Question

show that for a perfectly elastic collision co- efficient restitution is 1 . prove that.

Answer 1

   Let two bodies of masses  M and m travel with velocities  U and u before collision.  Let them collide elastically. so energy is conserved.  Let the potential energy remain same for each of them. Let them travel with  V and v after collision.

  Conservation of linear momentum:
       M U + m u = M V + m v
       M (V - U)  = m (u - v)        --- (1)

change in the momentum of one object is negative of the change in momentum of the other object.

  Conservation of energy :
      1/2 M U² + 1/2 m u² = 1/2 M V² + 1/2 m v²
      M(V² - U²) = m (u² - v²)          --- (2)

  divide  (2) by (1) to get:
       V + U  =  u + v
       V - v   =  - (U  -  u)
   => Relative velocity after collision = relative velocity before collision.

e = coefficient of restitution
  = relative velocity after collision / relative velocity before collision
  = 1