Question

A big ball of mass , moving with a velocity $ u $ strikes a small ball of mass $ m $ , which is at rest. Finally, the small ball attains velocity $ u $ and big ball $ v $ . Then what is the value of $ v $
(A) $ \dfrac{{{\rm M} - m}}{M}u $
(B) $ \dfrac{m}{{M + m}}u $
(C) $ \dfrac{{2m}}{{M + m}}u $
(D) $ \dfrac{{2M}}{{M + m}}u $

Answer 1

Hint: To solve this question, we need to calculate the total initial and the total final momenta of the system. Since no external force acts on the system, we can equate these to get the final value of the velocity.

Complete Step-by-Step solution:
On the system of the big ball and the small ball, no external force acts on the system. We know that the external force acting on a system is equal to the rate of change of momentum of the system. Since the external force is equal to zero, so the total momentum of the system will remain conserved. So the initial momentum of the system must be equal to the final momentum.
Initially, the big ball of mass $ M $ is moving with a velocity of $ u $ , while the small ball of mass $ m $ is at rest. So the total initial momentum of the system becomes
 $ {p_i} = Mu + m\left( 0 \right) $
 $ \Rightarrow {p_i} = Mu $ ......................(1)
Now, according to the question, after the collision the small ball moves with a velocity of $ u $ and the big ball moves with a velocity of $ v $ . So the total final momentum of the system becomes
 $ {p_f} = Mv + mu $ ......................(2)
Since the initial momentum of the system is equal to the final momentum, so equating (1) and (2) we get
 $ Mu = Mv + mu $
 $ \Rightarrow Mv = \left( {M - m} \right)u $
Dividing by $ M $ we finally get
 $ v = \dfrac{{{\rm M} - m}}{M}u $
Thus, the value of the velocity $ v $ is equal to $ \dfrac{{{\rm M} - m}}{M}u $ .
Hence, the correct answer is option A.

Note:
We do not have to worry about the direction of the velocities of the two balls. Since the bigger ball strikes the small ball at rest, it is clear that they both will move in the same direction as that of the initial velocity of the bigger ball.