Question

prove that in an elastic collision in one dimension the relative velocity of aaproach before imapct is equal to relative velocity of separation after impact

Answer 1

In an elastic collision, total linear momentum is conserved.

Therefore ,m1u1 + m2u2 = m1v1 + m2v2......(1)

Similarly, kinetic energy is also conserved.

Answer 2

Let u1, u2 , v1 , v2 be the velocities (positive if in +ve x direction).  m1 and m2 are masses.  There is a collision, so  v1 ≠ u1.

We apply the law of conservation of linear momentum.

        m1 u1 + m2 u2 = m1 v1 + m2 v2
   => m1 (u1 - v1) = m2 (v2 - u2)    ---- (1) 

Law of conservation of KE  (as the collision is elastic):
        1/2 m1 u1² + 1/2 m2 u2²  =  1/2 m1 v1²  + 1/2 m2 v2²
    =>  m1 (u1² - v1²)  = m2 (v2² - u2²)   ---- (2)

Divide (2) by (1): 
          u1 + v1  = v2 + u2
          u1 - u2   = v2 - v1

    u2 < u1  , otherwise there is no collision.
So the Relative velocity of m1 approaching m2 is equal to velocity of m2 more than that of m1 (separation of m2 from m1).