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Relative Velocities of Bodies Before and After an Elastic Collision
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, re-arranging 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 re-arranged 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.  
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