Question

A ball is dropped from a height ℎ to the ground. If the coefficient of restitution is 𝑒, the time required for the ball to stop bouncing is proportional to:

• A. \(\frac{2+e}{1-e}\)
• B. \(\frac{1+e}{1-e}\)
• C. \(\frac{1-e}{1+e}\)
• D. \(\frac{2-e}{1+e}\)

Answer 1

The Correct Option is B

The coefficient of restitution, \( e \), is the ratio of velocities after and before the collision. For a ball dropped from a height \( h \), the time between successive bounces is proportional to \( \sqrt{h} \).

Key Observations:

• After each bounce, the ball’s height reduces as \( e^2 \) (due to the loss of energy).
• The total time \( T \) taken by the ball to stop bouncing can be calculated as the sum of a geometric series.

Step 1: Geometric Series for Total Time

The total time \( T \) is proportional to:

\[ T \propto \sum_{n=0}^\infty e^n = \frac{1}{1 - e}, \quad \text{for } 0 < e < 1 \]

Step 2: Adding Initial Drop and Bounce Heights

When adding the effect of the initial drop time and the bounce heights, the proportionality constant includes \( 1 + e \) in the numerator and \( 1 - e \) in the denominator:

\[ T \propto \frac{1 + e}{1 - e} \]

Conclusion:

Thus, the total time taken by the ball to stop bouncing is proportional to \( \frac{1 + e}{1 - e} \).