A ball falls from a height h upon a fixed horizontal plane, e is the coefficient of restitution, the whole distance described by the ball before it comes to rest is

Options

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

Answer

To solve the problem of a ball falling from a height \( h \) onto a fixed horizontal plane with a coefficient of restitution \( e \), we need to analyze the motion of the ball as it bounces and comes to rest.

### Step-by-Step Solution:

1. **Initial Drop**:
The ball is dropped from a height \( h \). When it falls, it will hit the ground with a certain velocity which can be calculated using the equation of motion:
\[ v = \sqrt{2gh} \]
where \( g \) is the acceleration due to gravity.

**Hint**: Use the equation of motion to find the velocity just before impact.

2. **First Impact**:
Upon hitting the ground, the ball will compress and then rebound. The height to which it rebounds can be determined using the coefficient of restitution \( e \):
\[ h_1 = e^2 h \]
This is because the velocity after the bounce will be \( e \cdot v \), and the new height can be found using the same energy conservation principles.

**Hint**: Remember that the coefficient of restitution relates the velocities before and after the impact.

3. **Subsequent Bounces**:
After the first bounce, the ball will continue to bounce, and each subsequent height will be reduced by a factor of \( e^2 \):
\[ h_2 = e^2 h_1 = e^4 h \]
\[ h_3 = e^2 h_2 = e^6 h \]
Continuing this pattern, the height after \( n \) bounces can be expressed as:
\[ h_n = e^{2n} h \]

**Hint**: Recognize the pattern in the heights after each bounce.

4. **Total Distance Traveled**:
The total distance \( D \) traveled by the ball before coming to rest is the sum of the distances fallen and risen:
\[ D = h + 2(h_1 + h_2 + h_3 + \ldots) \]
The series \( h_1 + h_2 + h_3 + \ldots \) is a geometric series with the first term \( h_1 = e^2 h \) and the common ratio \( r = e^2 \):
\[ \text{Sum} = \frac{h_1}{1 - r} = \frac{e^2 h}{1 - e^2} \]

**Hint**: Use the formula for the sum of an infinite geometric series.

5. **Final Calculation**:
Substituting the sum back into the total distance formula:
\[ D = h + 2 \left( \frac{e^2 h}{1 - e^2} \right) \]
Simplifying this gives:
\[ D = h + \frac{2e^2 h}{1 - e^2} = h \left( 1 + \frac{2e^2}{1 - e^2} \right) \]

**Hint**: Factor out \( h \) to simplify the expression.

### Conclusion:
The total distance described by the ball before it comes to rest is:
\[ D = h \left( 1 + \frac{2e^2}{1 - e^2} \right) \]