A glass wind screen whose inclination with the vertical can be changed is mounted on a car. The moves horizontally with a speed of `2m//s`. At what angle `alpha` with the vertical should the wind screen placed so that the rain drops falling vertically downwards with velcoity `6m//s` strike the wind screen perpendicularly?

Options

A. `tan^(-1)(1//3)`
B. `tan^(-1)(3)`
C. `cos^(-1)(3)`
D. `sin^(-1)(1//3)`

Answer

To solve the problem, we need to determine the angle \( \alpha \) at which the windscreen should be inclined so that the raindrops, falling vertically downward with a velocity of \( 6 \, \text{m/s} \), strike the windscreen perpendicularly while the car is moving horizontally at a speed of \( 2 \, \text{m/s} \).

### Step-by-Step Solution:

1. **Understand the Problem**:
- The rain is falling vertically with a velocity \( V_r = 6 \, \text{m/s} \).
- The car is moving horizontally with a velocity \( V_c = 2 \, \text{m/s} \).
- We need to find the angle \( \alpha \) such that the relative velocity of the rain with respect to the car strikes the windscreen perpendicularly.

2. **Relative Velocity**:
- The relative velocity of the rain with respect to the car can be expressed as:
\[ V_{r \text{ (relative)}} = V_r - V_c \]
- Here, \( V_r \) is the vertical component (downward) and \( V_c \) is the horizontal component (to the right).

3. **Components of Relative Velocity**:
- The vertical component of the relative velocity is \( V_r = 6 \, \text{m/s} \) (downward).
- The horizontal component of the relative velocity is \( V_c = 2 \, \text{m/s} \) (to the right).

4. **Finding the Angle**:
- The angle \( \alpha \) can be found using the tangent function:
\[ \tan \alpha = \frac{\text{Vertical Component}}{\text{Horizontal Component}} = \frac{V_r}{V_c} \]
- Substituting the values:
\[ \tan \alpha = \frac{6}{2} = 3 \]

5. **Calculating the Angle**:
- To find \( \alpha \), we take the inverse tangent:
\[ \alpha = \tan^{-1}(3) \]

6. **Final Answer**:
- The angle \( \alpha \) with the vertical at which the windscreen should be placed is:
\[ \alpha = \tan^{-1}(3) \]