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

Concept: Momentum change in each bounce: \[ \Delta p = p(1+e) \] After each bounce, momentum reduces by factor \(e\), forming a GP.

Step 1: Momentum series.
\[ p(1+e) + pe(1+e) + pe^2(1+e) + \cdots \]

Step 2: Sum of GP.
\[ \text{Total} = (1+e)p \left(1 + e + e^2 + \cdots \right) \] \[ = (1+e)p \cdot \frac{1}{1-e} \] \[ = p\left(\frac{1+e}{1-e}\right) \]