Question

A ball falls vertically onto a floor with momentum \[p\] and then bounces repeatedly. If coefficient of restitution is \[e\], then the total momentum imparted by the ball to the floor is-

(A) \[p(1 + e)\]
(B) \[\dfrac{p}{{1 - e}}\]
(C) \[p(\dfrac{{1 - e}}{{1 + e}})\]
(D) \[p(\dfrac{{1 + e}}{{1 - e}})\]

Answer 1

Hint It is given that a ball falls vertically onto the ground with a momentum \[p\]and coefficient of restitution\[e\]. Understand that when a particle undergoes collision with a stationary object like wall or floor and bounces with a coefficient of restitution, the speed after bounce will be e times the speed before bounce.

Complete Step by Step Solution
It is given that a ball undergoes a free fall motion vertically and hits the ground with a momentum \[p\]and bounces repeatedly. Now, velocity of the object can be calculated by dividing the momentum of the object by its mass.
 Coefficient of restitution is defined as the ratio of velocity post the impact on the surface to the velocity of the object before the impact. This applies for a one dimensional collision and hence from the basic understanding we can conclude that our scenario is a one dimensional collision.
 Now, When a particle undergoes one dimensional collision with a stationary object, it bounces constantly until the velocity becomes zero. Now, it comes with a coefficient of restitution and thus the speed of the body after impact is the product of coefficient of restitution and speed before impact.
 Now, change in momentum for first bounce is given as
\[ \Rightarrow \Delta {p_1} = ep - ( - p)\]
\[ \Rightarrow \Delta {p_1} = p(1 + e)\]
For the second bounce, ep will be the initial momentum , whereas e(ep) is the final momentum
\[ \Rightarrow \Delta {p_2} = {e^2}p + ep\]
So for the nth bounce, we get
\[ \Rightarrow \Delta {p_n} = {e^n}p + {e^{n - 1}}p\]
Now summation of all the change in momentum is given as
\[ \Rightarrow \Delta {p_s} = p(1 + e)[1 + e + {e^2} + ....... + {e^n}]\]
Now the general term for the given series is \[\dfrac{1}{{1 - e}}\],substituting this in the given equation we get
\[ \Rightarrow \Delta {p_s} = p(\dfrac{{1 + e}}{{1 - e}})\]

Thus, option (d) is the right answer for the given question.

Note In physics, restitution is defined as the kinetic energy lost when a moving object collides with another moving object or another stationary object. Its coefficient is calculated as the ratio between the final and initial velocity before and after collision.