Question

In the collision between two bodies a heavier and the other lighter. Write the relation between the changes in momentum of two bodies

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        --- (3)

       V - v   =  - (U  -  u)      --- (4)
 Relative velocity after collision = relative velocity before collision.

For finding the change in momentum of the bodies:  we have to find V and v in terms of u and U.

     V = v + u - U  --- (5)

Substitute this in equation (1)....
      M (v + u - 2 U) = m u - m v
      v =  (m u - M u + 2 M U) / (M + m)
       v  = u  + 2 M (U - u) / (M+m)        ---  (6)

by (5)  then        V  =   2 u  -  2 [ M u + m U ] / (M + m)
                         V  =   [2 m u - m U + M U ] / (m + M)         --- (7)

The change in momentum of each body:

   M (V - U)  =  2 M m (u - U) / (M + m)
   m (v - u)  =  2 m M (U - u) / (m + M)  
        
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some notes.... if the masses are equal and then the initial velocities are equal and opposite...then:  ie.,  M = m and  U = -u  then:

   V - U =   u - U  => V = -  U
   v - u =   U - u  =  2 U  =>  v = - u