A man holds his umbrella vertically upward while walking due west with a constant velocity of magnitude `V_(m) = 1.5` m/sec in rain. To protect himself from rain, he has to rotate his umbrella through an angle `phi = 30^(@)` when he stops walking. Find the velocity of the rain.

Answer

To solve the problem, we need to analyze the situation involving the man, his umbrella, and the rain. The man is walking west with a velocity of \( V_m = 1.5 \, \text{m/s} \) and when he stops, he has to tilt his umbrella at an angle of \( \phi = 30^\circ \) to avoid getting wet. We need to find the velocity of the rain.

### Step-by-Step Solution:

1. **Understand the Situation**:
- The man walks west with a velocity of \( V_m = 1.5 \, \text{m/s} \).
- While walking, the rain appears to come vertically downwards relative to him.
- When he stops, he tilts his umbrella at an angle of \( 30^\circ \) to the vertical to avoid getting wet.

2. **Identify the Components**:
- Let \( V_{rg} \) be the velocity of the rain with respect to the ground.
- When the man is walking, the rain's velocity with respect to him is purely vertical. Thus, the horizontal component of the rain's velocity must equal the man's walking velocity, \( V_m \).

3. **Set Up the Right Triangle**:
- When the man stops and tilts the umbrella at \( 30^\circ \), we can visualize a right triangle where:
- The vertical side represents the vertical component of the rain's velocity (\( V_{rv} \)).
- The horizontal side represents the horizontal component of the rain's velocity, which is equal to the man's walking speed (\( V_m = 1.5 \, \text{m/s} \)).
- The hypotenuse represents the resultant velocity of the rain with respect to the ground (\( V_{rg} \)).

4. **Use Trigonometry**:
- From the triangle, we can use the sine function:
\[ \sin(30^\circ) = \frac{V_m}{V_{rg}} \]
- We know that \( \sin(30^\circ) = \frac{1}{2} \), so we can substitute:
\[ \frac{1}{2} = \frac{1.5}{V_{rg}} \]

5. **Solve for \( V_{rg} \)**:
- Rearranging the equation gives:
\[ V_{rg} = 2 \times 1.5 = 3 \, \text{m/s} \]

6. **Conclusion**:
- The velocity of the rain with respect to the ground is \( V_{rg} = 3 \, \text{m/s} \).

### Final Answer:
The velocity of the rain is \( 3 \, \text{m/s} \) at an angle of \( 30^\circ \) with respect to the vertical.