Elastic collision
An elastic collision is one in which the total kinetic energy of the system after the collision equals the total kinetic energy before it. Momentum is conserved too, but momentum is conserved in every collision, so kinetic energy is the test that identifies this case.
AP defines this case by energy, not by whether things bounce.
EK 4.4.A.1 is the whole definition: an elastic collision between objects is one in which the initial kinetic energy of the system is equal to the final kinetic energy of the system. Nothing is derived there; that is what the words mean.
Two things follow that students routinely get backwards.
- Conserved momentum does not make a collision elastic. Momentum is conserved in every collision, elastic or not, as long as the system is isolated. Kinetic energy is the quantity that tells the cases apart.
- Individual objects can still gain or lose kinetic energy. EK 4.4.A.2 notes that the final kinetic energies of each object may differ from their initial ones. Only the system total has to match.
Perfectly elastic collisions are an idealization. Steel balls, carts with magnetic bumpers and colliding gas molecules come close; two cars do not.
In one dimension an elastic collision has a second signature worth carrying: the relative speed of separation equals the relative speed of approach. It falls out of the two conservation conditions and is often faster than the quadratic.
Solving one means running conservation of momentum and conservation of kinetic energy together, which is a procedure rather than a definition. What is conserved in a collision works it through.