A man is going due east with a velocity of `5ms^(-1)`.It is vertically rainging downwards with a velocity of `4ms^(-1)`.At what angle should he hold the umbrella to the vertical so as to protect him self from the rain?

Options

A. `tan^(-1)(5/4)` in anti-clockwise direction
B. `tan^(-1)(5/4)` in clockwise direction
C. `tan^(-1)(5/4)`North of East
D. `tan^(-1)(5/4)`East of North

Explanation

To solve the problem of determining the angle at which the man should hold his umbrella to protect himself from the rain, we can follow these steps:

### Step 1: Understand the velocities involved
- The man is moving due east with a velocity \( V_m = 5 \, \text{m/s} \).
- The rain is falling vertically downwards with a velocity \( V_r = 4 \, \text{m/s} \).

### Step 2: Represent the velocities as vectors
- The velocity of the man can be represented as a vector pointing to the right (east).
- The velocity of the rain can be represented as a vector pointing downwards.

### Step 3: Determine the relative velocity of the rain with respect to the man
- The relative velocity of the rain with respect to the man is given by:
\[ V_{rm} = V_r - V_m \]
- Since the rain is falling downwards and the man is moving east, we can represent this as:
\[ V_{rm} = (0 \, \hat{i} - 4 \, \hat{j}) - (5 \, \hat{i} + 0 \, \hat{j}) = -5 \, \hat{i} - 4 \, \hat{j} \]
- This means the relative velocity vector points 5 m/s to the west and 4 m/s downwards.

### Step 4: Calculate the angle using trigonometry
- We can use the tangent function to find the angle \( \theta \) that the umbrella should make with the vertical:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{V_m}{V_r} = \frac{5}{4} \]
- Therefore, we can calculate \( \theta \):
\[ \theta = \tan^{-1}\left(\frac{5}{4}\right) \]

### Step 5: Calculate the angle
- Using a calculator, we find:
\[ \theta \approx 51.34^\circ \]

### Conclusion
- The man should hold the umbrella at an angle of approximately \( 51.34^\circ \) to the vertical to protect himself from the rain.

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