A ball is projected in a direction inclined to the vertical and bounces on a smooth horizontal plane. The range of one rebound is `R`. If the coefficient of restitution is `e`, then range of the next rebound is

Options

A. `R'=eR`
B. `R'=e^(2)R`
C. `R'=R/e`
D. R'=R`

Explanation

To solve the problem, we need to determine the range of the next rebound of a ball that has already bounced once, given the initial range \( R \) and the coefficient of restitution \( e \).

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- A ball is projected at an angle \( \theta \) to the horizontal.
- The range of the first rebound is given as \( R \).
- The coefficient of restitution \( e \) affects the vertical component of the velocity after the bounce.

2. **Range Formula**:
- The range \( R \) of a projectile is given by the formula:
\[ R = \frac{2u^2 \sin \theta \cos \theta}{g} \]
- Here, \( u \) is the initial velocity, \( g \) is the acceleration due to gravity, and \( \theta \) is the angle of projection.

3. **Components of Velocity**:
- The horizontal component of the initial velocity is:
\[ u_x = u \cos \theta \]
- The vertical component of the initial velocity is:
\[ u_y = u \sin \theta \]

4. **After the First Bounce**:
- Upon bouncing, the horizontal component of the velocity remains unchanged:
\[ u_x' = u \cos \theta \]
- The vertical component of the velocity after the bounce is reduced by the coefficient of restitution:
\[ u_y' = e \cdot (u \sin \theta) \]

5. **Calculating the Range of the Next Rebound**:
- The range \( R' \) for the next rebound can be calculated using the new vertical component:
\[ R' = \frac{2 (u_x')^2 \sin \theta \cos \theta}{g} \]
- Substituting \( u_x' \) and \( u_y' \):
\[ R' = \frac{2 (u \cos \theta)^2 (e \cdot (u \sin \theta)) \cos \theta}{g} \]
- This simplifies to:
\[ R' = \frac{2 (u^2 \cos^2 \theta) (e \cdot (u \sin \theta))}{g} \]
- Rearranging gives:
\[ R' = e \cdot R \]

6. **Final Result**:
- Therefore, the range of the next rebound is:
\[ R' = eR \]

### Summary:
The range of the next rebound is directly proportional to the range of the first rebound multiplied by the coefficient of restitution \( e \).