Multi-particle analysis of Momentum: Difference between revisions

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'''Claimed by mhelmstadter3
 
'''


==The Main Idea==
==The Main Idea==
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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.
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===
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:blue_interact.png|350px|center|thumb|Figure 1 <br> Note: Figure created by author]]
 
From the figure above, 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).


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.
'''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  
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  
Line 33: Line 56:


<math> \vec{F}_{12} = m_2\vec{a}_2 </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.  
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.  
Line 49: Line 71:
<math> \frac{d\vec{P}}{dt} =\vec{F}_{ext} </math>.
<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===
===Computational Models===
Line 55: Line 94:


[https://trinket.io/embed/glowscript/8271b15824 GlowScript]
[https://trinket.io/embed/glowscript/8271b15824 GlowScript]
    # GlowScript 1.1 VPython
        #people
        #note that the people has a trail
        human1=sphere(pos=vec(-40,0,0), radius=0.5, color=color.red, make_trail=true)
        human2=sphere(pos=vec(-20,0,0), radius=0.5, color=color.green, make_trail=true)
        human3=sphere(pos=vec(20,0,0), radius=0.5, color=color.blue, make_trail=true)
        human4=sphere(pos=vec(40,0,0), radius=0.5, color=color.white, make_trail=true)
        human1.m=0.1 #mass of the people in kg
        human2.m=0.1
        human3.m=0.1
        human4.m=0.1
        human1.p=human1.m*vec(0,0,0) #starting momentum mass times velocity
        human2.p=human2.m*vec(-2,0,0)
        human3.p=human3.m*vec(0,2,0)
        human4.p=human4.m*vec(-2,2,0)
        t=0
        dt=0.01 #size of the time step
        #putting the loop as "while True" means it runs forever
        #you could change this to while t< 10: or something
        while t<0.02:
            rate(100) #this says 100 calculations per second
   
        #given net force
        Fnet=vec(1,-6,-1)
   
        #update the momentum
        human1.p=human1.p+Fnet*dt
        human2.p=human2.p+Fnet*dt
        human3.p=human3.p+Fnet*dt
        human4.p=human4.p+Fnet*dt
   
        #update the position of the ball
        human1.pos= human1.pos +human1.p*dt/human1.m
        human2.pos= human2.pos +human2.p*dt/human2.m
        human3.pos= human3.pos +human3.p*dt/human3.m
        human4.pos= human4.pos +human4.p*dt/human4.m
   
        #this next line updates the spring
        #spring.axis=ball.pos-holder.pos
    print("Final Momentum of Human 1 = ",human1.p," kg*m/s")
    print("Final Momentum of Human 2 = ",human2.p," kg*m/s")
    print("Final Momentum of Human 3 = ",human3.p," kg*m/s")
    print("Final Momentum of Human 4 = ",human4.p," kg*m/s")
   


or check out this collision simulation to see a multi-particle analysis of it:
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]
[https://phet.colorado.edu/sims/collision-lab/collision-lab_en.html Collision Lab]


==Examples==
==Examples==


===Easy===
===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?  
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?  
Line 73: Line 162:
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.
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.


===Mid-Difficulty===
===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.
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.
Line 86: Line 175:
Initial momentum = Final momentum, therefore
Initial momentum = Final momentum, therefore
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
total final momentum = <math> p_1 + p_2 = m_1v_1 + m_2v_2</math>,
which is equal to 9.4e4N.  
which is equal to 9.4e4N.


===Difficult===
===Difficult===


**Angular Momentum**
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''?


Consider a pool ball of mass '''m''' moving with a velocity '''v'''. It is at position '''r''' and rotates so that it passes through the origin of the coordinate system. Symbolically determine the rate of change of the angular momentum of the pool ball.
'''Answer: <math> v_f = \frac{2m * v}{m_{people} + m_{bin}} </math>'''


'''Answer: <math> \frac{p^2}\{m} + r * f </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>


\frac{p^2}\{m} + r * f
Then solve for <math> v_f </math>.


'''Explanation:'''
==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.


==Connectedness==
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.


All over the world and at every point in time, interactions are continuously occurring, and I thought it was interesting to see how the Momentum Principle was the most fundamental principle that is used in starting to the analyze the different interactions. Although there is not a direct relationship between the concept of the Momentum Principle and my major in Biochemistry (which would have more connections with the Energy Principle), there are many industrial applications of the Momentum Principle. Again, the Momentum Principle is not directly connected to the applications, but it is used in the process (especially in the beginning) of industrial application. For example, when creating life saving airbags and seat belts for cars, the Momentum Principle is used. The final momentum of a car during an accident would be zero, or would stop, and the initial momentum would be based on the mass and velocity of the car. With the change in momentum fixed, the airbag and seat belt would focus on increasing the time taken for the body's momentum to reach zero (final momentum), which would consequently reduce the force of the collision and protect the body from getting as injured. With the Momentum Principle being applicable in so many areas of my life, I found the concept even more interesting.
==History==


==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.


Although the Momentum Principle is credited as Newton’s second law of motion, it is difficult to just credit Isaac Newton (1643AD – 1727AD) for the development of the principle. As the Momentum Principle is the quantitative and more in-depth representation of Newton’s first law of motion (“An object tends to be at rest or moves in a straight line and a constant speed except to the extent that it interact with other objects”), the development of the first law also serves an important role in the history of the Momentum Principle. Aristotle (384BC – 322BC) initially proposed that objects had the natural tendency to be at rest and that a push (or a force) was absolutely needed to keep the object moving. His proposal was challenged by Galileo (1564AD – 1642AD), who introduced the idea that objects had the natural tendency to travel in a straight line at constant speed unless something (or a force) was interacting with something. Likewise, Descartes (1596AD – 1650AD) also contributed as he proposed three laws of nature in his “Principle of Philosophy,” which actually outlined the later published Newton’s first law of motion. After studying Descartes, Newton adopted Descartes’ principles as his first law of motion, and alongside the famous story of Newton sitting under an apple, Newton was able to create the Momentum Principle, or his second law of motion.
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 ==
== See also ==


As the Momentum Principle is the first of three fundamental principles of mechanics, the next possible topics to examine would be the other fundamental principles, the [http://www.physicsbook.gatech.edu/The_Energy_Principle Energy Principle] and the [http://www.physicsbook.gatech.edu/The_Angular_Momentum_Principle Angular Momentum Principle]. Also, although the Momentum Principle is an extremely important concept that usually signals the start of a momentum related problem, the principle branches out into other momentum topics like [http://www.physicsbook.gatech.edu/Impulse_Momentum Impulse] and [http://www.physicsbook.gatech.edu/Iterative_Prediction Iterative Prediction], which are used to solve other types of problems.
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].


===External links===
Other information on multi-particle systems can be found in the Week 9 category of the Main Page:
[1] https://www.khanacademy.org/science/physics/linear-momentum/momentum-tutorial/v/introduction-to-momentum
[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!


[2] https://www.youtube.com/watch?v=ZvPrn3aBQG8


==References==
==References==
[1] Harris, William. "How Netwon's Laws of Motion Works." HowStuffWorks. HowStuffWorks.com, 29 July 2008. Web. 29 Nov. 2015. <http://science.howstuffworks.com/innovation/scientific-experiments/newton-law-of-motion.htm>
[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.


[2] 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.


[3] Fenton, Flavio. "Momentum and Second Newton's Law." 26 Aug. 2015. 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:Momentum]]
[[Category: Multiparticle Systems]]

Latest revision as of 11:18, 29 July 2019


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]\displaystyle{ ∆\vec{p} = \vec{F}_{net} * {∆t} }[/math]

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

[math]\displaystyle{ \vec{F}_{net} = \vec{F}_{1} + \vec{F}_{2} + \vec{F}_{3} + ... }[/math]

Figure 1
Note: Figure created by author

From the figure above, 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]\displaystyle{ \vec{F}_{1,2} = \vec{-F}_{2,1} }[/math]

[math]\displaystyle{ \vec{F}_{1,3} = \vec{-F}_{3,1} }[/math]

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

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

[math]\displaystyle{ P = p_1 + p_2 = m_1v_1 + m_2v_2 }[/math]

and deriving from [math]\displaystyle{ F = m * a }[/math], the rate of change is the sum of the vector forces, and therefore:

[math]\displaystyle{ \frac{d\vec{P}}{dt} = m_1\vec{a}_1 + m_2\vec{a}_2 }[/math].

Newton's third law gives that:

[math]\displaystyle{ \frac{d\vec{P}}{dt} = 0 }[/math],

so,

[math]\displaystyle{ m_1\vec{a}\_1 + m_2\vec{a}_2 = 0 }[/math].

[math]\displaystyle{ \vec{F}_{12} = -\vec{F}_{21} }[/math].

[math]\displaystyle{ \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]\displaystyle{ \vec{P} = \sum_{i=1}^N m_i\vec{v}_i }[/math] and [math]\displaystyle{ \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]\displaystyle{ \frac{d\vec{P}}{dt} =\vec{F}_{ext} }[/math].

See more on that at: [1]

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]\displaystyle{ \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]\displaystyle{ \vec{∆p}_{sys} = \vec{F}_{net,surr} * ∆t }[/math]

This equation can essentially be broken down further into this:

[math]\displaystyle{ \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,

GlowScript 

   # GlowScript 1.1 VPython

       #people
       #note that the people has a trail
       human1=sphere(pos=vec(-40,0,0), radius=0.5, color=color.red, make_trail=true)
       human2=sphere(pos=vec(-20,0,0), radius=0.5, color=color.green, make_trail=true)
       human3=sphere(pos=vec(20,0,0), radius=0.5, color=color.blue, make_trail=true)
       human4=sphere(pos=vec(40,0,0), radius=0.5, color=color.white, make_trail=true)

       human1.m=0.1 #mass of the people in kg
       human2.m=0.1
       human3.m=0.1
       human4.m=0.1

       human1.p=human1.m*vec(0,0,0) #starting momentum mass times velocity
       human2.p=human2.m*vec(-2,0,0)
       human3.p=human3.m*vec(0,2,0)
       human4.p=human4.m*vec(-2,2,0)

       t=0
       dt=0.01 #size of the time step

       #putting the loop as "while True" means it runs forever
       #you could change this to while t< 10: or something
       while t<0.02: 
           rate(100) #this says 100 calculations per second
   
       #given net force
       Fnet=vec(1,-6,-1)
   
       #update the momentum
       human1.p=human1.p+Fnet*dt
       human2.p=human2.p+Fnet*dt
       human3.p=human3.p+Fnet*dt
       human4.p=human4.p+Fnet*dt
   
       #update the position of the ball
       human1.pos= human1.pos +human1.p*dt/human1.m
       human2.pos= human2.pos +human2.p*dt/human2.m
       human3.pos= human3.pos +human3.p*dt/human3.m
       human4.pos= human4.pos +human4.p*dt/human4.m
   
       #this next line updates the spring
       #spring.axis=ball.pos-holder.pos

   print("Final Momentum of Human 1 = ",human1.p," kg*m/s")
   print("Final Momentum of Human 2 = ",human2.p," kg*m/s")
   print("Final Momentum of Human 3 = ",human3.p," kg*m/s")
   print("Final Momentum of Human 4 = ",human4.p," kg*m/s")

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

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]\displaystyle{ p_1 + p_2 = m_1v_1 + m_2v_2 }[/math]

Initial momentum = Final momentum, therefore total final momentum = [math]\displaystyle{ 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]\displaystyle{ v_f = \frac{2m * v}{m_{people} + m_{bin}} }[/math]

Explanation: Momentum before: 0, Momentum after: [math]\displaystyle{ (2m(v - v_f) - m_{bin} * v_f) * i }[/math] so, [math]\displaystyle{ 0 = 2m_{people}(v - v_f) - m_{bin} * v }[/math]

Then solve for [math]\displaystyle{ 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: [2] and [3]

See also

In order to better understand the multi-particle analysis of Momentum, it would be best to first visit: Momentum Principle

and to then explore the Conservation of Momentum page: [4].

Other information on multi-particle systems can be found in the Week 9 category of the 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.

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