Question

A ball falls vertically on to a floor, with momentum \(p\), and then bounces repeatedly, the coefficient of restitution is \(e\). The total momentum imparted by the ball to the floor is:

• A. \( p(1 + e) \)
• B. \( \frac{p}{1-e} \)
• C. \( p\left(\frac{1+e}{1-e}\right) \)
• D. \( p\left(1 - \frac{1}{e}\right) \)

Hint:

Repeated collisions form a geometric progression when coefficient of restitution is constant.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The momentum imparted to the floor in a single collision is the change in momentum of the ball.
The ball bounces multiple times, and with each bounce, its momentum decreases due to the coefficient of restitution \( e \).
: Key Formula or Approach:
1. Momentum change for first impact: \( \Delta p_1 = p - (-ep) = p(1+e) \).
2. Successive momentum changes form a geometric series.
Step 2: Detailed Explanation:
- 1st Impact: Initial momentum \( p \), Final momentum \( -ep \).
Imparted momentum \( = p - (-ep) = p(1+e) \).
- 2nd Impact: Ball hits with momentum \( ep \), rebounds with \( -e^2p \).
Imparted momentum \( = ep - (-e^2p) = ep(1+e) \).
- 3rd Impact: Imparted momentum \( = e^2p(1+e) \).
Total momentum imparted \( P = p(1+e) + ep(1+e) + e^2p(1+e) + \dots \)
This is an infinite geometric series with first term \( a = p(1+e) \) and common ratio \( r = e \).
\[ P = \frac{a}{1 - r} \]
\[ P = \frac{p(1+e)}{1 - e} \]
\[ P = p \left( \frac{1+e}{1-e} \right) \]
Step 3: Final Answer:
The total momentum imparted is \( p \left( \frac{1+e}{1-e} \right) \).