A rubber ball falls from a height h and rebounds to a height h/2. A rubber ball of double the mass falling from the same height h rebounds to a height

Options

A. h
B. h/2
C. 3h/4
D. 2h

Answer

To solve the problem, we need to analyze the behavior of the rubber balls when they fall and rebound.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a rubber ball of mass \( m \) that falls from a height \( h \) and rebounds to a height of \( \frac{h}{2} \).
- We are asked to find the rebound height of another rubber ball with double the mass \( 2m \) when it falls from the same height \( h \).

2. **Key Concept - Coefficient of Restitution**:
- The coefficient of restitution (e) is a measure of how much kinetic energy remains after a collision. It is defined as the ratio of the velocity after the collision to the velocity before the collision.
- The rebound height of a ball is directly related to the coefficient of restitution and the height from which it falls.

3. **Rebound Height Calculation**:
- For the first ball (mass \( m \)), it falls from height \( h \) and rebounds to height \( \frac{h}{2} \). This means that the coefficient of restitution \( e \) can be calculated as follows:
\[ e = \sqrt{\frac{\text{Rebound Height}}{\text{Initial Height}}} = \sqrt{\frac{\frac{h}{2}}{h}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

4. **Applying the Same Concept to the Second Ball**:
- For the second ball (mass \( 2m \)), it also falls from the same height \( h \). Since the coefficient of restitution is a property of the material and does not depend on the mass of the ball, the rebound height will be the same as for the first ball.
- Therefore, the rebound height for the second ball will also be:
\[ \text{Rebound Height} = e^2 \cdot h = \frac{1}{2} \cdot h = \frac{h}{2} \]

5. **Conclusion**:
- The rubber ball of double the mass \( 2m \) will rebound to the same height \( \frac{h}{2} \) as the first ball of mass \( m \).

### Final Answer:
The rubber ball of double the mass will rebound to a height of \( \frac{h}{2} \).

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