We want the speeds of the two balls after the collision. There are two things to find, and .
What we have is one conservation law, . Two unknowns and one equation do not settle anything.
Indeed there are endlessly many pairs with the same sum. The figure moves the two velocities while holding the total fixed. Every one of them satisfies the conservation law, and only one of them actually happens.
The missing equation ought to be the one that fixes how much the collision bounces. The quantity that measures this is the coefficient of restitution.
The coefficient of restitution is the speed of separation divided by the speed of approach. Written with the signs included, , it can be used directly as an equation.
Only relative velocities appear in it. Which ball you watch from makes no difference, because all it measures is the ratio of the separating speed to the approaching speed.
If the balls separate as fast as they approached. This is a perfectly elastic collision. If they do not separate at all: they collide and move on together, a perfectly inelastic collision.
is fixed by the materials of the two bodies and lies between and . This is the equation we were missing.
Now there are two equations, the conservation law and . All that is left is to solve them together.
The result is , and is the same expression with the subscripts exchanged.
This is not a formula to memorise. It is simply what comes out of solving the pair. What matters is that the conservation law plus one more equation is always enough.
As a check, put in and . Out come and . Two balls of equal mass in a head-on collision exchange velocities outright. That is the motion you see on a billiard table.
Momentum was conserved. What about the kinetic energy?
Working it out, the loss in the collision is , where is called the reduced mass.
If then is and nothing is lost. The smaller is the more is lost, and the loss is greatest at . The figure varies : the momentum bar holds still and only the energy bar shrinks.
What is lost has not vanished. It has gone into heat, into sound, into a dent. Momentum is conserved and kinetic energy is not. In one and the same collision, what holds depends on which quantity you ask about.