A man travelling at `10.8 kmph` in topless car on a rainy day. He holds an umbrella at angle of `37^(@)` with the vertical so that he does not wet. If rain drops falls vertically downwards what is rain velocity.

Options

A. `1 m//s`
B. `2 m//s`
C. `3 m//s`
D. `4 m//s`

Explanation

To solve the problem, we need to find the velocity of the rain given that a man is traveling in a topless car at a speed of 10.8 km/h and holds an umbrella at an angle of 37 degrees with the vertical.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- Speed of the man (Vm) = 10.8 km/h
- Angle of the umbrella with the vertical (θ) = 37 degrees
- The rain is falling vertically downwards.

2. **Convert the Speed of the Man to m/s:**
\[ Vm = 10.8 \, \text{km/h} = \frac{10.8 \times 1000}{3600} \, \text{m/s} = 3 \, \text{m/s} \]

3. **Set Up the Velocity Components:**
- Let Vr be the velocity of the rain (which we need to find).
- The velocity of the rain with respect to the man (Vrm) makes an angle of 37 degrees with the vertical.

4. **Break Down the Velocity of Rain:**
- The vertical component of the velocity of rain (Vr_vertical) can be expressed as:
\[ Vr_{\text{vertical}} = Vr \cdot \cos(37^\circ) \]
- The horizontal component of the velocity of rain (Vr_horizontal) can be expressed as:
\[ Vr_{\text{horizontal}} = Vr \cdot \sin(37^\circ) \]

5. **Relate the Components to the Man's Velocity:**
- Since the rain appears to fall vertically to the man, the horizontal component of the rain's velocity must equal the speed of the man:
\[ Vr_{\text{horizontal}} = Vm \]
\[ Vr \cdot \sin(37^\circ) = 3 \, \text{m/s} \]

6. **Calculate the Velocity of Rain:**
- From the above equation, we can solve for Vr:
\[ Vr = \frac{3}{\sin(37^\circ)} \]
- Using \(\sin(37^\circ) \approx 0.6018\):
\[ Vr \approx \frac{3}{0.6018} \approx 4.98 \, \text{m/s} \]

7. **Find the Vertical Component of the Rain:**
- Now, we can find the vertical component:
\[ Vr_{\text{vertical}} = Vr \cdot \cos(37^\circ) = 4.98 \cdot \cos(37^\circ) \]
- Using \(\cos(37^\circ) \approx 0.7986\):
\[ Vr_{\text{vertical}} \approx 4.98 \cdot 0.7986 \approx 3.98 \, \text{m/s} \]

8. **Final Result:**
- The magnitude of the rain's velocity is approximately:
\[ Vr \approx 4.98 \, \text{m/s} \]

### Conclusion:
The velocity of the rain is approximately **4.98 m/s** vertically downward.