It is raining at a speed of `5 ms^(-1)` at an angle `37^@` to vertical, towards east. A man is moving to west with a velocity of `5 ms^(-1)`. The angle with the vertical at which he has to hold the umbrella to protect himself from rain is

Options

A. `Tan^(-1) (2)` to west
B. `Tan^(-1) (2)` to east
C. `Tan^(-1) (1//2)` to south
D. `Tan^(-1) (1//2) ` to east

Answer

To solve the problem, we need to determine the angle at which a man should hold his umbrella to protect himself from the rain. The rain is falling at a speed of \(5 \, \text{m/s}\) at an angle of \(37^\circ\) to the vertical towards the east, while the man is moving west at a speed of \(5 \, \text{m/s}\).

### Step-by-Step Solution:

1. **Identify the Components of Rain Velocity**:
- The rain is falling at an angle of \(37^\circ\) to the vertical.
- The vertical component of the rain's velocity (\(V_{r_y}\)) can be calculated as:
\[ V_{r_y} = 5 \cos(37^\circ) \]
- The horizontal component of the rain's velocity (\(V_{r_x}\)) towards the east can be calculated as:
\[ V_{r_x} = 5 \sin(37^\circ) \]

2. **Calculate the Components**:
- Using the trigonometric values:
\[ \cos(37^\circ) \approx 0.8 \quad \text{and} \quad \sin(37^\circ) \approx 0.6 \]
- Thus:
\[ V_{r_y} = 5 \times 0.8 = 4 \, \text{m/s} \quad \text{(downward)} \]
\[ V_{r_x} = 5 \times 0.6 = 3 \, \text{m/s} \quad \text{(towards east)} \]

3. **Determine the Man's Velocity**:
- The man is moving west at \(5 \, \text{m/s}\). This velocity can be represented as:
\[ V_m = -5 \, \text{m/s} \quad \text{(towards west)} \]

4. **Calculate the Relative Velocity of Rain with Respect to the Man**:
- The effective horizontal component of the rain's velocity with respect to the man is:
\[ V_{rm_x} = V_{r_x} - V_m = 3 - (-5) = 3 + 5 = 8 \, \text{m/s} \quad \text{(towards east)} \]
- The vertical component remains the same:
\[ V_{rm_y} = V_{r_y} = 4 \, \text{m/s} \quad \text{(downward)} \]

5. **Determine the Angle with the Vertical**:
- The angle \(\theta\) at which the man should hold the umbrella can be found using the tangent function:
\[ \tan(\theta) = \frac{V_{rm_x}}{V_{rm_y}} = \frac{8}{4} = 2 \]
- Therefore, the angle \(\theta\) is:
\[ \theta = \tan^{-1}(2) \]

6. **Conclusion**:
- The angle at which the man should hold the umbrella to protect himself from the rain is \(\tan^{-1}(2)\) towards the east.