In what follows, we assume that there are no external forces acting
on the colliding bodies. 

The velocities before the collision are represented by u and those after by
v. 



The definition of an elastic collision is
that the total kinetic energy is the same before and after the
collision. 

We say that KE is conserved during the interaction. 

This can be expressed as 



Let's forget about those halves, so, we have 



and, rearranging this equation gives 



the "difference of two squares" in each of the brackets above can be
replaced as below 



equation 1 

As we have seen (here), in any collision, during which
no external forces act, momentum is conserved. 

In other words, the total momentum is the same
before and after the interaction. 

This is expressed in the following equation 



which, when rearranged gives 



equation 2 

Now we simply divide equation 1 by equation 2 to find 



so 



in which we
see, on the left, the velocity of A relative to B after the collision and,
on the right, the velocity of B relative to A before the collision (see the
relative speed page if necessary). 

In other words, on the right we see the relative velocity of approach
of the two colliding bodies and on the left, their relative
velocity of separation. 



These two relative velocities have therefore been shown to be equal
during an elastic collision. 
