Question

A lead ball and a rubber ball having the same mass, strike normally on a smooth vertical wall with the same velocity. The lead ball falls down after striking but the rubber ball bounces back. Then:
A. Momentum of lead ball is greater than that of rubber ball
B. Momentum of rubber ball is greater than that of lead ball
C. The rubber ball suffers a greater change in momentum as compared to the lead ball
D. Both the balls suffer an equal change in momentum

Answer 1

- Hint: We are given that both the lead and rubber ball are striking the wall with the same velocity, and one of them falls straight down while the other bounces right back. In other words, the only difference between them is their final velocities, and their final moments. So, calculate the initial and final momenta of both the lead and rubber balls to see if we can rule out the first two options, and subsequently calculate their change in momenta, which we can use to check for the last two options.

Formula Used:
Change in momentum $\Delta p = p_i -p_f$ where, $p_i$ is the initial momentum and $p_f$ is the final momentum of a body.

Complete step-by-step solution
Let us deconstruct the question for an easier understanding. So, let us try and think of this problem in the context of momenta.
Let the mass of the lead ball and the rubber ball be $m_{lead} = m_{rubber} = m$
They both strike the wall with the same velocities $u_{lead} = u_{rubber} = u$
Their initial momenta will be $p_{i_{lead}} = p_{i_{rubber}} = mu$ each.
What happened now is different for each.
The lead ball is said to fall down upon impact $\Rightarrow$ its final velocity $v_{lead} = 0$
The final momentum of the
Whereas, the rubber ball bounces back with the same impact velocity $\Rightarrow v_{rubber} = u_{rubber} = u$ but in the opposite direction, $\Rightarrow v_{rubber} = -u$
Their final momenta will be $p_{f_{lead}} = 0 $ and $p_{f_{rubber}} =-mu$ respectively.
The negative sign indicates that since it bounced back the momentum was directed opposite to the incident motion.
Change in momentum is given as the difference between the initial and final momentum.
Therefore, the change in momentum for lead ball $\Delta p_{lead} = mu – 0 = mu $
And the change in momentum for the rubber ball $\Delta p_{rubber} = mu –(-mu) = 2mu $
Thus, the correct choice would be C. The rubber ball suffers a greater change in momentum as compared to the lead ball.

Note: Do not forget to account for the signs to indicate change in direction, since momentum is ultimately a vector quantity whose magnitude and direction are co-dependent.
A consequence of change in momenta is the impulse produced. In a collision, an object experiences a force F for a given amount of time that results in the object undergoing a change in velocity. The impulse experienced by an object is always equal to the change in its momentum. Quantitatively, we can summarize this as:
Impulse $I = F \times t \Rightarrow m\Delta v = F \times t$.
For example, when you hit a ball with a bat, you apply a force for a short period of time which causes a sufficient change in momentum in the ball via the force applied by the bat.
This is called the impulse-momentum change theorem.