OpenStax-CNX module: m26666
1
Two-Body Collision Problem
∗
Stephen Wong
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 3.0†
Abstract
The derivation of the kinetics of a perfectly elastic the collision of two spherical objects of dierent
mass
Consider the perfectly elastic collision between two massive point objects moving with individual velocities:
∗ Version
1.1: Jul 2, 2009 6:22 pm -0500
† http://creativecommons.org/licenses/by/3.0/
http://cnx.org/content/m26666/1.1/
OpenStax-CNX module: m26666
2
General Two-Body Collision
Figure 1: Perfectly elastic collision of two spherical bodies with dierent masses and velocities.
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 nd the motion of the center of mass by utilizing the
denition that the net momentum is equal to the total mass times the velocity of the center of mass:
We can thus do the transformation to the center-of-mass coordinate system:
http://cnx.org/content/m26666/1.1/
OpenStax-CNX module: m26666
3
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:
General Two-Body Collision in the Center-of-Mass frame
Figure 2: In the center of mass frame, the two bodies have exactly opposite momenta.
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 nal momenta (or velocities) are the original momenta (or velocities) which is the non-interacting
case or that the nal 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.
http://cnx.org/content/m26666/1.1/
OpenStax-CNX module: m26666
4
It is nice to talk about the impulse of the collision, which is the change in momentum due to the
collision:
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.
The change in velocity is thus
To check this answer, consider the case where the masses are equal:
http://cnx.org/content/m26666/1.1/
OpenStax-CNX module: m26666
5
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 dierence
between their original velocities, i.e. it simply elastically bounces o the other ball. Ball #2 on the other
hand, has a change of velocity that tends towards zero, that is, it is unaected by the much smaller ball
bouncing o of it.
http://cnx.org/content/m26666/1.1/