A ball strikes a wall with a velocity `vecu` at an angle `theta` with the normal to the wall surface. and rebounds from it at an angle `beta` with the surface. Then

Options

A. `(theta+beta)lt90^(@)` if the wall is smooth
B. if the wall is rough , coefficient of restitution `=tanbeta//costheta`
C. if the wall is rough, coefficient of restition
`lttanbeta//cottheta`
D. none of these

Explanation

To solve the problem of a ball striking a wall and rebounding, we can break it down into several steps:

### Step 1: Understand the situation
A ball strikes a wall with a velocity \( \vec{u} \) at an angle \( \theta \) with the normal to the wall. After striking the wall, it rebounds at an angle \( \beta \) with the surface. The angles \( \theta \) and \( \beta \) are defined with respect to the normal and the surface, respectively.

### Step 2: Resolve the velocity into components
The initial velocity \( \vec{u} \) can be resolved into two components:
- Perpendicular to the wall (normal component): \( u \cos \theta \)
- Parallel to the wall (tangential component): \( u \sin \theta \)

### Step 3: Apply the coefficient of restitution
The coefficient of restitution \( e \) is defined as the ratio of the velocity of separation to the velocity of approach. For a smooth wall, there is no friction, and the tangential component remains unchanged. Thus, we have:
\[ e = \frac{\text{Velocity of separation}}{\text{Velocity of approach}} = \frac{V_{\perpendicular}}{u \cos \theta} \]
Where \( V_{\perpendicular} \) is the component of the rebound velocity in the direction normal to the wall.

### Step 4: Calculate the rebound velocity
From the above equation, we can express \( V_{\perpendicular} \):
\[ V_{\perpendicular} = e \cdot u \cos \theta \]

### Step 5: Analyze the tangential component
Since the wall is smooth, the tangential component of the velocity does not change:
\[ u \sin \theta = V \sin \beta \]
Where \( V \) is the magnitude of the rebound velocity.

### Step 6: Relate angles using geometry
From the geometry of the problem, we know that:
\[ \alpha + \beta = 90^\circ \]
Where \( \alpha \) is the angle made by the rebound velocity with the normal.

### Step 7: Substitute and simplify
Using the relationship between angles:
\[ \sin \beta = \cos \alpha \quad \text{and} \quad \cos \beta = \sin \alpha \]
We can substitute these into our equations to find relationships between \( \theta \), \( \beta \), and \( e \).

### Step 8: Analyze the conditions for a rough wall
If the wall is rough, the coefficient of restitution changes, and we can express it in terms of the angles:
\[ e = \frac{\tan \beta}{\cos \theta} \]

### Step 9: Conclusion
From the analysis, we conclude that:
- If the wall is smooth, \( \theta + \beta < 90^\circ \) holds true.
- If the wall is rough, the coefficient of restitution is given by \( e = \frac{\tan \beta}{\cos \theta} \).