Momentum is always conserved. Always.
Energy is also always conserved -- if we look closely enough. Often, tho', large-scale, macroscopic Kinetic Energy is turned into heat, the random motion of molecules. This is still energy -- on a microscopic scale.
A totally elastic collision is one in which the Kinetic Energy is conserved.
KEi = KEf
KEi = (1/2) m1 v1i2 + (1/2) m2 v2i2 =
= (1/2) m1 v1f2 + (1/2) m2 v2f2 = KEf
m1 v1i2 + m2 v2i2 = m1 v1f2 + m2 v2f2
Notice that this is a single, scalar equation. Kinetic Energy -- like any kind of energy -- is a scalar.
From momentum conservation, since momentum is a vector, we have two equations. For a totally elastic collision, we have this additional equation, for a total of three equations. This means we can solve for three unknowns. Given the initial conditions, and that we have an elastic collision,we can solve for three unknowns in the final system.
Three unknowns? For two objects, there are four unknowns in the final system -- the two final velocities and their directions. That means we must determine one of the unknowns in some other way or, given one of the final velocities or directions we can then determine the other three unknowns.
Inelastic Collisions A Special Case Return to ToC, Work and Energy (c) 2002, Doug Davis; all rights reserved