \vec{p}_{2} + \vec{p}_{3}</math>

And for a multi-particle system, since internal forces cancel, to find the total force on on the system, we add only the external forces applied to each object within the system:

And use the sum of the forces in the following equation:

<math>\vec{∆p}_{sys} = \vec{F}_{net,surr} * ∆t </math>

This equation can essentially be broken down further into this:

<math>\vec{∆p}_{1} + \vec{∆p}_{2} + \vec{∆p}_{3} = (\vec{F}_{1,surr} + \vec{F}_{2,surr} + \vec{F}_{3,surr}) * ∆t</math>

Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as

<math>{\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net}</math> (or <math>∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

<math> P = p_1 + p_2 = m_1v_1 + m_2v_2</math>

and deriving from <math> F = m * a </math>, the rate of change is the sum of the vector forces, and therefore:

<math> \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 </math>.

Newton's third law gives that:

<math> \frac{d\vec{P}}{dt} = 0 </math>,

'''so,'''

<math> m_1\vec{a}\_1 + m_2\vec{a}_2 = 0</math>.

<math>\vec{F}_{12} = -\vec{F}_{21} </math>.

<math> \vec{F}_{12} = m_2\vec{a}_2 </math>.

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

<math> \vec{P} = \sum_{i=1}^N m_i\vec{v}_i </math>
'''and'''
<math> \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} </math>.

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.

See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]

===Computational Models===

Click on the link below to understand the basics of the momentum principle in VPython,

[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

or check out this collision simulation to see a multi-particle analysis of it:

[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]

==Examples==

===Simple===

A loaded cart is travelling with momentum Mcart, and a brick is falling with momentum Mbrick. As the cart advances forward, the brick lands on it and the now heavier cart continues to move forward. The initial momentum of the system includes that of the cart and of the brick. Symbolically, what is the final momentum of the system?

'''Answer: Mcart + Mbrick'''

'''Explanation:'''

As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

===Middling===

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

'''Answer: 9.4e4 N'''

'''Explanation:'''

With the car as particle 1 and the truck as particle 2,
total initial momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>

Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.

===Difficult===

A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

'''Explanation:'''
Momentum before: 0,
Momentum after: <math> (2m(v - v_f) - m_{bin} * v_f) * i </math>
so,
<math> 0 = 2m_{people}(v - v_f) - m_{bin} * v</math>

Then solve for <math> v_f </math>.

==Connectedness==

Multi-particle analysis of momentum is just one way of analyzing a particular system, but multi-particle analysis is a way to examine other systems as well, and the momentum principle is very important for a variety of professional fields.

One application includes its use by automobile safety engineers, who must take into account the physics of momentum in order to safely design vehicles for their customers' use. However, this is by no means the only application--momentum is an applicable concept for literally anything that moves and has mass: figure skaters, roller coasters, pool balls...the list goes on.

For computer scientists, it's possible to create simulations like those listed in the "Computational Models" section of this article! In order to design those it is necessary for the creator to understand not only computer science concepts, but also the physics concepts that they are attempting to represent. In any field, it is possible that momentum comes into play.

==History==

The scientists most credited for the introduction of the concept of momentum to their field are Rene Descartes and Isaac Newton. Newton is more acknowledged for his Laws, from which the more specific concept of Conservation of Momentum can be derived, while Descartes holds the original "discovery" of momentum.

More information on the history of momentum can be found at:
[http://www.physicsbook.gatech.edu/Momentum_Principle#Connectedness] and
[http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html]

== See also ==

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit:
[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]

and to then explore the Conservation of Momentum page:
[http://www.physicsbook.gatech.edu/Conservation_of_Momentum].

Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[http://www.physicsbook.gatech.edu/Main_Page# Main Page].

Check out the links in the references below for more information on this subject as well as other physics concepts!

==References==
[1] Fitzpatrick, R. (2006, February 2). Angular momentum of a point particle. Retrieved November 27, 2016, from http://farside.ph.utexas.edu/teaching/301/lectures/node118.html

[2] Torre, Charles. "Newton's Third Law. Multi-Particle Systems." Newton’s Third Law. Multi-particle Systems. Conservation of Momentum. Physics 3550, Fall 2012 Newton’s Third Law. Multi-particle Systems. Relevant Sections in Text: §1.5, 3.1, 3.2, 3.3 (2012): n. pag. Intermediate Classical Mechanics. Utah State University, Sept. 2012. Web. 27 Nov. 2016.

[3] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 27 Nov. 2016. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[4] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[5] Fenton, Flavio. "Momentum and Second Newton's Law." 27 Nov. 2016. Lecture.

[6] Henderson, Tom. "Momentum and Its Conservation." Momentum and Its Conservation. The Physics Clasroom, 31 Dec. 2012. Web. 27 Nov. 2016.

[7] Dourmashkin, Peter. Lister, J. David. Pritchard, David E. Surrow, Bernd. "Momentum and Collisions." Massachusetts Institute of Technology. Sep. 2004. Lecture.

[[Category: Multiparticle Systems]]

Multi-particle analysis of Momentum

2018-04-18T20:00:19Z

Sdhal3: /* A Mathematical Model */

'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' '''Claimed by Simran Dhal (sdhal3) (Spring 2018)'''

==The Main Idea==

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

A multi-particle analysis of momentum can be applied to systems with multiple objects that may be interacting. Using the same principles of momentum applied on single objects, we apply these on the multiple objects that may make up a chosen system. However, one interesting characteristic of the multi-particle analysis of momentum is that it only factors in external forces applied on the object as opposed to internal forces which would be applied by objects on one another within a system even though objects within the system are interacting.

'''Multi-particle System Momentum Principle'''

<math> ∆\vec{p} = \vec{F}_{net} * {∆t}</math>

Where Fnet is the sum of all external forces on the system. Therefore:

<math>\vec{F}_{net} = \vec{F}_{1} + \vec{F}_{2} + \vec{F}_{3} + ...</math>

[[File:ExternalForces.png|300px|thumb|left|Figure 1]]

From Figure 1 we can see that there are internal forces within the system since each object contributes a force to its surrounding objects. However, when applying the multi-particle momentum principle, we only look at the external forces. So in this case, to sum the forces in order to find the change in momentum we would add F1,surr + F2,surr + F3,surr + F4,surr to sum up to an Ftotal that we would multiply by the time elapsed in order to determine the change in momentum.

The reason that the internal forces are not used within the momentum principle is because the internal forces would cancel due to the principle of reciprocity. Since each object exerts an equal and opposite force on each other, all of the internal forces would cancel, leaving only the external forces. This makes solving the momentum for the system, which would have been a complicated to solve for with a multitude of forces, much simpler.

Theory of reciprocity results in this:

<math>\vec{F}_{1,2} = \vec{-F}_{2,1}</math>

<math>\vec{F}_{1,3} = \vec{-F}_{3,1}</math>

<math>\vec{F}_{1,4} = \vec{-F}_{1,1}</math>

And this would be the case for every object in the system (objects 2,3 and 4).

===A Mathematical Model===

Mathematically, momentum is the product of a body's (in our case, a particle's) mass and velocity. It is also defined as the product of mass, net force, and change in time.

Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as

<math>{\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net}</math> (or <math>∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

<math> P = p_1 + p_2 = m_1v_1 + m_2v_2</math>

and deriving from <math> F = m * a </math>, the rate of change is the sum of the vector forces, and therefore:

<math> \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 </math>.

Newton's third law gives that:

<math> \frac{d\vec{P}}{dt} = 0 </math>,

'''so,'''

<math> m_1\vec{a}\_1 + m_2\vec{a}_2 = 0</math>.

<math>\vec{F}_{12} = -\vec{F}_{21} </math>.

<math> \vec{F}_{12} = m_2\vec{a}_2 </math>.

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

<math> \vec{P} = \sum_{i=1}^N m_i\vec{v}_i </math>
'''and'''
<math> \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} </math>.

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.

See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]

===Computational Models===

Click on the link below to understand the basics of the momentum principle in VPython,

[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

or check out this collision simulation to see a multi-particle analysis of it:

[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]

==Examples==

===Simple===

A loaded cart is travelling with momentum Mcart, and a brick is falling with momentum Mbrick. As the cart advances forward, the brick lands on it and the now heavier cart continues to move forward. The initial momentum of the system includes that of the cart and of the brick. Symbolically, what is the final momentum of the system?

'''Answer: Mcart + Mbrick'''

'''Explanation:'''

As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

===Middling===

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

'''Answer: 9.4e4 N'''

'''Explanation:'''

With the car as particle 1 and the truck as particle 2,
total initial momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>

Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.

===Difficult===

A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

'''Explanation:'''
Momentum before: 0,
Momentum after: <math> (2m(v - v_f) - m_{bin} * v_f) * i </math>
so,
<math> 0 = 2m_{people}(v - v_f) - m_{bin} * v</math>

Then solve for <math> v_f </math>.

==Connectedness==

Multi-particle analysis of momentum is just one way of analyzing a particular system, but multi-particle analysis is a way to examine other systems as well, and the momentum principle is very important for a variety of professional fields.

One application includes its use by automobile safety engineers, who must take into account the physics of momentum in order to safely design vehicles for their customers' use. However, this is by no means the only application--momentum is an applicable concept for literally anything that moves and has mass: figure skaters, roller coasters, pool balls...the list goes on.

For computer scientists, it's possible to create simulations like those listed in the "Computational Models" section of this article! In order to design those it is necessary for the creator to understand not only computer science concepts, but also the physics concepts that they are attempting to represent. In any field, it is possible that momentum comes into play.

==History==

The scientists most credited for the introduction of the concept of momentum to their field are Rene Descartes and Isaac Newton. Newton is more acknowledged for his Laws, from which the more specific concept of Conservation of Momentum can be derived, while Descartes holds the original "discovery" of momentum.

More information on the history of momentum can be found at:
[http://www.physicsbook.gatech.edu/Momentum_Principle#Connectedness] and
[http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html]

== See also ==

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit:
[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]

and to then explore the Conservation of Momentum page:
[http://www.physicsbook.gatech.edu/Conservation_of_Momentum].

Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[http://www.physicsbook.gatech.edu/Main_Page# Main Page].

Check out the links in the references below for more information on this subject as well as other physics concepts!

==References==
[1] Fitzpatrick, R. (2006, February 2). Angular momentum of a point particle. Retrieved November 27, 2016, from http://farside.ph.utexas.edu/teaching/301/lectures/node118.html

[2] Torre, Charles. "Newton's Third Law. Multi-Particle Systems." Newton’s Third Law. Multi-particle Systems. Conservation of Momentum. Physics 3550, Fall 2012 Newton’s Third Law. Multi-particle Systems. Relevant Sections in Text: §1.5, 3.1, 3.2, 3.3 (2012): n. pag. Intermediate Classical Mechanics. Utah State University, Sept. 2012. Web. 27 Nov. 2016.

[3] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 27 Nov. 2016. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[4] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[5] Fenton, Flavio. "Momentum and Second Newton's Law." 27 Nov. 2016. Lecture.

[6] Henderson, Tom. "Momentum and Its Conservation." Momentum and Its Conservation. The Physics Clasroom, 31 Dec. 2012. Web. 27 Nov. 2016.

[7] Dourmashkin, Peter. Lister, J. David. Pritchard, David E. Surrow, Bernd. "Momentum and Collisions." Massachusetts Institute of Technology. Sep. 2004. Lecture.

[[Category: Multiparticle Systems]]

Multi-particle analysis of Momentum

2018-04-18T19:54:25Z

Sdhal3:

'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' '''Claimed by Simran Dhal (sdhal3) (Spring 2018)'''

==The Main Idea==

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

A multi-particle analysis of momentum can be applied to systems with multiple objects that may be interacting. Using the same principles of momentum applied on single objects, we apply these on the multiple objects that may make up a chosen system. However, one interesting characteristic of the multi-particle analysis of momentum is that it only factors in external forces applied on the object as opposed to internal forces which would be applied by objects on one another within a system even though objects within the system are interacting.

'''Multi-particle System Momentum Principle'''

<math> ∆\vec{p} = \vec{F}_{net} * {∆t}</math>

Where Fnet is the sum of all external forces on the system. Therefore:

<math>\vec{F}_{net} = \vec{F}_{1} + \vec{F}_{2} + \vec{F}_{3} + ...</math>

[[File:ExternalForces.png|300px|thumb|left|Figure 1]]

From Figure 1 we can see that there are internal forces within the system since each object contributes a force to its surrounding objects. However, when applying the multi-particle momentum principle, we only look at the external forces. So in this case, to sum the forces in order to find the change in momentum we would add F1,surr + F2,surr + F3,surr + F4,surr to sum up to an Ftotal that we would multiply by the time elapsed in order to determine the change in momentum.

The reason that the internal forces are not used within the momentum principle is because the internal forces would cancel due to the principle of reciprocity. Since each object exerts an equal and opposite force on each other, all of the internal forces would cancel, leaving only the external forces. This makes solving the momentum for the system, which would have been a complicated to solve for with a multitude of forces, much simpler.

Theory of reciprocity results in this:

<math>\vec{F}_{1,2} = \vec{-F}_{2,1}</math>

<math>\vec{F}_{1,3} = \vec{-F}_{3,1}</math>

<math>\vec{F}_{1,4} = \vec{-F}_{1,1}</math>

And this would be the case for every object in the system (objects 2,3 and 4).

===A Mathematical Model===

Mathematically, momentum is the product of a body's (in our case, a particle's) mass and velocity. It is also defined as the product of mass, net force, and change in time.

Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as

<math>{\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net}</math> (or <math>∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

<math> P = p_1 + p_2 = m_1v_1 + m_2v_2</math>

and deriving from <math> F = m * a </math>, the rate of change is the sum of the vector forces, and therefore:

<math> \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 </math>.

Newton's third law gives that:

<math> \frac{d\vec{P}}{dt} = 0 </math>,

'''so,'''

<math> m_1\vec{a}\_1 + m_2\vec{a}_2 = 0</math>.

<math>\vec{F}_{12} = -\vec{F}_{21} </math>.

<math> \vec{F}_{12} = m_2\vec{a}_2 </math>.

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

<math> \vec{P} = \sum_{i=1}^N m_i\vec{v}_i </math>
'''and'''
<math> \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} </math>.

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.

See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]

===Computational Models===

Click on the link below to understand the basics of the momentum principle in VPython,

[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

or check out this collision simulation to see a multi-particle analysis of it:

[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]

==Examples==

===Simple===

A loaded cart is travelling with momentum Mcart, and a brick is falling with momentum Mbrick. As the cart advances forward, the brick lands on it and the now heavier cart continues to move forward. The initial momentum of the system includes that of the cart and of the brick. Symbolically, what is the final momentum of the system?

'''Answer: Mcart + Mbrick'''

'''Explanation:'''

As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

===Middling===

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

'''Answer: 9.4e4 N'''

'''Explanation:'''

With the car as particle 1 and the truck as particle 2,
total initial momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>

Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.

===Difficult===

A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

'''Explanation:'''
Momentum before: 0,
Momentum after: <math> (2m(v - v_f) - m_{bin} * v_f) * i </math>
so,
<math> 0 = 2m_{people}(v - v_f) - m_{bin} * v</math>

Then solve for <math> v_f </math>.

==Connectedness==

Multi-particle analysis of momentum is just one way of analyzing a particular system, but multi-particle analysis is a way to examine other systems as well, and the momentum principle is very important for a variety of professional fields.

One application includes its use by automobile safety engineers, who must take into account the physics of momentum in order to safely design vehicles for their customers' use. However, this is by no means the only application--momentum is an applicable concept for literally anything that moves and has mass: figure skaters, roller coasters, pool balls...the list goes on.

For computer scientists, it's possible to create simulations like those listed in the "Computational Models" section of this article! In order to design those it is necessary for the creator to understand not only computer science concepts, but also the physics concepts that they are attempting to represent. In any field, it is possible that momentum comes into play.

==History==

The scientists most credited for the introduction of the concept of momentum to their field are Rene Descartes and Isaac Newton. Newton is more acknowledged for his Laws, from which the more specific concept of Conservation of Momentum can be derived, while Descartes holds the original "discovery" of momentum.

More information on the history of momentum can be found at:
[http://www.physicsbook.gatech.edu/Momentum_Principle#Connectedness] and
[http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html]

== See also ==

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit:
[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]

and to then explore the Conservation of Momentum page:
[http://www.physicsbook.gatech.edu/Conservation_of_Momentum].

Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[http://www.physicsbook.gatech.edu/Main_Page# Main Page].

Check out the links in the references below for more information on this subject as well as other physics concepts!

==References==
[1] Fitzpatrick, R. (2006, February 2). Angular momentum of a point particle. Retrieved November 27, 2016, from http://farside.ph.utexas.edu/teaching/301/lectures/node118.html

[2] Torre, Charles. "Newton's Third Law. Multi-Particle Systems." Newton’s Third Law. Multi-particle Systems. Conservation of Momentum. Physics 3550, Fall 2012 Newton’s Third Law. Multi-particle Systems. Relevant Sections in Text: §1.5, 3.1, 3.2, 3.3 (2012): n. pag. Intermediate Classical Mechanics. Utah State University, Sept. 2012. Web. 27 Nov. 2016.

[3] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 27 Nov. 2016. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[4] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[5] Fenton, Flavio. "Momentum and Second Newton's Law." 27 Nov. 2016. Lecture.

[6] Henderson, Tom. "Momentum and Its Conservation." Momentum and Its Conservation. The Physics Clasroom, 31 Dec. 2012. Web. 27 Nov. 2016.

[7] Dourmashkin, Peter. Lister, J. David. Pritchard, David E. Surrow, Bernd. "Momentum and Collisions." Massachusetts Institute of Technology. Sep. 2004. Lecture.

[[Category: Multiparticle Systems]]

Multi-particle analysis of Momentum

2018-04-18T19:43:21Z

Sdhal3: /* The Main Idea */

'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' "Claimed by Simran Dhal (Spring 2018)"

==The Main Idea==

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

A multi-particle analysis of momentum can be applied to systems with multiple objects that may be interacting. Using the same principles of momentum applied on single objects, we apply these on the multiple objects that may make up a chosen system. However, one interesting characteristic of the multi-particle analysis of momentum is that it only factors in external forces applied on the object as opposed to internal forces which would be applied by objects on one another within a system even though objects within the system are interacting.

"Multi-particle System Momentum Principle"

<math> ∆\vec{p} = \vec{F}_{net} * {∆t}</math>

Where Fnet is the sum of all external forces on the system. Therefore:

<math>\vec{F}_{net} = \vec{F}_{1} + \vec{F}_{2} + \vec{F}_{3} + ...</math>

[[File:ExternalForces.png|300px|thumb|left|Figure 1]]

From Figure 1 we can see that there are internal forces within the system since each object contributes a force to its surrounding objects. However, when applying the multi-particle momentum principle, we only look at the external forces. So in this case, to sum the forces in order to find the change in momentum we would add F1,surr + F2,surr + F3,surr + F4,surr to sum up to an Ftotal that we would multiply by the time elapsed in order to determine the change in momentum.

The reason that the internal forces are not used within the momentum principle is because the internal forces would cancel due to the principle of reciprocity. Since each object exerts an equal and opposite force on each other, all of the internal forces would cancel, leaving only the external forces. This makes solving the momentum for the system, which would have been a complicated to solve for with a multitude of forces, much simpler.

Theory of reciprocity results in this:

<math>\vec{F}_{1,2} = \vec{-F}_{2,1}</math>

<math>\vec{F}_{1,3} = \vec{-F}_{3,1}</math>

<math>\vec{F}_{1,4} = \vec{-F}_{1,1}</math>

And this would be the case for every object in the system (objects 2,3 and 4).

===A Mathematical Model===

Mathematically, momentum is the product of a body's (in our case, a particle's) mass and velocity. It is also defined as the product of mass, net force, and change in time.

Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as

<math>{\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net}</math> (or <math>∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

<math> P = p_1 + p_2 = m_1v_1 + m_2v_2</math>

and deriving from <math> F = m * a </math>, the rate of change is the sum of the vector forces, and therefore:

<math> \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 </math>.

Newton's third law gives that:

<math> \frac{d\vec{P}}{dt} = 0 </math>,

'''so,'''

<math> m_1\vec{a}\_1 + m_2\vec{a}_2 = 0</math>.

<math>\vec{F}_{12} = -\vec{F}_{21} </math>.

<math> \vec{F}_{12} = m_2\vec{a}_2 </math>.

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

<math> \vec{P} = \sum_{i=1}^N m_i\vec{v}_i </math>
'''and'''
<math> \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} </math>.

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.

See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]

===Computational Models===

Click on the link below to understand the basics of the momentum principle in VPython,

[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

or check out this collision simulation to see a multi-particle analysis of it:

[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]

==Examples==

===Simple===

A loaded cart is travelling with momentum Mcart, and a brick is falling with momentum Mbrick. As the cart advances forward, the brick lands on it and the now heavier cart continues to move forward. The initial momentum of the system includes that of the cart and of the brick. Symbolically, what is the final momentum of the system?

'''Answer: Mcart + Mbrick'''

'''Explanation:'''

As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

===Middling===

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

'''Answer: 9.4e4 N'''

'''Explanation:'''

With the car as particle 1 and the truck as particle 2,
total initial momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>

Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.

===Difficult===

A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

'''Explanation:'''
Momentum before: 0,
Momentum after: <math> (2m(v - v_f) - m_{bin} * v_f) * i </math>
so,
<math> 0 = 2m_{people}(v - v_f) - m_{bin} * v</math>

Then solve for <math> v_f </math>.

==Connectedness==

Multi-particle analysis of momentum is just one way of analyzing a particular system, but multi-particle analysis is a way to examine other systems as well, and the momentum principle is very important for a variety of professional fields.

One application includes its use by automobile safety engineers, who must take into account the physics of momentum in order to safely design vehicles for their customers' use. However, this is by no means the only application--momentum is an applicable concept for literally anything that moves and has mass: figure skaters, roller coasters, pool balls...the list goes on.

For computer scientists, it's possible to create simulations like those listed in the "Computational Models" section of this article! In order to design those it is necessary for the creator to understand not only computer science concepts, but also the physics concepts that they are attempting to represent. In any field, it is possible that momentum comes into play.

==History==

The scientists most credited for the introduction of the concept of momentum to their field are Rene Descartes and Isaac Newton. Newton is more acknowledged for his Laws, from which the more specific concept of Conservation of Momentum can be derived, while Descartes holds the original "discovery" of momentum.

More information on the history of momentum can be found at:
[http://www.physicsbook.gatech.edu/Momentum_Principle#Connectedness] and
[http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html]

== See also ==

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit:
[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]

and to then explore the Conservation of Momentum page:
[http://www.physicsbook.gatech.edu/Conservation_of_Momentum].

Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[http://www.physicsbook.gatech.edu/Main_Page# Main Page].

Check out the links in the references below for more information on this subject as well as other physics concepts!

==References==
[1] Fitzpatrick, R. (2006, February 2). Angular momentum of a point particle. Retrieved November 27, 2016, from http://farside.ph.utexas.edu/teaching/301/lectures/node118.html

[2] Torre, Charles. "Newton's Third Law. Multi-Particle Systems." Newton’s Third Law. Multi-particle Systems. Conservation of Momentum. Physics 3550, Fall 2012 Newton’s Third Law. Multi-particle Systems. Relevant Sections in Text: §1.5, 3.1, 3.2, 3.3 (2012): n. pag. Intermediate Classical Mechanics. Utah State University, Sept. 2012. Web. 27 Nov. 2016.

[3] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 27 Nov. 2016. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[4] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[5] Fenton, Flavio. "Momentum and Second Newton's Law." 27 Nov. 2016. Lecture.

[6] Henderson, Tom. "Momentum and Its Conservation." Momentum and Its Conservation. The Physics Clasroom, 31 Dec. 2012. Web. 27 Nov. 2016.

[7] Dourmashkin, Peter. Lister, J. David. Pritchard, David E. Surrow, Bernd. "Momentum and Collisions." Massachusetts Institute of Technology. Sep. 2004. Lecture.

[[Category: Multiparticle Systems]]

Multi-particle analysis of Momentum

2018-04-18T19:41:40Z

Sdhal3: /* The Main Idea */

'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' "Claimed by Simran Dhal (Spring 2018)"

==The Main Idea==

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

A multi-particle analysis of momentum can be applied to systems with multiple objects that may be interacting. Using the same principles of momentum applied on single objects, we apply these on the multiple objects that may make up a chosen system. However, one interesting characteristic of the multi-particle analysis of momentum is that it only factors in external forces applied on the object as opposed to internal forces which would be applied by objects on one another within a system even though objects within the system are interacting.

"Multi-particle System Momentum Principle"

<math> ∆\vec{p} = \vec{F}_{net} * {∆t}</math>

Where Fnet is the sum of all external forces on the system. Therefore:

<math>\vec{F}_{net} = \vec{F}_{1} + \vec{F}_{2} + \vec{F}_{3} + ...</math>

[[File:ExternalForces.png|300px|thumb|left|Figure 1]]

From Figure 1 we can see that there are internal forces within the system since each object contributes a force to its surrounding objects. However, when applying the multi-particle momentum principle, we only look at the external forces. So in this case, to sum the forces in order to find the change in momentum we would add F1,surr + F2,surr + F3,surr + F4,surr to sum up to an Ftotal that we would multiply by the time elapsed in order to determine the change in momentum.

The reason that the internal forces are not used within the momentum principle is because the internal forces would cancel due to the principle of reciprocity. Since each object exerts an equal and opposite force on each other, all of the internal forces would cancel, leaving only the external forces.

Theory of reciprocity results in this:

<math>\vec{F}_{1,2} = \vec{-F}_{2,1}</math>

<math>\vec{F}_{1,3} = \vec{-F}_{3,1}</math>

<math>\vec{F}_{1,4} = \vec{-F}_{1,1}</math>

And this would be the case for every object in the system (objects 2,3 and 4).

===A Mathematical Model===

Mathematically, momentum is the product of a body's (in our case, a particle's) mass and velocity. It is also defined as the product of mass, net force, and change in time.

Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as

<math>{\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net}</math> (or <math>∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

<math> P = p_1 + p_2 = m_1v_1 + m_2v_2</math>

and deriving from <math> F = m * a </math>, the rate of change is the sum of the vector forces, and therefore:

<math> \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 </math>.

Newton's third law gives that:

<math> \frac{d\vec{P}}{dt} = 0 </math>,

'''so,'''

<math> m_1\vec{a}\_1 + m_2\vec{a}_2 = 0</math>.

<math>\vec{F}_{12} = -\vec{F}_{21} </math>.

<math> \vec{F}_{12} = m_2\vec{a}_2 </math>.

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

<math> \vec{P} = \sum_{i=1}^N m_i\vec{v}_i </math>
'''and'''
<math> \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} </math>.

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.

See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]

===Computational Models===

Click on the link below to understand the basics of the momentum principle in VPython,

[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

or check out this collision simulation to see a multi-particle analysis of it:

[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]

==Examples==

===Simple===

A loaded cart is travelling with momentum Mcart, and a brick is falling with momentum Mbrick. As the cart advances forward, the brick lands on it and the now heavier cart continues to move forward. The initial momentum of the system includes that of the cart and of the brick. Symbolically, what is the final momentum of the system?

'''Answer: Mcart + Mbrick'''

'''Explanation:'''

As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

===Middling===

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

'''Answer: 9.4e4 N'''

'''Explanation:'''

With the car as particle 1 and the truck as particle 2,
total initial momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>

Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.

===Difficult===

A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

'''Explanation:'''
Momentum before: 0,
Momentum after: <math> (2m(v - v_f) - m_{bin} * v_f) * i </math>
so,
<math> 0 = 2m_{people}(v - v_f) - m_{bin} * v</math>

Then solve for <math> v_f </math>.

==Connectedness==

Multi-particle analysis of momentum is just one way of analyzing a particular system, but multi-particle analysis is a way to examine other systems as well, and the momentum principle is very important for a variety of professional fields.

One application includes its use by automobile safety engineers, who must take into account the physics of momentum in order to safely design vehicles for their customers' use. However, this is by no means the only application--momentum is an applicable concept for literally anything that moves and has mass: figure skaters, roller coasters, pool balls...the list goes on.

For computer scientists, it's possible to create simulations like those listed in the "Computational Models" section of this article! In order to design those it is necessary for the creator to understand not only computer science concepts, but also the physics concepts that they are attempting to represent. In any field, it is possible that momentum comes into play.

==History==

The scientists most credited for the introduction of the concept of momentum to their field are Rene Descartes and Isaac Newton. Newton is more acknowledged for his Laws, from which the more specific concept of Conservation of Momentum can be derived, while Descartes holds the original "discovery" of momentum.

More information on the history of momentum can be found at:
[http://www.physicsbook.gatech.edu/Momentum_Principle#Connectedness] and
[http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html]

== See also ==

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit:
[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]

and to then explore the Conservation of Momentum page:
[http://www.physicsbook.gatech.edu/Conservation_of_Momentum].

Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[http://www.physicsbook.gatech.edu/Main_Page# Main Page].

Check out the links in the references below for more information on this subject as well as other physics concepts!

==References==
[1] Fitzpatrick, R. (2006, February 2). Angular momentum of a point particle. Retrieved November 27, 2016, from http://farside.ph.utexas.edu/teaching/301/lectures/node118.html

[2] Torre, Charles. "Newton's Third Law. Multi-Particle Systems." Newton’s Third Law. Multi-particle Systems. Conservation of Momentum. Physics 3550, Fall 2012 Newton’s Third Law. Multi-particle Systems. Relevant Sections in Text: §1.5, 3.1, 3.2, 3.3 (2012): n. pag. Intermediate Classical Mechanics. Utah State University, Sept. 2012. Web. 27 Nov. 2016.

[3] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 27 Nov. 2016. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[4] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[5] Fenton, Flavio. "Momentum and Second Newton's Law." 27 Nov. 2016. Lecture.

[6] Henderson, Tom. "Momentum and Its Conservation." Momentum and Its Conservation. The Physics Clasroom, 31 Dec. 2012. Web. 27 Nov. 2016.

[7] Dourmashkin, Peter. Lister, J. David. Pritchard, David E. Surrow, Bernd. "Momentum and Collisions." Massachusetts Institute of Technology. Sep. 2004. Lecture.

[[Category: Multiparticle Systems]]

Multi-particle analysis of Momentum

2018-04-18T19:39:48Z

Sdhal3: /* The Main Idea */

'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' "Claimed by Simran Dhal (Spring 2018)"

==The Main Idea==

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

A multi-particle analysis of momentum can be applied to systems with multiple objects that may be interacting. Using the same principles of momentum applied on single objects, we apply these on the multiple objects that may make up a chosen system. However, one interesting characteristic of the multi-particle analysis of momentum is that it only factors in external forces applied on the object as opposed to internal forces which would be applied by objects on one another within a system even though objects within the system are interacting.

"Multi-particle System Momentum Principle"

<math> ∆\vec{p} = \vec{F}_{net} * {∆t}</math>

Where Fnet is the sum of all external forces on the system. Therefore:

<math>\vec{F}_{net} = \vec{F}_{1} + \vec{F}_{2} + \vec{F}_{3} + ...</math>

[[File:ExternalForces.png|300px|thumb|left|Figure 1]]

From Figure 1 we can see that there are internal forces within the system since each object contributes a force to its surrounding objects. However, when applying the multi-particle momentum principle, we only look at the external forces. So in this case, to sum the forces in order to find the change in momentum we would add F1,surr + F2,surr + F3,surr + F4,surr to sum up to an Ftotal that we would multiply by the time elapsed in order to determine the change in momentum.

The reason that the internal forces are not used within the momentum principle is because the internal forces would cancel due to the principle of reciprocity. Since each object exerts an equal and opposite force on each other, all of the internal forces would cancel, leaving only the external forces.

Theory of reciprocity results in this:
<math>\vec{F}_{1,2} = \vec{-F}_{2,1}</math>
<math>\vec{F}_{1,3} = \vec{-F}_{3,1}</math>

===A Mathematical Model===

Mathematically, momentum is the product of a body's (in our case, a particle's) mass and velocity. It is also defined as the product of mass, net force, and change in time.

Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as

<math>{\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net}</math> (or <math>∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

<math> P = p_1 + p_2 = m_1v_1 + m_2v_2</math>

and deriving from <math> F = m * a </math>, the rate of change is the sum of the vector forces, and therefore:

<math> \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 </math>.

Newton's third law gives that:

<math> \frac{d\vec{P}}{dt} = 0 </math>,

'''so,'''

<math> m_1\vec{a}\_1 + m_2\vec{a}_2 = 0</math>.

<math>\vec{F}_{12} = -\vec{F}_{21} </math>.

<math> \vec{F}_{12} = m_2\vec{a}_2 </math>.

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

<math> \vec{P} = \sum_{i=1}^N m_i\vec{v}_i </math>
'''and'''
<math> \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} </math>.

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.

See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]

===Computational Models===

Click on the link below to understand the basics of the momentum principle in VPython,

[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

or check out this collision simulation to see a multi-particle analysis of it:

[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]

==Examples==

===Simple===

A loaded cart is travelling with momentum Mcart, and a brick is falling with momentum Mbrick. As the cart advances forward, the brick lands on it and the now heavier cart continues to move forward. The initial momentum of the system includes that of the cart and of the brick. Symbolically, what is the final momentum of the system?

'''Answer: Mcart + Mbrick'''

'''Explanation:'''

As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

===Middling===

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

'''Answer: 9.4e4 N'''

'''Explanation:'''

With the car as particle 1 and the truck as particle 2,
total initial momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>

Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.

===Difficult===

A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

'''Explanation:'''
Momentum before: 0,
Momentum after: <math> (2m(v - v_f) - m_{bin} * v_f) * i </math>
so,
<math> 0 = 2m_{people}(v - v_f) - m_{bin} * v</math>

Then solve for <math> v_f </math>.

==Connectedness==

Multi-particle analysis of momentum is just one way of analyzing a particular system, but multi-particle analysis is a way to examine other systems as well, and the momentum principle is very important for a variety of professional fields.

One application includes its use by automobile safety engineers, who must take into account the physics of momentum in order to safely design vehicles for their customers' use. However, this is by no means the only application--momentum is an applicable concept for literally anything that moves and has mass: figure skaters, roller coasters, pool balls...the list goes on.

For computer scientists, it's possible to create simulations like those listed in the "Computational Models" section of this article! In order to design those it is necessary for the creator to understand not only computer science concepts, but also the physics concepts that they are attempting to represent. In any field, it is possible that momentum comes into play.

==History==

The scientists most credited for the introduction of the concept of momentum to their field are Rene Descartes and Isaac Newton. Newton is more acknowledged for his Laws, from which the more specific concept of Conservation of Momentum can be derived, while Descartes holds the original "discovery" of momentum.

More information on the history of momentum can be found at:
[http://www.physicsbook.gatech.edu/Momentum_Principle#Connectedness] and
[http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html]

== See also ==

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit:
[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]

and to then explore the Conservation of Momentum page:
[http://www.physicsbook.gatech.edu/Conservation_of_Momentum].

Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[http://www.physicsbook.gatech.edu/Main_Page# Main Page].

Check out the links in the references below for more information on this subject as well as other physics concepts!

==References==
[1] Fitzpatrick, R. (2006, February 2). Angular momentum of a point particle. Retrieved November 27, 2016, from http://farside.ph.utexas.edu/teaching/301/lectures/node118.html

[2] Torre, Charles. "Newton's Third Law. Multi-Particle Systems." Newton’s Third Law. Multi-particle Systems. Conservation of Momentum. Physics 3550, Fall 2012 Newton’s Third Law. Multi-particle Systems. Relevant Sections in Text: §1.5, 3.1, 3.2, 3.3 (2012): n. pag. Intermediate Classical Mechanics. Utah State University, Sept. 2012. Web. 27 Nov. 2016.

[3] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 27 Nov. 2016. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[4] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[5] Fenton, Flavio. "Momentum and Second Newton's Law." 27 Nov. 2016. Lecture.

[6] Henderson, Tom. "Momentum and Its Conservation." Momentum and Its Conservation. The Physics Clasroom, 31 Dec. 2012. Web. 27 Nov. 2016.

[7] Dourmashkin, Peter. Lister, J. David. Pritchard, David E. Surrow, Bernd. "Momentum and Collisions." Massachusetts Institute of Technology. Sep. 2004. Lecture.

[[Category: Multiparticle Systems]]

Multi-particle analysis of Momentum

2018-04-18T19:21:32Z

Sdhal3: /* The Main Idea */

'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' "Claimed by Simran Dhal (Spring 2018)"

==The Main Idea==

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

A multi-particle analysis of momentum can be applied to systems with multiple objects that may be interacting. Using the same principles of momentum applied on single objects, we apply these on the multiple objects that may make up a chosen system. However, one interesting characteristic of the multi-particle analysis of momentum is that it only factors in external forces applied on the object as opposed to internal forces which would be applied by objects on one another within a system even though objects within the system are interacting.

"Multi-particle System Momentum Principle"

<math> ∆\vec{p} = \vec{F}_{net} * {∆t}</math>

Where Fnet is the sum of all external forces on the system. Therefore:

<math>\vec{F}_{net} = \vec{F}_{1} + \vec{F}_{2} + \vec{F}_{3} + ...</math>

[[File:ExternalForces.png|300px|thumb|left|Figure 1]]

From Figure 1 we can see that there are internal forces within the system since each object contributes a force to its surrounding objects. However, when applying the multi-particle momentum principle, we only look at the external forces. So in this case, to sum the forces in order to find the change in momentum we would add F1,surr + F2,surr + F3,surr + F4,surr to sum up to an Ftotal that we would multiply by the time elapsed in order to determine the change in momentum.

===A Mathematical Model===

Mathematically, momentum is the product of a body's (in our case, a particle's) mass and velocity. It is also defined as the product of mass, net force, and change in time.

Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as

<math>{\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net}</math> (or <math>∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

<math> P = p_1 + p_2 = m_1v_1 + m_2v_2</math>

and deriving from <math> F = m * a </math>, the rate of change is the sum of the vector forces, and therefore:

<math> \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 </math>.

Newton's third law gives that:

<math> \frac{d\vec{P}}{dt} = 0 </math>,

'''so,'''

<math> m_1\vec{a}\_1 + m_2\vec{a}_2 = 0</math>.

<math>\vec{F}_{12} = -\vec{F}_{21} </math>.

<math> \vec{F}_{12} = m_2\vec{a}_2 </math>.

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

<math> \vec{P} = \sum_{i=1}^N m_i\vec{v}_i </math>
'''and'''
<math> \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} </math>.

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.

See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]

===Computational Models===

Click on the link below to understand the basics of the momentum principle in VPython,

[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

or check out this collision simulation to see a multi-particle analysis of it:

[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]

==Examples==

===Simple===

A loaded cart is travelling with momentum Mcart, and a brick is falling with momentum Mbrick. As the cart advances forward, the brick lands on it and the now heavier cart continues to move forward. The initial momentum of the system includes that of the cart and of the brick. Symbolically, what is the final momentum of the system?

'''Answer: Mcart + Mbrick'''

'''Explanation:'''

As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

===Middling===

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

'''Answer: 9.4e4 N'''

'''Explanation:'''

With the car as particle 1 and the truck as particle 2,
total initial momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>

Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.

===Difficult===

A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

'''Explanation:'''
Momentum before: 0,
Momentum after: <math> (2m(v - v_f) - m_{bin} * v_f) * i </math>
so,
<math> 0 = 2m_{people}(v - v_f) - m_{bin} * v</math>

Then solve for <math> v_f </math>.

==Connectedness==

Multi-particle analysis of momentum is just one way of analyzing a particular system, but multi-particle analysis is a way to examine other systems as well, and the momentum principle is very important for a variety of professional fields.

One application includes its use by automobile safety engineers, who must take into account the physics of momentum in order to safely design vehicles for their customers' use. However, this is by no means the only application--momentum is an applicable concept for literally anything that moves and has mass: figure skaters, roller coasters, pool balls...the list goes on.

For computer scientists, it's possible to create simulations like those listed in the "Computational Models" section of this article! In order to design those it is necessary for the creator to understand not only computer science concepts, but also the physics concepts that they are attempting to represent. In any field, it is possible that momentum comes into play.

==History==

The scientists most credited for the introduction of the concept of momentum to their field are Rene Descartes and Isaac Newton. Newton is more acknowledged for his Laws, from which the more specific concept of Conservation of Momentum can be derived, while Descartes holds the original "discovery" of momentum.

More information on the history of momentum can be found at:
[http://www.physicsbook.gatech.edu/Momentum_Principle#Connectedness] and
[http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html]

== See also ==

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit:
[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]

and to then explore the Conservation of Momentum page:
[http://www.physicsbook.gatech.edu/Conservation_of_Momentum].

Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[http://www.physicsbook.gatech.edu/Main_Page# Main Page].

Check out the links in the references below for more information on this subject as well as other physics concepts!

==References==
[1] Fitzpatrick, R. (2006, February 2). Angular momentum of a point particle. Retrieved November 27, 2016, from http://farside.ph.utexas.edu/teaching/301/lectures/node118.html

[2] Torre, Charles. "Newton's Third Law. Multi-Particle Systems." Newton’s Third Law. Multi-particle Systems. Conservation of Momentum. Physics 3550, Fall 2012 Newton’s Third Law. Multi-particle Systems. Relevant Sections in Text: §1.5, 3.1, 3.2, 3.3 (2012): n. pag. Intermediate Classical Mechanics. Utah State University, Sept. 2012. Web. 27 Nov. 2016.

[3] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 27 Nov. 2016. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[4] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[5] Fenton, Flavio. "Momentum and Second Newton's Law." 27 Nov. 2016. Lecture.

[6] Henderson, Tom. "Momentum and Its Conservation." Momentum and Its Conservation. The Physics Clasroom, 31 Dec. 2012. Web. 27 Nov. 2016.

[7] Dourmashkin, Peter. Lister, J. David. Pritchard, David E. Surrow, Bernd. "Momentum and Collisions." Massachusetts Institute of Technology. Sep. 2004. Lecture.

[[Category: Multiparticle Systems]]

File:ExternalForces.png

2018-04-18T19:13:03Z

Sdhal3:

Conservation of Momentum

2018-04-18T19:09:58Z

Sdhal3: /* Medium Difficulty */

Claimed by Emily Dunford Edited by Therese Stanley (tstanley6)
* I couldn't figure out how to do in text citations in the wiki so the links at the end of sentences link to the actual links the information was paraphrased from but the references are below in the reference section **

The conservation of momentum is one of the fundamental laws of physics. Within the definitions of a problem, the total momentum of the system stays constant [https://www.grc.nasa.gov/www/k-12/airplane/conmo.html]. Much like the conservation of mass or the conservation of energy, the momentum of the objects before the collision is the same as the momentum of the objects after the collision. The momentum is changed through the action of forces [https://www.grc.nasa.gov/www/k-12/airplane/conmo.html] as in Newton’s law of motion. This is a powerful idea to solving problems.

==The Main Idea==

Conservation refers to something that doesn’t change. Conservation of momentum is the idea that momentum is the same before and after an event [http://www.physicsclassroom.com/class/momentum/Lesson-2/Momentum-Conservation-Principle]. Two common events that will observe the conservation of momentum are elastic and inelastic collisions. This is true in an isolated system, meaning that the system is not acted on by an external force. An elastic collision is one in which the objects collide and do not stick together, an inelastic collision is when the objects that collided stick together in their final state [http://www.physicsclassroom.com/mmedia/momentum/cthoi.cfm]. During an inelastic collision the momentum lost by the first object is equal to the momentum gained by the second object [http://www.physicsclassroom.com/class/momentum/Lesson-2/Momentum-Conservation-Principle]. However, this does not mean that other aspects of the two objects do not change. In inelastic collisions, the momentums of the two objects in their initial states is equal to the momentum of both objects in their final state. If the two objects are of equal mass, the velocity of the combined objects will be halved, though the momentum remains conserved because the mass has doubled. The main difference between the conservation of momentum and the conservation of mass or the conservation of energy is that the momentum is a vector quantity, meaning the momentum is conserved in the x, y, and z directions [https://www.grc.nasa.gov/www/k-12/airplane/conmo.html].

The law of conservation of momentum can be logically derived from Newton’s Third Law [http://www.physicsclassroom.com/class/momentum/Lesson-2/Momentum-Conservation-Principle]. When 2 objects collide, the force on object 1 on object 2 (<math> \begin{align} F_1 \end{align} </math>) is equal in magnitude and opposite in direction to the force on object 2 on object 1 (<math> \begin{align} F_1 \end{align} </math>). So: <math> \begin{align} F_1=-F_2 \end{align} </math>

The objects collide during a certain time period (<math> \begin{align} \Delta t \end{align} </math>) so that the force acting on object 1 and the force acting on object 2 act over the this time period (Δt). So: <math> \begin{align} F_1* \Delta 1 =-F_2* \Delta 1 \end{align} </math>
We know that <math> \begin{align} F*t \end{align} </math> is the formula for impulse and since the change in impulse is equal to the change in momentum: <math> \begin{align} m_1 * \Delta v_1 = -m_2 * \Delta v_2 \end{align} </math> (The Law of Conservation of Momentum).

===A Mathematical Model===

<math> \begin{align} \Delta p_1 = m_1 * \Delta v_1 = -m_2 * \Delta v_2 = \Delta p_2 \,. \end{align} </math>

<math> \begin{align} p_1 \end{align} </math> is the momentum of the first object, <math> \begin{align} m_1 \end{align} </math> is the mass of the first object, <math> \begin{align} v_1 \end{align} </math> is the velocity of the first object. <math> \begin{align} p_2 \end{align} </math> is the momentum of the second object, <math> \begin{align} m_2 \end{align} </math> is the mass of the second object, <math> \begin{align} v_2 \end{align} </math> is the velocity of the second object.

===A Computational Model===

The link below shows a simulation of an elastic collision where one object transfers all of its momentum to the second object after the collision.

[https://trinket.io/embed/glowscript/7bc4cb8e9a]

==Examples==

===Simple Difficulty===

You are playing pool with your friends at Tech Rec. Two cue balls collide in a head-on collision. Both cue balls have equal mass of 0.165 kg. Before the collision, the first ball is travelling at 9.8 meters per second and the second ball is stationary, 0 meters per second. After the collision, the first ball travels at a velocity of 2 meters per second. What is the velocity of the second ball?

Solution:
<math> \begin{align}
m_1 = m_2 = m = 0.165 \,. \end{align} </math>

<math> \begin{align}
v_1i = 9.8 \,. \end{align} </math>

<math> \begin{align}
v_1f = -2 \,. \end{align} </math>

<math> \begin{align}
V_2i = 0 \,. \end{align} </math>

<math> \begin{align}
m * (v_1f - v_1i)= m (v_2f - v_2i) \,. \end{align} </math>

<math> \begin{align}
v_2f = v_1f - v_1i + v_2i = 9.8 + 2 - 0 = 11.8 \,. \end{align} </math>
The velocity of the second ball is 11.8 meters per second in the positive x direction.

There is a perfectly inelastic collision between a big fish and a little fish. The big fish is initially going 2 m/s and the little fish is going 10 m/s. The big fish’s mass is 5 kg and the little fish is 0.5 kg. What is the velocity of the system after the collision?

Solution: Because there are no forces acting on the system, the system’s total momentum before the collision is equal to the momentum after the collision.
<math> \begin{align}
m_1 * v_1i + m_2 * v_2i = (m_1 + m_2) * v_f \,. \end{align} </math>

<math> \begin{align}
5 kg * 2 m/s + 0.5 kg * 10 m/s = (5 kg + 0.5 kg) * v_f \,. \end{align} </math>
The final velocity of the big and little fish system is 0.909 m/s

===Medium Difficulty===

You are looking out the window of your dorm when you see a person on a Hoverboard collide with a person on a bike. The person on bike was going west with velocity 9 m/s and the person on the Hoverboard was traveling at 30 degrees north of west with velocity 5 m/s. After the collision, both the person on the Hoverboard and the person on the bike become entangled and travel together at some velocity <math> \begin{align} v_f \, \end{align} </math>. Find <math> \begin{align} v_f \, \end{align} </math> (See Figure 1 Below)

[[File: Middle Problem2.jpg|200px|thumb|left|Figure 1]]

Solution:
<math> \begin{align}
v_H=[9*cos(30), 9*sin(30), 0] \, \end{align} </math>
<math> \begin{align}
v_B=[-5,0,0] \, \end{align} </math>

<math> \begin{align}
m_H = 60 kg \, \end{align} </math>
<math> \begin{align}
m_B = 50 kg \, \end{align} </math>

<math> \begin{align}
m_H*v_H + m_bike*v_B = (m_H+m_B)*v_final \, \end{align} </math>

<math> \begin{align}
v_f= (m_H*v_H + m_B*v_B)/ (m_H+m_B) \, \end{align} </math>

<math> \begin{align}
v_f = [-1.5155, -4.8503, 0]
\,. \end{align} </math> m/s

An 80 kg running back runs in the -y direction at 1.5 m/s before getting tackled by a 95 kg linebacker traveling at 2.0 m/s in the +y direction. Both players bounce off each other after the collision. If the linebacker continues moving in the same direction at 0.5 m/s, what is the velocity and direction of the running back?

Solution: This is an elastic collision, and the total momentum is conserved.

<math> \begin{align}
m_1 * v_1i + m_2 * v_2i = m_1 * v_1f + m_2 * v_2f \,. \end{align} </math> m/s

<math> \begin{align}
80 kg * [0,-1.5,0] m/s + 95 kg * [0,2,0] m/s = 80 kg * v_1f + 95 kg * [0,.5,0] \,. \end{align} </math> m/s

Then get v1f by itself and you find out that the linebacker is going [0,.282,0] m/s.

===Difficult===

During a baseball game the 0.145 kg baseball is thrown straight upward with a velocity of 40 m/s. What is the recoil velocity of the earth? Why don’t we notice that the earth has gained velocity?

Solution:
<math> \begin{align}
m_b = 0.145 kg \, \end{align} </math>
<math> \begin{align}
m_E = 5.972 e 24 kg\, \end{align} </math>
<math> \begin{align}
v_b = 40 m/s \, \end{align} </math>

The system is the baseball plus the earth. So, the total momentum of the system must be conserved (i.e. the momentum before must be equal to the momentum after.)

<math> \begin{align}
m_b*v_b=-m_E*v_E .\, \end{align} </math>

<math> \begin{align}
v_e= - (m_b*v_b)/m_E .\, \end{align} </math>

<math> \begin{align}
v_e=- 9.7120e-25 m/s .\, \end{align} </math>

Even though the magnitude of the momentum of the ball equals the magnitude of the momentum of the Earth, the Earth’s mass is so massive that the Earth recoils with a velocity so small we don’t feel it.

A cannon is rigidly attached to a carriage, which can move along horizontal rails but is connected to a post by a large spring, initially upstretched and with a spring constant k = 2.0e4 N/M. The cannon fires a 200 kg projectile at a velocity of 125 m/s directed 45 degrees above the horizontal. If the mass of the cannon and the carriage is 5,000 kg find the recoil speed of the cannon.

Solution:
The initial velocity of the cannon and the projectile are both 0 m/s.
<math> \begin{align}
0 = 200 kg * 125 m/s * cos(45) + 5000 kg * v_x .\, \end{align} </math>

v_x = -3.54 m/s

==Real World Connection==

The Conservation of Momentum is prticularly important in fluid flow and transport. In the field of biomedical engineering, engineers are often times working with fluids flowing through the body. So when a new design for a pacemaker is made, engineers need to find out how the new pacemaker diverts fluid flow in the body. To solve a problem like this, one would use Reynold's Transport Theorem, as well as the principle of conservation of momentum as a result of the basis for Reynold's Transport Theorem being the principles of Conservation of Mass and the Conservation of Momentum. No new device can be made for the human body without such an analysis. In addition, in certain disease states such as atherosclerosis, when certain arteries become clogged, the fluid flow is diverted and there is a change in pressure. To model certain situations, engineers would start from basic principles such as Conservation of Mass and Conservation of Momentum.

The conservation of momentum is used to successfully propels a rocket through space. When a rocket is started it sends exhaust gases downward at a high velocity. The gases have mass and therefore have momentum. For the momentum to be conserved the rocket then moves upward at a velocity equivalent to the original momentum divided by the mass of the rocket. Nothing actually pushes the rocket while it's moving, it's just the momentum of the gases working with the momentum of the rocket itself [http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3-Physics-Vol-1/Conservation-Laws-Real-life-applications.html].

In this way, the Conservation of Momentum (as well as the Conservation of Mass and the Conservation of Energy) plays an important roles in engineering fields, such as Biomedical Engineering. These conservation law are also important in other engineering fields. To name a few examples, Chemical engineers are concerned with fluid flowing through a pipe, and mechanical engineers are concerned with fluid flowing through an engine.

==History==

Newton established the Conservation of Momentum along with the other Conservation Laws (except Conservation of Energy). Newton published his theories in 1687 in Philosophiæ Naturalis Principia Mathematica (linked below in external links) [http://plato.stanford.edu/entries/newton-principia/]. When Newton was publishing his work, the challenge that he faced was describing his theories without using calculus. Newton did not publish his La Methode Dex Fluxions until 1736 [http://www.maa.org/press/periodicals/convergence/mathematical-treasure-newtons-method-of-fluxions]. Newton's work on momentum was mostly focused on forces rather than energy and vectors but his work known as Newton's law of motion implies the conservation of momentum [http://plato.stanford.edu/entries/newton-principia/].

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===
For extra concept quetions see: [http://homepage.smc.edu/morse_peter/phy14/MotionForces/MOMENTUM%20ANSWERS.pdf]

===External links===
Newton's Philosophiæ Naturalis Principia Mathematica: [https://archive.org/stream/100878576#page/84/mode/2up]

==References==
1. "Conservation of Momentum." Conservation of Momentum. NASA, 05 May 2015. Web. 05 Dec. 2015. <https://www.grc.nasa.gov/www/k-12/airplane/conmo.html>.

2. "Momentum Conservation Principle." Momentum Conservation Principle. The Physics Classroom, n.d. Web. 05 Dec. 2015. <http://www.physicsclassroom.com/class/momentum/Lesson-2/Momentum-Conservation-Principle>.

3. Smith, George. "Newton's Philosophiae Naturalis Principia Mathematica." Stanford University. Stanford University, 20 Dec. 2007. Web. 05 Dec. 2015. <http://plato.stanford.edu/entries/newton-principia/>.

4. "Mathematical Treasure: Newton's Method of Fluxions." Mathematical Treasure: Newton's Method of Fluxions. Mathematical Association of America, n.d. Web. 05 Dec. 2015. <http://www.maa.org/press/periodicals/convergence/mathematical-treasure-newtons-method-of-fluxions>.

5. "Momentum Conservation Principle." Momentum Conservation Principle. N.p., n.d. Web. 26 Nov. 2016.

6. "Conservation Laws - Real-life Applications." Real-life Applications - Conservation Laws - Conservation of Linear Momentum, Firing a Rifle. N.p., n.d. Web. 26 Nov. 2016.

[[Category:Momentum]]

Multi-particle analysis of Momentum

2018-04-18T18:48:17Z

Sdhal3: /* The Main Idea */

'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' "Claimed by Simran Dhal (Spring 2018)"

==The Main Idea==

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

A multi-particle analysis of momentum can be applied to systems with multiple objects that may be interacting. Using the same principles of momentum applied on single objects, we apply these on the multiple objects that may make up a chosen system. However, one interesting characteristic of the multi-particle analysis of momentum is that it only factors in external forces applied on the object as opposed to internal forces which would be applied by objects on one another within a system even though objects within the system are interacting.

"Multi-particle System Momentum Principle"

<math> ∆\vec{p} = \vec{F}_{net} * {∆t}</math>

Where Fnet is the sum of all external forces on the system. Therefore:

<math>\vec{F}_{net} = \vec{F}_{1} + \vec{F}_{2} + \vec{F}_{3} + ...</math>

===A Mathematical Model===

Mathematically, momentum is the product of a body's (in our case, a particle's) mass and velocity. It is also defined as the product of mass, net force, and change in time.

Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as

<math>{\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net}</math> (or <math>∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

<math> P = p_1 + p_2 = m_1v_1 + m_2v_2</math>

and deriving from <math> F = m * a </math>, the rate of change is the sum of the vector forces, and therefore:

<math> \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 </math>.

Newton's third law gives that:

<math> \frac{d\vec{P}}{dt} = 0 </math>,

'''so,'''

<math> m_1\vec{a}\_1 + m_2\vec{a}_2 = 0</math>.

<math>\vec{F}_{12} = -\vec{F}_{21} </math>.

<math> \vec{F}_{12} = m_2\vec{a}_2 </math>.

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

<math> \vec{P} = \sum_{i=1}^N m_i\vec{v}_i </math>
'''and'''
<math> \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} </math>.

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.

See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]

===Computational Models===

Click on the link below to understand the basics of the momentum principle in VPython,

[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

or check out this collision simulation to see a multi-particle analysis of it:

[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]

==Examples==

===Simple===

A loaded cart is travelling with momentum Mcart, and a brick is falling with momentum Mbrick. As the cart advances forward, the brick lands on it and the now heavier cart continues to move forward. The initial momentum of the system includes that of the cart and of the brick. Symbolically, what is the final momentum of the system?

'''Answer: Mcart + Mbrick'''

'''Explanation:'''

As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

===Middling===

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

'''Answer: 9.4e4 N'''

'''Explanation:'''

With the car as particle 1 and the truck as particle 2,
total initial momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>

Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.

===Difficult===

A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

'''Explanation:'''
Momentum before: 0,
Momentum after: <math> (2m(v - v_f) - m_{bin} * v_f) * i </math>
so,
<math> 0 = 2m_{people}(v - v_f) - m_{bin} * v</math>

Then solve for <math> v_f </math>.

==Connectedness==

Multi-particle analysis of momentum is just one way of analyzing a particular system, but multi-particle analysis is a way to examine other systems as well, and the momentum principle is very important for a variety of professional fields.

One application includes its use by automobile safety engineers, who must take into account the physics of momentum in order to safely design vehicles for their customers' use. However, this is by no means the only application--momentum is an applicable concept for literally anything that moves and has mass: figure skaters, roller coasters, pool balls...the list goes on.

For computer scientists, it's possible to create simulations like those listed in the "Computational Models" section of this article! In order to design those it is necessary for the creator to understand not only computer science concepts, but also the physics concepts that they are attempting to represent. In any field, it is possible that momentum comes into play.

==History==

The scientists most credited for the introduction of the concept of momentum to their field are Rene Descartes and Isaac Newton. Newton is more acknowledged for his Laws, from which the more specific concept of Conservation of Momentum can be derived, while Descartes holds the original "discovery" of momentum.

More information on the history of momentum can be found at:
[http://www.physicsbook.gatech.edu/Momentum_Principle#Connectedness] and
[http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html]

== See also ==

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit:
[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]

and to then explore the Conservation of Momentum page:
[http://www.physicsbook.gatech.edu/Conservation_of_Momentum].

Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[http://www.physicsbook.gatech.edu/Main_Page# Main Page].

Check out the links in the references below for more information on this subject as well as other physics concepts!

==References==
[1] Fitzpatrick, R. (2006, February 2). Angular momentum of a point particle. Retrieved November 27, 2016, from http://farside.ph.utexas.edu/teaching/301/lectures/node118.html

[2] Torre, Charles. "Newton's Third Law. Multi-Particle Systems." Newton’s Third Law. Multi-particle Systems. Conservation of Momentum. Physics 3550, Fall 2012 Newton’s Third Law. Multi-particle Systems. Relevant Sections in Text: §1.5, 3.1, 3.2, 3.3 (2012): n. pag. Intermediate Classical Mechanics. Utah State University, Sept. 2012. Web. 27 Nov. 2016.

[3] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 27 Nov. 2016. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[4] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[5] Fenton, Flavio. "Momentum and Second Newton's Law." 27 Nov. 2016. Lecture.

[6] Henderson, Tom. "Momentum and Its Conservation." Momentum and Its Conservation. The Physics Clasroom, 31 Dec. 2012. Web. 27 Nov. 2016.

[7] Dourmashkin, Peter. Lister, J. David. Pritchard, David E. Surrow, Bernd. "Momentum and Collisions." Massachusetts Institute of Technology. Sep. 2004. Lecture.

[[Category: Multiparticle Systems]]

Multi-particle analysis of Momentum

2018-04-18T18:44:50Z

Sdhal3: /* The Main Idea */

'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' "Claimed by Simran Dhal (Spring 2018)"

==The Main Idea==

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

A multi-particle analysis of momentum can be applied to systems with multiple objects that may be interacting. Using the same principles of momentum applied on single objects, we apply these on the multiple objects that may make up a chosen system. However, one interesting characteristic of the multi-particle analysis of momentum is that it only factors in external forces applied on the object as opposed to internal forces which would be applied by objects on one another within a system even though objects within the system are interacting.

"Multi-particle System Momentum Principle"

<math> ∆\vec{p} = \vec{F}_{net} * {∆t}</math>

===A Mathematical Model===

Mathematically, momentum is the product of a body's (in our case, a particle's) mass and velocity. It is also defined as the product of mass, net force, and change in time.

Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as

<math>{\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net}</math> (or <math>∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

<math> P = p_1 + p_2 = m_1v_1 + m_2v_2</math>

and deriving from <math> F = m * a </math>, the rate of change is the sum of the vector forces, and therefore:

<math> \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 </math>.

Newton's third law gives that:

<math> \frac{d\vec{P}}{dt} = 0 </math>,

'''so,'''

<math> m_1\vec{a}\_1 + m_2\vec{a}_2 = 0</math>.

<math>\vec{F}_{12} = -\vec{F}_{21} </math>.

<math> \vec{F}_{12} = m_2\vec{a}_2 </math>.

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

<math> \vec{P} = \sum_{i=1}^N m_i\vec{v}_i </math>
'''and'''
<math> \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} </math>.

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.

See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]

===Computational Models===

Click on the link below to understand the basics of the momentum principle in VPython,

[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

or check out this collision simulation to see a multi-particle analysis of it:

[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]

==Examples==

===Simple===

A loaded cart is travelling with momentum Mcart, and a brick is falling with momentum Mbrick. As the cart advances forward, the brick lands on it and the now heavier cart continues to move forward. The initial momentum of the system includes that of the cart and of the brick. Symbolically, what is the final momentum of the system?

'''Answer: Mcart + Mbrick'''

'''Explanation:'''

As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

===Middling===

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

'''Answer: 9.4e4 N'''

'''Explanation:'''

With the car as particle 1 and the truck as particle 2,
total initial momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>

Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.

===Difficult===

A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

'''Explanation:'''
Momentum before: 0,
Momentum after: <math> (2m(v - v_f) - m_{bin} * v_f) * i </math>
so,
<math> 0 = 2m_{people}(v - v_f) - m_{bin} * v</math>

Then solve for <math> v_f </math>.

==Connectedness==

Multi-particle analysis of momentum is just one way of analyzing a particular system, but multi-particle analysis is a way to examine other systems as well, and the momentum principle is very important for a variety of professional fields.

One application includes its use by automobile safety engineers, who must take into account the physics of momentum in order to safely design vehicles for their customers' use. However, this is by no means the only application--momentum is an applicable concept for literally anything that moves and has mass: figure skaters, roller coasters, pool balls...the list goes on.

For computer scientists, it's possible to create simulations like those listed in the "Computational Models" section of this article! In order to design those it is necessary for the creator to understand not only computer science concepts, but also the physics concepts that they are attempting to represent. In any field, it is possible that momentum comes into play.

==History==

The scientists most credited for the introduction of the concept of momentum to their field are Rene Descartes and Isaac Newton. Newton is more acknowledged for his Laws, from which the more specific concept of Conservation of Momentum can be derived, while Descartes holds the original "discovery" of momentum.

More information on the history of momentum can be found at:
[http://www.physicsbook.gatech.edu/Momentum_Principle#Connectedness] and
[http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html]

== See also ==

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit:
[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]

and to then explore the Conservation of Momentum page:
[http://www.physicsbook.gatech.edu/Conservation_of_Momentum].

Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[http://www.physicsbook.gatech.edu/Main_Page# Main Page].

Check out the links in the references below for more information on this subject as well as other physics concepts!

==References==
[1] Fitzpatrick, R. (2006, February 2). Angular momentum of a point particle. Retrieved November 27, 2016, from http://farside.ph.utexas.edu/teaching/301/lectures/node118.html

[2] Torre, Charles. "Newton's Third Law. Multi-Particle Systems." Newton’s Third Law. Multi-particle Systems. Conservation of Momentum. Physics 3550, Fall 2012 Newton’s Third Law. Multi-particle Systems. Relevant Sections in Text: §1.5, 3.1, 3.2, 3.3 (2012): n. pag. Intermediate Classical Mechanics. Utah State University, Sept. 2012. Web. 27 Nov. 2016.

[3] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 27 Nov. 2016. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[4] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[5] Fenton, Flavio. "Momentum and Second Newton's Law." 27 Nov. 2016. Lecture.

[6] Henderson, Tom. "Momentum and Its Conservation." Momentum and Its Conservation. The Physics Clasroom, 31 Dec. 2012. Web. 27 Nov. 2016.

[7] Dourmashkin, Peter. Lister, J. David. Pritchard, David E. Surrow, Bernd. "Momentum and Collisions." Massachusetts Institute of Technology. Sep. 2004. Lecture.

[[Category: Multiparticle Systems]]

Multi-particle analysis of Momentum

2018-04-18T18:44:41Z

Sdhal3: /* The Main Idea */

'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' "Claimed by Simran Dhal (Spring 2018)"

==The Main Idea==

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

A multi-particle analysis of momentum can be applied to systems with multiple objects that may be interacting. Using the same principles of momentum applied on single objects, we apply these on the multiple objects that may make up a chosen system. However, one interesting characteristic of the multi-particle analysis of momentum is that it only factors in external forces applied on the object as opposed to internal forces which would be applied by objects on one another within a system even though objects within the system are interacting.

"Multi-particle System Momentum Principle"

<math> ∆\vec{p} = \vec{F}_{net} * {∆t}</math>.

===A Mathematical Model===

Mathematically, momentum is the product of a body's (in our case, a particle's) mass and velocity. It is also defined as the product of mass, net force, and change in time.

Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as

<math>{\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net}</math> (or <math>∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

<math> P = p_1 + p_2 = m_1v_1 + m_2v_2</math>

and deriving from <math> F = m * a </math>, the rate of change is the sum of the vector forces, and therefore:

<math> \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 </math>.

Newton's third law gives that:

<math> \frac{d\vec{P}}{dt} = 0 </math>,

'''so,'''

<math> m_1\vec{a}\_1 + m_2\vec{a}_2 = 0</math>.

<math>\vec{F}_{12} = -\vec{F}_{21} </math>.

<math> \vec{F}_{12} = m_2\vec{a}_2 </math>.

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

<math> \vec{P} = \sum_{i=1}^N m_i\vec{v}_i </math>
'''and'''
<math> \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} </math>.

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.

See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]

===Computational Models===

Click on the link below to understand the basics of the momentum principle in VPython,

[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

or check out this collision simulation to see a multi-particle analysis of it:

[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]

==Examples==

===Simple===

A loaded cart is travelling with momentum Mcart, and a brick is falling with momentum Mbrick. As the cart advances forward, the brick lands on it and the now heavier cart continues to move forward. The initial momentum of the system includes that of the cart and of the brick. Symbolically, what is the final momentum of the system?

'''Answer: Mcart + Mbrick'''

'''Explanation:'''

As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

===Middling===

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

'''Answer: 9.4e4 N'''

'''Explanation:'''

With the car as particle 1 and the truck as particle 2,
total initial momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>

Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.

===Difficult===

A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

'''Explanation:'''
Momentum before: 0,
Momentum after: <math> (2m(v - v_f) - m_{bin} * v_f) * i </math>
so,
<math> 0 = 2m_{people}(v - v_f) - m_{bin} * v</math>

Then solve for <math> v_f </math>.

==Connectedness==

Multi-particle analysis of momentum is just one way of analyzing a particular system, but multi-particle analysis is a way to examine other systems as well, and the momentum principle is very important for a variety of professional fields.

One application includes its use by automobile safety engineers, who must take into account the physics of momentum in order to safely design vehicles for their customers' use. However, this is by no means the only application--momentum is an applicable concept for literally anything that moves and has mass: figure skaters, roller coasters, pool balls...the list goes on.

For computer scientists, it's possible to create simulations like those listed in the "Computational Models" section of this article! In order to design those it is necessary for the creator to understand not only computer science concepts, but also the physics concepts that they are attempting to represent. In any field, it is possible that momentum comes into play.

==History==

The scientists most credited for the introduction of the concept of momentum to their field are Rene Descartes and Isaac Newton. Newton is more acknowledged for his Laws, from which the more specific concept of Conservation of Momentum can be derived, while Descartes holds the original "discovery" of momentum.

More information on the history of momentum can be found at:
[http://www.physicsbook.gatech.edu/Momentum_Principle#Connectedness] and
[http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html]

== See also ==

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit:
[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]

and to then explore the Conservation of Momentum page:
[http://www.physicsbook.gatech.edu/Conservation_of_Momentum].

Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[http://www.physicsbook.gatech.edu/Main_Page# Main Page].

Check out the links in the references below for more information on this subject as well as other physics concepts!

==References==
[1] Fitzpatrick, R. (2006, February 2). Angular momentum of a point particle. Retrieved November 27, 2016, from http://farside.ph.utexas.edu/teaching/301/lectures/node118.html

[2] Torre, Charles. "Newton's Third Law. Multi-Particle Systems." Newton’s Third Law. Multi-particle Systems. Conservation of Momentum. Physics 3550, Fall 2012 Newton’s Third Law. Multi-particle Systems. Relevant Sections in Text: §1.5, 3.1, 3.2, 3.3 (2012): n. pag. Intermediate Classical Mechanics. Utah State University, Sept. 2012. Web. 27 Nov. 2016.

[3] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 27 Nov. 2016. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[4] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[5] Fenton, Flavio. "Momentum and Second Newton's Law." 27 Nov. 2016. Lecture.

[6] Henderson, Tom. "Momentum and Its Conservation." Momentum and Its Conservation. The Physics Clasroom, 31 Dec. 2012. Web. 27 Nov. 2016.

[7] Dourmashkin, Peter. Lister, J. David. Pritchard, David E. Surrow, Bernd. "Momentum and Collisions." Massachusetts Institute of Technology. Sep. 2004. Lecture.

[[Category: Multiparticle Systems]]

Multi-particle analysis of Momentum

2018-04-18T18:44:18Z

Sdhal3: /* The Main Idea */

'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' "Claimed by Simran Dhal (Spring 2018)"

==The Main Idea==

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

A multi-particle analysis of momentum can be applied to systems with multiple objects that may be interacting. Using the same principles of momentum applied on single objects, we apply these on the multiple objects that may make up a chosen system. However, one interesting characteristic of the multi-particle analysis of momentum is that it only factors in external forces applied on the object as opposed to internal forces which would be applied by objects on one another within a system even though objects within the system are interacting.

"Multi-particle System Momentum Principle"

<math> (∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

===A Mathematical Model===

Mathematically, momentum is the product of a body's (in our case, a particle's) mass and velocity. It is also defined as the product of mass, net force, and change in time.

Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as

<math>{\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net}</math> (or <math>∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

<math> P = p_1 + p_2 = m_1v_1 + m_2v_2</math>

and deriving from <math> F = m * a </math>, the rate of change is the sum of the vector forces, and therefore:

<math> \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 </math>.

Newton's third law gives that:

<math> \frac{d\vec{P}}{dt} = 0 </math>,

'''so,'''

<math> m_1\vec{a}\_1 + m_2\vec{a}_2 = 0</math>.

<math>\vec{F}_{12} = -\vec{F}_{21} </math>.

<math> \vec{F}_{12} = m_2\vec{a}_2 </math>.

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

<math> \vec{P} = \sum_{i=1}^N m_i\vec{v}_i </math>
'''and'''
<math> \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} </math>.

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.

See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]

===Computational Models===

Click on the link below to understand the basics of the momentum principle in VPython,

[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

or check out this collision simulation to see a multi-particle analysis of it:

[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]

==Examples==

===Simple===

A loaded cart is travelling with momentum Mcart, and a brick is falling with momentum Mbrick. As the cart advances forward, the brick lands on it and the now heavier cart continues to move forward. The initial momentum of the system includes that of the cart and of the brick. Symbolically, what is the final momentum of the system?

'''Answer: Mcart + Mbrick'''

'''Explanation:'''

As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

===Middling===

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

'''Answer: 9.4e4 N'''

'''Explanation:'''

With the car as particle 1 and the truck as particle 2,
total initial momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>

Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.

===Difficult===

A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

'''Explanation:'''
Momentum before: 0,
Momentum after: <math> (2m(v - v_f) - m_{bin} * v_f) * i </math>
so,
<math> 0 = 2m_{people}(v - v_f) - m_{bin} * v</math>

Then solve for <math> v_f </math>.

==Connectedness==

Multi-particle analysis of momentum is just one way of analyzing a particular system, but multi-particle analysis is a way to examine other systems as well, and the momentum principle is very important for a variety of professional fields.

One application includes its use by automobile safety engineers, who must take into account the physics of momentum in order to safely design vehicles for their customers' use. However, this is by no means the only application--momentum is an applicable concept for literally anything that moves and has mass: figure skaters, roller coasters, pool balls...the list goes on.

For computer scientists, it's possible to create simulations like those listed in the "Computational Models" section of this article! In order to design those it is necessary for the creator to understand not only computer science concepts, but also the physics concepts that they are attempting to represent. In any field, it is possible that momentum comes into play.

==History==

The scientists most credited for the introduction of the concept of momentum to their field are Rene Descartes and Isaac Newton. Newton is more acknowledged for his Laws, from which the more specific concept of Conservation of Momentum can be derived, while Descartes holds the original "discovery" of momentum.

More information on the history of momentum can be found at:
[http://www.physicsbook.gatech.edu/Momentum_Principle#Connectedness] and
[http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html]

== See also ==

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit:
[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]

and to then explore the Conservation of Momentum page:
[http://www.physicsbook.gatech.edu/Conservation_of_Momentum].

Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[http://www.physicsbook.gatech.edu/Main_Page# Main Page].

Check out the links in the references below for more information on this subject as well as other physics concepts!

==References==
[1] Fitzpatrick, R. (2006, February 2). Angular momentum of a point particle. Retrieved November 27, 2016, from http://farside.ph.utexas.edu/teaching/301/lectures/node118.html

[2] Torre, Charles. "Newton's Third Law. Multi-Particle Systems." Newton’s Third Law. Multi-particle Systems. Conservation of Momentum. Physics 3550, Fall 2012 Newton’s Third Law. Multi-particle Systems. Relevant Sections in Text: §1.5, 3.1, 3.2, 3.3 (2012): n. pag. Intermediate Classical Mechanics. Utah State University, Sept. 2012. Web. 27 Nov. 2016.

[3] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 27 Nov. 2016. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[4] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[5] Fenton, Flavio. "Momentum and Second Newton's Law." 27 Nov. 2016. Lecture.

[6] Henderson, Tom. "Momentum and Its Conservation." Momentum and Its Conservation. The Physics Clasroom, 31 Dec. 2012. Web. 27 Nov. 2016.

[7] Dourmashkin, Peter. Lister, J. David. Pritchard, David E. Surrow, Bernd. "Momentum and Collisions." Massachusetts Institute of Technology. Sep. 2004. Lecture.

[[Category: Multiparticle Systems]]

Multi-particle analysis of Momentum

2018-04-18T18:43:55Z

Sdhal3: /* The Main Idea */

'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' "Claimed by Simran Dhal (Spring 2018)"

==The Main Idea==

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

A multi-particle analysis of momentum can be applied to systems with multiple objects that may be interacting. Using the same principles of momentum applied on single objects, we apply these on the multiple objects that may make up a chosen system. However, one interesting characteristic of the multi-particle analysis of momentum is that it only factors in external forces applied on the object as opposed to internal forces which would be applied by objects on one another within a system even though objects within the system are interacting.

"Multi-particle System Momentum Principle"

<math> (∆\vec{p_sys} = \vec{F}_{net} * {∆t}</math>).

===A Mathematical Model===

Mathematically, momentum is the product of a body's (in our case, a particle's) mass and velocity. It is also defined as the product of mass, net force, and change in time.

Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as

<math>{\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net}</math> (or <math>∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

<math> P = p_1 + p_2 = m_1v_1 + m_2v_2</math>

and deriving from <math> F = m * a </math>, the rate of change is the sum of the vector forces, and therefore:

<math> \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 </math>.

Newton's third law gives that:

<math> \frac{d\vec{P}}{dt} = 0 </math>,

'''so,'''

<math> m_1\vec{a}\_1 + m_2\vec{a}_2 = 0</math>.

<math>\vec{F}_{12} = -\vec{F}_{21} </math>.

<math> \vec{F}_{12} = m_2\vec{a}_2 </math>.

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

<math> \vec{P} = \sum_{i=1}^N m_i\vec{v}_i </math>
'''and'''
<math> \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} </math>.

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.

See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]

===Computational Models===

Click on the link below to understand the basics of the momentum principle in VPython,

[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

or check out this collision simulation to see a multi-particle analysis of it:

[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]

==Examples==

===Simple===

A loaded cart is travelling with momentum Mcart, and a brick is falling with momentum Mbrick. As the cart advances forward, the brick lands on it and the now heavier cart continues to move forward. The initial momentum of the system includes that of the cart and of the brick. Symbolically, what is the final momentum of the system?

'''Answer: Mcart + Mbrick'''

'''Explanation:'''

As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

===Middling===

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

'''Answer: 9.4e4 N'''

'''Explanation:'''

With the car as particle 1 and the truck as particle 2,
total initial momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>

Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.

===Difficult===

A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

'''Explanation:'''
Momentum before: 0,
Momentum after: <math> (2m(v - v_f) - m_{bin} * v_f) * i </math>
so,
<math> 0 = 2m_{people}(v - v_f) - m_{bin} * v</math>

Then solve for <math> v_f </math>.

==Connectedness==

Multi-particle analysis of momentum is just one way of analyzing a particular system, but multi-particle analysis is a way to examine other systems as well, and the momentum principle is very important for a variety of professional fields.

One application includes its use by automobile safety engineers, who must take into account the physics of momentum in order to safely design vehicles for their customers' use. However, this is by no means the only application--momentum is an applicable concept for literally anything that moves and has mass: figure skaters, roller coasters, pool balls...the list goes on.

For computer scientists, it's possible to create simulations like those listed in the "Computational Models" section of this article! In order to design those it is necessary for the creator to understand not only computer science concepts, but also the physics concepts that they are attempting to represent. In any field, it is possible that momentum comes into play.

==History==

The scientists most credited for the introduction of the concept of momentum to their field are Rene Descartes and Isaac Newton. Newton is more acknowledged for his Laws, from which the more specific concept of Conservation of Momentum can be derived, while Descartes holds the original "discovery" of momentum.

More information on the history of momentum can be found at:
[http://www.physicsbook.gatech.edu/Momentum_Principle#Connectedness] and
[http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html]

== See also ==

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit:
[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]

and to then explore the Conservation of Momentum page:
[http://www.physicsbook.gatech.edu/Conservation_of_Momentum].

Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[http://www.physicsbook.gatech.edu/Main_Page# Main Page].

Check out the links in the references below for more information on this subject as well as other physics concepts!

==References==
[1] Fitzpatrick, R. (2006, February 2). Angular momentum of a point particle. Retrieved November 27, 2016, from http://farside.ph.utexas.edu/teaching/301/lectures/node118.html

[2] Torre, Charles. "Newton's Third Law. Multi-Particle Systems." Newton’s Third Law. Multi-particle Systems. Conservation of Momentum. Physics 3550, Fall 2012 Newton’s Third Law. Multi-particle Systems. Relevant Sections in Text: §1.5, 3.1, 3.2, 3.3 (2012): n. pag. Intermediate Classical Mechanics. Utah State University, Sept. 2012. Web. 27 Nov. 2016.

[3] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 27 Nov. 2016. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[4] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[5] Fenton, Flavio. "Momentum and Second Newton's Law." 27 Nov. 2016. Lecture.

[6] Henderson, Tom. "Momentum and Its Conservation." Momentum and Its Conservation. The Physics Clasroom, 31 Dec. 2012. Web. 27 Nov. 2016.

[7] Dourmashkin, Peter. Lister, J. David. Pritchard, David E. Surrow, Bernd. "Momentum and Collisions." Massachusetts Institute of Technology. Sep. 2004. Lecture.

[[Category: Multiparticle Systems]]

Multi-particle analysis of Momentum

2018-04-18T18:43:33Z

Sdhal3: /* The Main Idea */

'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' "Claimed by Simran Dhal (Spring 2018)"

==The Main Idea==

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

A multi-particle analysis of momentum can be applied to systems with multiple objects that may be interacting. Using the same principles of momentum applied on single objects, we apply these on the multiple objects that may make up a chosen system. However, one interesting characteristic of the multi-particle analysis of momentum is that it only factors in external forces applied on the object as opposed to internal forces which would be applied by objects on one another within a system even though objects within the system are interacting.

"Multi-particle System Momentum Principle"

<math> (∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

===A Mathematical Model===

Mathematically, momentum is the product of a body's (in our case, a particle's) mass and velocity. It is also defined as the product of mass, net force, and change in time.

Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as

<math>{\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net}</math> (or <math>∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

<math> P = p_1 + p_2 = m_1v_1 + m_2v_2</math>

and deriving from <math> F = m * a </math>, the rate of change is the sum of the vector forces, and therefore:

<math> \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 </math>.

Newton's third law gives that:

<math> \frac{d\vec{P}}{dt} = 0 </math>,

'''so,'''

<math> m_1\vec{a}\_1 + m_2\vec{a}_2 = 0</math>.

<math>\vec{F}_{12} = -\vec{F}_{21} </math>.

<math> \vec{F}_{12} = m_2\vec{a}_2 </math>.

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

<math> \vec{P} = \sum_{i=1}^N m_i\vec{v}_i </math>
'''and'''
<math> \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} </math>.

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.

See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]

===Computational Models===

Click on the link below to understand the basics of the momentum principle in VPython,

[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

or check out this collision simulation to see a multi-particle analysis of it:

[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]

==Examples==

===Simple===

A loaded cart is travelling with momentum Mcart, and a brick is falling with momentum Mbrick. As the cart advances forward, the brick lands on it and the now heavier cart continues to move forward. The initial momentum of the system includes that of the cart and of the brick. Symbolically, what is the final momentum of the system?

'''Answer: Mcart + Mbrick'''

'''Explanation:'''

As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

===Middling===

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

'''Answer: 9.4e4 N'''

'''Explanation:'''

With the car as particle 1 and the truck as particle 2,
total initial momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>

Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.

===Difficult===

A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

'''Explanation:'''
Momentum before: 0,
Momentum after: <math> (2m(v - v_f) - m_{bin} * v_f) * i </math>
so,
<math> 0 = 2m_{people}(v - v_f) - m_{bin} * v</math>

Then solve for <math> v_f </math>.

==Connectedness==

Multi-particle analysis of momentum is just one way of analyzing a particular system, but multi-particle analysis is a way to examine other systems as well, and the momentum principle is very important for a variety of professional fields.

One application includes its use by automobile safety engineers, who must take into account the physics of momentum in order to safely design vehicles for their customers' use. However, this is by no means the only application--momentum is an applicable concept for literally anything that moves and has mass: figure skaters, roller coasters, pool balls...the list goes on.

For computer scientists, it's possible to create simulations like those listed in the "Computational Models" section of this article! In order to design those it is necessary for the creator to understand not only computer science concepts, but also the physics concepts that they are attempting to represent. In any field, it is possible that momentum comes into play.

==History==

The scientists most credited for the introduction of the concept of momentum to their field are Rene Descartes and Isaac Newton. Newton is more acknowledged for his Laws, from which the more specific concept of Conservation of Momentum can be derived, while Descartes holds the original "discovery" of momentum.

More information on the history of momentum can be found at:
[http://www.physicsbook.gatech.edu/Momentum_Principle#Connectedness] and
[http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html]

== See also ==

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit:
[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]

and to then explore the Conservation of Momentum page:
[http://www.physicsbook.gatech.edu/Conservation_of_Momentum].

Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[http://www.physicsbook.gatech.edu/Main_Page# Main Page].

Check out the links in the references below for more information on this subject as well as other physics concepts!

==References==
[1] Fitzpatrick, R. (2006, February 2). Angular momentum of a point particle. Retrieved November 27, 2016, from http://farside.ph.utexas.edu/teaching/301/lectures/node118.html

[2] Torre, Charles. "Newton's Third Law. Multi-Particle Systems." Newton’s Third Law. Multi-particle Systems. Conservation of Momentum. Physics 3550, Fall 2012 Newton’s Third Law. Multi-particle Systems. Relevant Sections in Text: §1.5, 3.1, 3.2, 3.3 (2012): n. pag. Intermediate Classical Mechanics. Utah State University, Sept. 2012. Web. 27 Nov. 2016.

[3] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 27 Nov. 2016. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[4] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[5] Fenton, Flavio. "Momentum and Second Newton's Law." 27 Nov. 2016. Lecture.

[6] Henderson, Tom. "Momentum and Its Conservation." Momentum and Its Conservation. The Physics Clasroom, 31 Dec. 2012. Web. 27 Nov. 2016.

[7] Dourmashkin, Peter. Lister, J. David. Pritchard, David E. Surrow, Bernd. "Momentum and Collisions." Massachusetts Institute of Technology. Sep. 2004. Lecture.

[[Category: Multiparticle Systems]]

Multi-particle analysis of Momentum

2018-04-18T18:30:00Z

Sdhal3: /* The Main Idea */

'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' "Claimed by Simran Dhal (Spring 2018)"

==The Main Idea==

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

A multi-particle analysis of momentum can be applied to systems with multiple objects that may be interacting. Using the same principles of momentum applied on single objects, we apply these on the multiple objects that may make up a chosen system. However, one interesting characteristic of the multi-particle analysis of momentum is that it only factors in external forces applied on the object as opposed to internal forces which would be applied by objects on one another within a system even though objects within the system are interacting.

===A Mathematical Model===

Mathematically, momentum is the product of a body's (in our case, a particle's) mass and velocity. It is also defined as the product of mass, net force, and change in time.

Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as

<math>{\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net}</math> (or <math>∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

<math> P = p_1 + p_2 = m_1v_1 + m_2v_2</math>

and deriving from <math> F = m * a </math>, the rate of change is the sum of the vector forces, and therefore:

<math> \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 </math>.

Newton's third law gives that:

<math> \frac{d\vec{P}}{dt} = 0 </math>,

'''so,'''

<math> m_1\vec{a}\_1 + m_2\vec{a}_2 = 0</math>.

<math>\vec{F}_{12} = -\vec{F}_{21} </math>.

<math> \vec{F}_{12} = m_2\vec{a}_2 </math>.

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

<math> \vec{P} = \sum_{i=1}^N m_i\vec{v}_i </math>
'''and'''
<math> \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} </math>.

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.

See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]

===Computational Models===

Click on the link below to understand the basics of the momentum principle in VPython,

[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

or check out this collision simulation to see a multi-particle analysis of it:

[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]

==Examples==

===Simple===

A loaded cart is travelling with momentum Mcart, and a brick is falling with momentum Mbrick. As the cart advances forward, the brick lands on it and the now heavier cart continues to move forward. The initial momentum of the system includes that of the cart and of the brick. Symbolically, what is the final momentum of the system?

'''Answer: Mcart + Mbrick'''

'''Explanation:'''

As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

===Middling===

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

'''Answer: 9.4e4 N'''

'''Explanation:'''

With the car as particle 1 and the truck as particle 2,
total initial momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>

Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.

===Difficult===

A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

'''Explanation:'''
Momentum before: 0,
Momentum after: <math> (2m(v - v_f) - m_{bin} * v_f) * i </math>
so,
<math> 0 = 2m_{people}(v - v_f) - m_{bin} * v</math>

Then solve for <math> v_f </math>.

==Connectedness==

Multi-particle analysis of momentum is just one way of analyzing a particular system, but multi-particle analysis is a way to examine other systems as well, and the momentum principle is very important for a variety of professional fields.

One application includes its use by automobile safety engineers, who must take into account the physics of momentum in order to safely design vehicles for their customers' use. However, this is by no means the only application--momentum is an applicable concept for literally anything that moves and has mass: figure skaters, roller coasters, pool balls...the list goes on.

For computer scientists, it's possible to create simulations like those listed in the "Computational Models" section of this article! In order to design those it is necessary for the creator to understand not only computer science concepts, but also the physics concepts that they are attempting to represent. In any field, it is possible that momentum comes into play.

==History==

The scientists most credited for the introduction of the concept of momentum to their field are Rene Descartes and Isaac Newton. Newton is more acknowledged for his Laws, from which the more specific concept of Conservation of Momentum can be derived, while Descartes holds the original "discovery" of momentum.

More information on the history of momentum can be found at:
[http://www.physicsbook.gatech.edu/Momentum_Principle#Connectedness] and
[http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html]

== See also ==

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit:
[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]

and to then explore the Conservation of Momentum page:
[http://www.physicsbook.gatech.edu/Conservation_of_Momentum].

Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[http://www.physicsbook.gatech.edu/Main_Page# Main Page].

Check out the links in the references below for more information on this subject as well as other physics concepts!

==References==
[1] Fitzpatrick, R. (2006, February 2). Angular momentum of a point particle. Retrieved November 27, 2016, from http://farside.ph.utexas.edu/teaching/301/lectures/node118.html

[2] Torre, Charles. "Newton's Third Law. Multi-Particle Systems." Newton’s Third Law. Multi-particle Systems. Conservation of Momentum. Physics 3550, Fall 2012 Newton’s Third Law. Multi-particle Systems. Relevant Sections in Text: §1.5, 3.1, 3.2, 3.3 (2012): n. pag. Intermediate Classical Mechanics. Utah State University, Sept. 2012. Web. 27 Nov. 2016.

[3] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 27 Nov. 2016. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[4] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[5] Fenton, Flavio. "Momentum and Second Newton's Law." 27 Nov. 2016. Lecture.

[6] Henderson, Tom. "Momentum and Its Conservation." Momentum and Its Conservation. The Physics Clasroom, 31 Dec. 2012. Web. 27 Nov. 2016.

[7] Dourmashkin, Peter. Lister, J. David. Pritchard, David E. Surrow, Bernd. "Momentum and Collisions." Massachusetts Institute of Technology. Sep. 2004. Lecture.

[[Category: Multiparticle Systems]]

Multi-particle analysis of Momentum

2018-04-18T18:17:02Z

Sdhal3:

'''Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)''' "Claimed by Simran Dhal (Spring 2018)"

==The Main Idea==

Newton's third law states (on a basic level) that for every action, there is an equal and opposite reaction. The application and importance of this law in terms of momentum is that in a "closed system," momentum is conserved. The meaning of "closed" is that within the system--the actions and reactions on the involved objects--all forces are accounted for. We can model conservation of momentum by looking at the bodies of a system as particles, which is what will be discussed for the remainder of this article. The momentum principle and the multi-particle analysis of it make it easier to analyze situations such as collisions, and can make it much easier to solve problems.

===A Mathematical Model===

Mathematically, momentum is the product of a body's (in our case, a particle's) mass and velocity. It is also defined as the product of mass, net force, and change in time.

Where '''p''' is momentum, '''F''' is the net force from the surroundings, and '''t''' is time, measured in seconds, The Momentum Principle is formulaically defined as

<math>{\frac{d\vec{p}}{dt}}_{system}= \vec{F}_{net}</math> (or <math>∆\vec{p} = \vec{F}_{net} * {∆t}</math>).

Taking two particles, particle 1 and particle 2, we can define the total momentum as:

<math> P = p_1 + p_2 = m_1v_1 + m_2v_2</math>

and deriving from <math> F = m * a </math>, the rate of change is the sum of the vector forces, and therefore:

<math> \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 </math>.

Newton's third law gives that:

<math> \frac{d\vec{P}}{dt} = 0 </math>,

'''so,'''

<math> m_1\vec{a}\_1 + m_2\vec{a}_2 = 0</math>.

<math>\vec{F}_{12} = -\vec{F}_{21} </math>.

<math> \vec{F}_{12} = m_2\vec{a}_2 </math>.

The total momentum does not change despite the particles' change in position, velocity, and acceleration. This does not mean that each individual particle will retain the same momentum, but that the total momentum of the system is conserved.

If i and j are the first two particles out of total N particles,

<math> \vec{P} = \sum_{i=1}^N m_i\vec{v}_i </math>
'''and'''
<math> \frac{d\vec{P}}{dt} = \sum_{j=1}^N m_j\vec{a}_j = \sum_{j=1}^N\sum_{i=1}^N F_{ij} </math>.

Each term in this sum cancels with another due to the application of "equal and opposite."

If the system is subject to a force that is not included in the system (external),

<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.

See more on that at: [http://www.physicsbook.gatech.edu/Momentum_with_respect_to_external_Forces]

===Computational Models===

Click on the link below to understand the basics of the momentum principle in VPython,

[https://trinket.io/embed/glowscript/8271b15824 GlowScript]

or check out this collision simulation to see a multi-particle analysis of it:

[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]

==Examples==

===Simple===

A loaded cart is travelling with momentum Mcart, and a brick is falling with momentum Mbrick. As the cart advances forward, the brick lands on it and the now heavier cart continues to move forward. The initial momentum of the system includes that of the cart and of the brick. Symbolically, what is the final momentum of the system?

'''Answer: Mcart + Mbrick'''

'''Explanation:'''

As explained through the mathematical and computational models above, Newton's Third Law allows us to assume that momentum is conserved within the system before and after a collision, meaning that the initial and final momentum of the system are the same. The initial momentum here is Mcart + Mbrick, therefore the final momentum is the same.

===Middling===

Given a car with a mass of 200kg and an initial velocity of 90m/s and a truck with a mass of 950kg and an initial velocity of 80m/s, find the TOTAL, FINAL momentum of the system after the car and truck collide. Include both the car and truck in the system.

'''Answer: 9.4e4 N'''

'''Explanation:'''

With the car as particle 1 and the truck as particle 2,
total initial momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>

Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.

===Difficult===

A woman, Katherine, and a man, Robert, each have a mass of m, and decide to have some fun in the theatre before putting away curtains. They stand on top of a curtain bin. Katherine and Robert jump off of one side of the bin at a velocity of v relative to the bin, and when they do so, the bin rolls in the direction opposite of which they jumped. Symbolically, what is the final velocity of the bin if both Katherine and Robert jump at the ''same time''?

'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''

'''Explanation:'''
Momentum before: 0,
Momentum after: <math> (2m(v - v_f) - m_{bin} * v_f) * i </math>
so,
<math> 0 = 2m_{people}(v - v_f) - m_{bin} * v</math>

Then solve for <math> v_f </math>.

==Connectedness==

Multi-particle analysis of momentum is just one way of analyzing a particular system, but multi-particle analysis is a way to examine other systems as well, and the momentum principle is very important for a variety of professional fields.

One application includes its use by automobile safety engineers, who must take into account the physics of momentum in order to safely design vehicles for their customers' use. However, this is by no means the only application--momentum is an applicable concept for literally anything that moves and has mass: figure skaters, roller coasters, pool balls...the list goes on.

For computer scientists, it's possible to create simulations like those listed in the "Computational Models" section of this article! In order to design those it is necessary for the creator to understand not only computer science concepts, but also the physics concepts that they are attempting to represent. In any field, it is possible that momentum comes into play.

==History==

The scientists most credited for the introduction of the concept of momentum to their field are Rene Descartes and Isaac Newton. Newton is more acknowledged for his Laws, from which the more specific concept of Conservation of Momentum can be derived, while Descartes holds the original "discovery" of momentum.

More information on the history of momentum can be found at:
[http://www.physicsbook.gatech.edu/Momentum_Principle#Connectedness] and
[http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html]

== See also ==

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit:
[http://www.physicsbook.gatech.edu/Momentum_Principle Momentum Principle]

and to then explore the Conservation of Momentum page:
[http://www.physicsbook.gatech.edu/Conservation_of_Momentum].

Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[http://www.physicsbook.gatech.edu/Main_Page# Main Page].

Check out the links in the references below for more information on this subject as well as other physics concepts!

==References==
[1] Fitzpatrick, R. (2006, February 2). Angular momentum of a point particle. Retrieved November 27, 2016, from http://farside.ph.utexas.edu/teaching/301/lectures/node118.html

[2] Torre, Charles. "Newton's Third Law. Multi-Particle Systems." Newton’s Third Law. Multi-particle Systems. Conservation of Momentum. Physics 3550, Fall 2012 Newton’s Third Law. Multi-particle Systems. Relevant Sections in Text: §1.5, 3.1, 3.2, 3.3 (2012): n. pag. Intermediate Classical Mechanics. Utah State University, Sept. 2012. Web. 27 Nov. 2016.

[3] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 27 Nov. 2016. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>

[4] Sherwood, Bruce A. "2.1 The Momentum Principle." Matter & Interactions. By Ruth W. Chabay. 4th ed. Vol. 1. N.p.: John Wiley & Sons, 2015. 45-50. Print. Modern Mechanics.

[5] Fenton, Flavio. "Momentum and Second Newton's Law." 27 Nov. 2016. Lecture.

[6] Henderson, Tom. "Momentum and Its Conservation." Momentum and Its Conservation. The Physics Clasroom, 31 Dec. 2012. Web. 27 Nov. 2016.

[7] Dourmashkin, Peter. Lister, J. David. Pritchard, David E. Surrow, Bernd. "Momentum and Collisions." Massachusetts Institute of Technology. Sep. 2004. Lecture.

[[Category: Multiparticle Systems]]