Two-Body Collision Problem

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Summary: The derivation of the kinetics of a perfectly elastic the collision of two spherical objects of different mass

Consider the perfectly elastic collision between two massive point objects moving with individual velocities:

Figure 1: Perfectly elastic collision of two spherical bodies with different masses and velocities.
General Two-Body Collision
Two bodies colliding

The problem is much more easily solved in the center of mass coordinate system because the total momentum in that coordinate system is zero. We can find the motion of the center of mass by utilizing the definition that the net momentum is equal to the total mass times the velocity of the center of mass:

Center of mass calculation

We can thus do the transformation to the center-of-mass coordinate system:

Center of mass calculation

Note that the total momentum in the center-of-mass frame is indeed zero. The mass factor in the momentum equations is called the “reduced mass”.

In this coordinate system, we have the following picture:

Figure 2: In the center of mass frame, the two bodies have exactly opposite momenta.
General Two-Body Collision in the Center-of-Mass frame
Two bodies colliding in center of mass frame

The solution to the elastic recoil problem is trivial now. Each mass simply rebounds with the negative of its original velocity. There are a number of ways to prove this, all of which come down to the same statement: To conserve both momentum and energy, the only solutions when the total initial momentum is zero are that the final momenta (or velocities) are the original momenta (or velocities) which is the non-interacting case or that the final momenta (or velocities) are the negatives of the original momenta (or velocities). This assumes, of course, that mass is conserved on a particle by particle basis.

Energy and momentum conservation

It is nice to talk about the “impulse” of the collision, which is the change in momentum due to the collision:

momentum impulse

The last equation is simply a statement of Newton’s Third Law of Motion.(“For every action there is an equal an opposite reaction”).

Since the impulse is a change in momentum, it is invariant under the Galilean transformation back to the laboratory (non-center-of-mass) coordinates.

momentum impulse in lab coordinates

The change in velocity is thus

change in velocity

To check this answer, consider the case where the masses are equal:

equal mass case

Another interesting limit is if one mass is much, much greater than the other, say here, mass #2 is much, much greater than mass #1. In this limit, the change in velocity for ball #1 is 2 times time difference between their original velocities, i.e. it simply elastically bounces off the other ball. Ball #2 on the other hand, has a change of velocity that tends towards zero, that is, it is unaffected by the much smaller ball bouncing off of it.



Originally published in Connexions (CNX):

© 2023 by Stephen Wong