Multi-particle analysis of Momentum

Written by Madeline Helmstadter; mhelmstadter3 (Fall 2016)

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]\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]

Computational Models

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

GlowScript

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.

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]\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.