A steel ball strikes a fixed smooth steel plate placed on a horizontal surface atan angle `theta` with the vertical. If the coefficient of restitution is e, the angle at which the rebound will take place is:

Options

A. `theta`
B. `tan^(-1) [ ( tan theta )/( e ) ]`
C. `e tan theta`
D. `tan^(-1) [( e )/( tan theta )]`

Answer

To find the angle at which the steel ball rebounds after striking a fixed smooth steel plate, we can follow these steps:

### Step 1: Understanding the Problem
The steel ball strikes the plate at an angle \( \theta \) with the vertical. The coefficient of restitution \( e \) determines how much of the vertical component of the velocity is retained after the collision.

### Step 2: Break Down the Velocity Components
Let the initial velocity of the steel ball be \( u \). We can break this velocity into two components:
- The horizontal component \( u_x = u \sin \theta \)
- The vertical component \( u_y = u \cos \theta \)

### Step 3: Analyze the Collision
Since the plate is smooth and fixed, the horizontal component of the velocity \( u_x \) remains unchanged after the collision:
- \( v_x = u_x = u \sin \theta \)

The vertical component of the velocity changes due to the collision. According to the coefficient of restitution \( e \):
- \( v_y = -e \cdot u_y = -e \cdot (u \cos \theta) \)

### Step 4: Find the Rebound Angle
The rebound angle \( \alpha \) can be found using the tangent function, which relates the vertical and horizontal components of the velocity after the collision:
\[ \tan \alpha = \frac{v_y}{v_x} \]

Substituting the values we found:
\[ \tan \alpha = \frac{-e \cdot (u \cos \theta)}{u \sin \theta} \]

### Step 5: Simplify the Expression
The \( u \) cancels out:
\[ \tan \alpha = \frac{-e \cdot \cos \theta}{\sin \theta} \]
This can be rewritten as:
\[ \tan \alpha = -e \cdot \cot \theta \]

### Step 6: Find the Angle \( \alpha \)
To find \( \alpha \), we take the arctangent:
\[ \alpha = \tan^{-1}(-e \cdot \cot \theta) \]

### Final Result
Thus, the angle at which the rebound will take place is:
\[ \alpha = \tan^{-1} \left( \frac{\tan \theta}{e} \right) \]