Question

Show that coefficient of restitution for one dimensional elastic collision is equal to one. 

Answer 1

Let there be two bodies of mass M and m, let the body M be moving with velocity V and the body m be at rest. 
Now when the body (M) will collide with the body (m) the KE would be conserved hence the body M will come to rest and body m will move with velocity V.
Putting the value of variables in the equation below;
e=$\frac{V-v}{U-u}$
 where V is velocity of separation of body M ,
            v is velocity of separation of body m,
            U is velocity of approach of body M,
            u is velocity of approach of body m.
⇒ e= $\frac{0-v}{U-0}$
since, v=U
⇒ e=1

Hence Proved

Answer 2

The coefficient of restitution e is a measure of the elasticity in a one-dimensional collision.* Its origin arises from the fact that during a perfectly elastic collision of two bodies, the velocity of approach is always equal to the velocity of separation, so that e = 1 in elastic collisions. In a perfectly inelastic collision the velocity of separation is zero, so that e = 0 in a totally inelastic collisions. 
Type 
Kinetic Energy 
Restitution 
Perfectly Elastic 
Conserved 
e = 1 
Partially Elastic 
Not Conserved 
0 < e < 1 
Perfectly Inelastic 
Maximum Possible Loss 
e = 0 
Hyperelastic 
Energy Gained 
e > 1