Elastic Collisions in 1 Dimension

Using the Center of Mass Reference Frame

Proof

The Problem:

First, switch to a frame of reference in which m2 is at rest. Now a particle of mass m1 and velocity v collides elastically (in one dimension) with a stationary particle of mass m2. What are the velocities of m1 and m2 after the collision?

Before diagram
after diagram

A particle of mass m1 and velocity v collides elastically with a particle of mass m2, initially at rest.

After the collision, m1 has velocity v1, and m2 has velocity v2. What are v1 and v2?

Here is a proof that the center of mass method works for one-dimensional elastic collisions:

  1. Find the velocity of the system center of mass:
    1. v_cm = (m1/(m1 + m2))v
  2. Switch to the center of mass reference frame. To do this, simply subtract vcm from each particle's velocity.
    1. v1_cm and v2_cm
  3. Have the collision. The particles' velocities reverse.
    1. velocities after in cm frame
  4. Switch back to the original frame of reference, by adding vcm to each particle's velocity.
    1. v1 and v2 after the collision
  5. Smile, you're done. Note the v1 and v2 shown in step 3 match the results derived for the general solution. Q.E.D.
last update September 28, 2004 by JL Stanbrough