Elastic Collision

When one body hits another, their velocities change. However, the total momentum remains the same, so the final and initial velocities are related by this equation for conservation of momentum:

m1u1 + m2u2 = m1v1 + m2v2

where

m1 = mass of body 1
u1 = initial velocity of body 1
v1 = final velocity of body 1

m2 = mass of body 2
u2 = initial velocity of body 2
v2 = final velocity of body 2

Physically, the final velocities can be different because the bodies are made of different materials. A tennis ball will bounce more, but a plasticine wall may stick to the other body. During collision, some kinetic energy could get changed to heat. A tennis ball is likely to retain most of its kinetic energy when it bounces back from a wall. We call this elastic collision.

We consider the ideal case in which no energy is lost to heat. This is then the total kinetic energy after the collision of two bodies remains the same. This is called perfectly eleastic collision. So the sum of the initial kinetic energies is equal to the sum of the final kinetic energies. This conservation of kinetic energy can be written down as an equation:

m1u12/2 + m2u22/2 = m1v12/2 + m2v22/2

With some algebra that I shall not go into, the two equations above can be combined to give

u1 - u2 = -(v1 - v2)


Copyright 2011 by Kai Hock. All rights reserved.
Last updated: 11 March 2011