A ball is bouncing down a set of stairs. The coefficient of restitution is e. The height of each step is d and the ball bounces one step at each bounce. After each bounce the ball rebounds to a height h above the next lower step. Neglect width of each step in comparison to h and assume the impacts to be effectively head on. Which of the following relative is correct ?

Options

A. `(h)/(d) = 1 - e^(2)`
B. `(h)/(d) = 1 - e`
C. `(h)/(d) = (1)/(1 - e^(2))`
D. `(h)/(d) = (1)/(1 - e)`

Explanation

To solve the problem of a ball bouncing down a set of stairs, we need to analyze the situation step by step, using the given information about the coefficient of restitution, the height of the steps, and the rebound height of the ball.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- We have a ball bouncing down a staircase.
- The coefficient of restitution is denoted as \( e \).
- Each step has a height \( d \).
- After each bounce, the ball rebounds to a height \( h \) above the next lower step.

2. **Analyzing the Collision:**
- When the ball strikes the step, it has a velocity \( V \) (velocity of approach).
- After the bounce, the velocity of the ball is \( V_s = eV \) (velocity of separation).

3. **Relating Heights and Velocities:**
- The ball rebounds to a height \( h \) above the next step, which means the total height it reaches after the bounce is \( h \).
- The distance the ball rises after the bounce is \( h - d \) (since it starts from the height of the step).

4. **Using Kinematic Equations:**
- The height \( h - d \) can be related to the velocity of separation using the kinematic equation:
\[ h - d = \frac{V_s^2}{2g} \]
- Substituting \( V_s = eV \):
\[ h - d = \frac{(eV)^2}{2g} = \frac{e^2 V^2}{2g} \]

5. **Finding the Initial Drop Height:**
- When the ball falls from height \( h \) to the step, we can use the same kinematic equation:
\[ h = \frac{V^2}{2g} \]

6. **Relating the Two Equations:**
- From the two equations we derived:
- \( h - d = \frac{e^2 V^2}{2g} \)
- \( h = \frac{V^2}{2g} \)

- We can substitute \( V^2 \) from the second equation into the first:
\[ h - d = e^2 \left(\frac{V^2}{2g}\right) \]
\[ h - d = e^2 h \]

7. **Rearranging the Equation:**
- Rearranging gives:
\[ h - e^2 h = d \]
\[ h(1 - e^2) = d \]

8. **Final Relation:**
- Dividing both sides by \( d \):
\[ \frac{h}{d} = \frac{1}{1 - e^2} \]

### Conclusion:
The correct relationship derived from the analysis is:
\[ \frac{h}{d} = \frac{1}{1 - e^2} \]