Question

A ball at rest falls vertically on ground from a height of 5m. The coefficient of restitution is 0.4. The maximum height of the ball after the first rebound is

• A. 0.8 m
• B. 0.6 m
• C. 0.2 m
• D. 0.4 m

Hint:

In problems involving the coefficient of restitution, always use the relationship between the velocities before and after the collision to calculate the new maximum height.

Answer 1

The Correct Option is A

Step 1: Understanding the coefficient of restitution.
The coefficient of restitution (\( e \)) is the ratio of the speed of separation to the speed of approach. When the ball hits the ground, the velocity of the ball just before the impact is given by \( v = \sqrt{2gh} \), where \( h = 5 \, \text{m} \) is the height and \( g = 10 \, \text{m/s}^2 \) is the acceleration due to gravity. After the collision, the velocity is reduced by the coefficient of restitution. Therefore, the rebound height will be related to the square of \( e \).
Step 2: Calculating the rebound height.
The velocity just before the rebound is: \[ v = \sqrt{2gh} = \sqrt{2 \times 10 \times 5} = \sqrt{100} = 10 \, \text{m/s} \] The rebound velocity is reduced by the factor \( e = 0.4 \), so the rebound velocity is: \[ v_{\text{rebound}} = e \times v = 0.4 \times 10 = 4 \, \text{m/s} \] Now, the maximum height after the rebound is given by: \[ h_{\text{rebound}} = \frac{v_{\text{rebound}}^2}{2g} = \frac{4^2}{2 \times 10} = \frac{16}{20} = 0.8 \, \text{m} \] Step 3: Conclusion.
Thus, the maximum height of the ball after the first rebound is 0.8 m, corresponding to option (A).