A ball is let fall from a height `h_(0)`. There are `n` collisions with the earth. If the velocity of rebound after `n` collision is `v_(n)` and the ball rises to a height `h_(n)` then coefficient of restitution `e` is given by

Options

A. `e^n = sqrt((h_n)/(h_0))`
B. `e^n = sqrt((h_0)/(h_n))`
C. `n e = sqrt((h_n)/(h_0))`
D. `sqrt(n e) = sqrt((h_n)/(h_0))`

Explanation

To solve the problem, we need to derive the expression for the coefficient of restitution \( e \) based on the given information about the ball's motion and its rebounds after \( n \) collisions.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- The ball is dropped from an initial height \( h_0 \).
- The velocity of the ball just before it hits the ground can be calculated using the formula for free fall:
\[ v_0 = \sqrt{2gh_0} \]
where \( g \) is the acceleration due to gravity.

**Hint**: Remember that the velocity just before impact can be derived from the equations of motion for free fall.

2. **Velocity After \( n \) Collisions**:
- After \( n \) collisions, the velocity of the ball when it rebounds is denoted as \( v_n \).
- The relationship between the velocity after \( n \) collisions and the initial velocity can be expressed using the coefficient of restitution \( e \):
\[ v_n = e^n v_0 \]

**Hint**: The coefficient of restitution relates the velocities before and after a collision.

3. **Height Reached After \( n \) Collisions**:
- The height \( h_n \) to which the ball rebounds after \( n \) collisions can be expressed in terms of the rebound velocity \( v_n \):
\[ h_n = \frac{v_n^2}{2g} \]

**Hint**: Use the kinetic energy to potential energy conversion to find the height reached after the rebound.

4. **Substituting for \( v_n \)**:
- Substitute \( v_n = e^n v_0 \) into the equation for \( h_n \):
\[ h_n = \frac{(e^n v_0)^2}{2g} = \frac{e^{2n} v_0^2}{2g} \]

5. **Substituting for \( v_0 \)**:
- Now substitute \( v_0 = \sqrt{2gh_0} \) into the equation for \( h_n \):
\[ h_n = \frac{e^{2n} (2gh_0)}{2g} = e^{2n} h_0 \]

6. **Finding the Coefficient of Restitution \( e \)**:
- Rearranging the equation gives:
\[ e^{2n} = \frac{h_n}{h_0} \]
- Taking the square root of both sides:
\[ e^n = \sqrt{\frac{h_n}{h_0}} \]
- Therefore, the coefficient of restitution \( e \) can be expressed as:
\[ e = \left(\frac{h_n}{h_0}\right)^{1/(2n)} \]

### Final Answer:
The coefficient of restitution \( e \) is given by:
\[ e = \left(\frac{h_n}{h_0}\right)^{1/(2n)} \]