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

Options

A. `1 ms^(-1)`
B. `2ms^(-1)`
C. `3 ms^(-1)`
D. `4 ms^(-1)`

Answer

To solve the problem, we need to determine the velocity of the rain (Vr) given that the man is traveling at a speed (Vm) of 10.8 km/h and holds his umbrella at an angle of 37° with the vertical.

### Step-by-Step Solution:

1. **Convert the speed of the man from km/h to m/s:**
\[ Vm = 10.8 \text{ km/h} = 10.8 \times \frac{1000 \text{ m}}{3600 \text{ s}} = 3 \text{ m/s} \]

2. **Identify the relationship between the velocities:**
The umbrella is held at an angle of 37° with the vertical. The vertical component of the rain's velocity (Vr) and the horizontal component (Vm) can be related using trigonometric functions. Since the rain falls vertically, we can use the tangent of the angle:
\[ \tan(37°) = \frac{Vm}{Vr} \]

3. **Substituting the known values:**
We know that \(\tan(37°) = \frac{3}{4}\). Therefore, we can write:
\[ \frac{3}{4} = \frac{3 \text{ m/s}}{Vr} \]

4. **Rearranging the equation to find Vr:**
Cross-multiplying gives:
\[ 3 \cdot 4 = 3 \cdot Vr \]
\[ 12 = 3 \cdot Vr \]
\[ Vr = \frac{12}{3} = 4 \text{ m/s} \]

5. **Conclusion:**
The velocity of the rain (Vr) is \(4 \text{ m/s}\).

### Final Answer:
The rain velocity is \(4 \text{ m/s}\).

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