Momentum & Collisions

Momentum, p = mv, is conserved in every collision, without exception, provided no external force acts. That is a direct consequence of Newton's third law: whatever impulse A exerts on B, B exerts an equal and opposite impulse on A, so the total is unchanged. Momentum is a vector, so direction matters and velocities in opposite senses carry opposite signs.

Kinetic energy is a different matter. It is conserved only in a perfectly elastic collision, where the bodies separate at the same relative speed at which they approached. In every real collision some energy goes into permanent deformation, heat, sound and vibration, so kinetic energy is lost even though momentum is not. The extreme case is a perfectly inelastic collision, where the bodies stick together and move off with a common velocity; that condition loses the most kinetic energy of any collision with the same momentum.

The coefficient of restitution e measures where a collision sits between those extremes: e = 1 is perfectly elastic, e = 0 is perfectly inelastic, and real materials fall somewhere between. This is exactly the reasoning behind crumple zones and energy-absorbing structures, which are designed to be inelastic on purpose so that the energy is dissipated by the structure rather than transmitted to the occupants.

• Try this: Compare elastic and perfectly inelastic with the same masses and speeds. Which quantity changes between them, and which does not?
• Try this: Set the masses equal and make B stationary, then run an elastic collision. Why does A stop dead and B move off at A's original speed?
• Try this: Give the two bodies equal and opposite momentum, then make the collision inelastic. Why do both end up stationary, and where did the kinetic energy go?