Question

Prove that in an elastic collision in one dimension the relative velocity of approach before impact is equal to the relative velocity of separation after impact.​

Answer 1

Question :-

Prove that in an elastic collision in one dimension the relative velocity of approach before impact is equal to the relative velocity of separation after impact.​

Answer :-

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).

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