## Elastic Collisions

Momentum is always conserved. Always.Energy is also

alwaysconserved -- if we look closely enough. Often, tho', large-scale, macroscopic Kinetic Energy is turned intoheat,the random motion of molecules. This is still energy -- on a microscopic scale.A

totally elastic collisionis one in which theKinetic Energy is conserved.KE _{i}= KE_{f}KE

_{i}= (^{1}/_{2}) m_{1}v_{1i}^{2}+ (^{1}/_{2}) m_{2}v_{2i}^{2}== (

^{1}/_{2}) m_{1}v_{1f}^{2}+ (^{1}/_{2}) m_{2}v_{2f}^{2}= KE_{f}m

_{1}v_{1i}^{2}+ m_{2}v_{2i}^{2}= m_{1}v_{1f}^{2}+ m_{2}v_{2f}^{2}Notice that this is a single,

scalarequation. Kinetic Energy -- like any kind of energy -- is ascalar.From

momentum conservation, sincemomentum is a vector, we havetwo equations.For a totallyelastic collision, we have thisadditional equation, for a total ofthree equations.This means we can solve forthree unknowns.Given the initial conditions, and that we have anelastic collision,we can solve forthree unknownsin the final system.

Three unknowns?For two objects, there arefour unknownsin the final system -- the two final velocities and their directions. That means we must determine one of the unknowns in some other way or, givenoneof the final velocities or directions we canthendetermine the other three unknowns.

Inelastic CollisionsA Special CaseReturn to ToC, Work and Energy(c) 2002, Doug Davis; all rights reserved