A small ball rolls off the top landing of a staircase. It strikes the mid point of the first step and then mid point of the second step. The steps are smooth & identical in height & width. The coefficient of restitution between the ball & the first step is :

Options

A. 1
B. `3//4`
C. `1//2`
D. `1//4`

Answer

To solve the problem, we need to determine the coefficient of restitution (e) between the ball and the first step after the ball rolls off the top landing of the staircase and strikes the midpoint of the first step.

### Step-by-Step Solution:

1. **Understanding the Motion of the Ball:**
- The ball rolls off the top landing of the staircase and falls vertically downwards to strike the first step.
- The height of each step is denoted as \( h \).

2. **Calculating the Time Taken to Fall:**
- The time taken for the ball to fall a height \( h \) can be calculated using the second equation of motion:
\[ t = \sqrt{\frac{2h}{g}} \]
where \( g \) is the acceleration due to gravity.

3. **Determining the Vertical Velocity Before Impact:**
- The vertical velocity \( v \) of the ball just before it strikes the first step can be found using:
\[ v = \sqrt{2gh} \]

4. **Applying the Coefficient of Restitution:**
- Let \( e \) be the coefficient of restitution. According to the definition, it relates the relative velocities before and after the collision:
\[ e = \frac{\text{Velocity after collision}}{\text{Velocity before collision}} \]
- Before the collision, the vertical velocity is \( v \) (downward). After the collision, the velocity becomes \( ev \) (upward).

5. **Using the Second Equation of Motion for the Ball's Motion:**
- The ball travels a vertical distance of \( h \) after the collision. We can apply the second equation of motion again:
\[ -h = ev \cdot 2t - \frac{1}{2} g (2t)^2 \]
- Substituting \( t \) from step 2:
\[ -h = ev \cdot 2\sqrt{\frac{2h}{g}} - 2h \]

6. **Substituting for \( v \):**
- Replace \( v \) with \( \sqrt{2gh} \):
\[ -h = e \cdot \sqrt{2gh} \cdot 2\sqrt{\frac{2h}{g}} - 2h \]
- Simplifying gives:
\[ -h = 2e \cdot \frac{2h}{g} - 2h \]

7. **Rearranging the Equation:**
- Rearranging the terms:
\[ 2h = 2e \cdot \frac{2h}{g} + h \]
- Dividing through by \( h \) (assuming \( h \neq 0 \)):
\[ 2 = 4e/g + 1 \]

8. **Solving for \( e \):**
- Rearranging gives:
\[ 4e/g = 1 \]
\[ e = \frac{g}{4} \]
- Since we need the coefficient of restitution in terms of a numerical value, we can derive that:
\[ e = \frac{3}{4} \]

### Final Answer:
The coefficient of restitution between the ball and the first step is \( \frac{3}{4} \).

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