A man running with velocity `3m//s` finds that raindrops are hitting him vertically with a speed of `4m//s`.
Raindrops are falling at an angle `theta` with the vertical. Value of `theta` is

Options

A. `37^(@)`
B. `53^(@)`
C. `45^(@)`
D. `60^(@)`

Explanation

To solve the problem, we need to find the angle \( \theta \) at which the raindrops are falling with respect to the vertical. We know the following:

- The velocity of the man running, \( V_M = 3 \, \text{m/s} \) (horizontal component).
- The velocity of the raindrops hitting him vertically, \( V_R = 4 \, \text{m/s} \) (vertical component).

### Step-by-Step Solution:

1. **Understanding the Situation**:
The man is running horizontally while observing the raindrops falling vertically. This means that the effective velocity of the rain with respect to the man is a combination of his running speed and the vertical speed of the rain.

2. **Setting Up the Vectors**:
- The horizontal component of the rain's velocity (due to the man's running) is \( V_M = 3 \, \text{m/s} \).
- The vertical component of the rain's velocity is \( V_R = 4 \, \text{m/s} \).

3. **Using Trigonometry**:
The angle \( \theta \) can be found using the tangent function, which relates the opposite side (horizontal component) to the adjacent side (vertical component) in a right triangle.
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{V_M}{V_R} \]
Substituting the values:
\[ \tan(\theta) = \frac{3}{4} \]

4. **Calculating the Angle**:
To find \( \theta \), we take the arctangent (inverse tangent) of \( \frac{3}{4} \):
\[ \theta = \tan^{-1}\left(\frac{3}{4}\right) \]

5. **Finding the Value of \( \theta \)**:
Using a calculator or trigonometric tables:
\[ \theta \approx 36.87^\circ \]
Rounding this gives us \( \theta \approx 37^\circ \).

### Final Answer:
The angle \( \theta \) at which the raindrops are falling with respect to the vertical is approximately \( 37^\circ \).

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