Multi-particle analysis of Momentum

Written by Madeline Helmstadter; mhelmstadter3

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].

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

Easy

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.

Mid-Difficulty

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

• • Angular Momentum**

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]\displaystyle{ \frac{p x p}{m} + (r x f) }[/math] where p is linear momentum, and f is rate of change of linear momentum.

Explanation:

Angular momentum is the cross product of position and linear momentum, and an angle Theta, which is given as the angle between the directions of position and momentum.

[math]\displaystyle{ \frac{dl}{dt} = r x p + (r x p) }[/math] [math]\displaystyle{ r = v = \frac{p}{m} }[/math] [math]\displaystyle{ p = f }[/math] [math]\displaystyle{ \frac{dl}{dt} = \frac{p x p}{m} + (r x f) }[/math]

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 Energy Principle and the 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 Impulse and Iterative Prediction, which are used to solve other types of problems.

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. 29 Nov. 2015. <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." 26 Aug. 2015. Lecture.