A steel ball strikes a fixed smooth steel plate placed on a horizontal surface at an 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)]`

Explanation

To solve the problem of a steel ball striking a fixed smooth steel plate at an angle θ with the vertical and rebounding at an angle φ, we can follow these steps:

### Step 1: Understand the Components of Velocity
When the steel ball strikes the plate, we can break its initial velocity \( u \) into two components:
- The vertical component: \( u \sin \theta \)
- The horizontal component: \( u \cos \theta \)

### Step 2: Analyze the Collision
Since the plate is smooth and fixed, there is no friction. The vertical component of the velocity will remain unchanged, while the horizontal component will change due to the collision.

### Step 3: Apply the Coefficient of Restitution
The coefficient of restitution \( e \) relates the velocities before and after the collision. It is defined as:
\[ e = \frac{\text{Velocity of separation}}{\text{Velocity of approach}} \]
- The velocity of approach (along the line of impact) is \( u \cos \theta \).
- The velocity of separation (after the collision) will be \( e u \cos \theta \).

### Step 4: Determine the Rebound Angle
After the collision, the vertical component of the velocity remains \( u \sin \theta \), and the horizontal component becomes \( e u \cos \theta \).

Using these components, we can find the angle \( \phi \) at which the ball rebounds:
\[ \tan \phi = \frac{\text{Vertical component}}{\text{Horizontal component}} = \frac{u \sin \theta}{e u \cos \theta} \]
This simplifies to:
\[ \tan \phi = \frac{\sin \theta}{e \cos \theta} \]

### Step 5: Find the Angle \( \phi \)
To find \( \phi \), we take the inverse tangent:
\[ \phi = \tan^{-1} \left( \frac{\sin \theta}{e \cos \theta} \right) \]

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