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. We will use vector addition to find the velocity of the rain.

### Step-by-Step Solution:

1. **Understanding the Situation:**
- The man is walking due west with a velocity \( V_m = 1.5 \, \text{m/s} \).
- The umbrella is initially held vertically upward.
- When the man stops walking, he has to tilt his umbrella at an angle \( \phi = 30^\circ \) to protect himself from the rain.

2. **Setting Up the Vectors:**
- Let \( V_r \) be the velocity of the rain with respect to the ground.
- The velocity of the man \( V_m \) is directed towards the west (let's assume this is the negative x-direction).
- The velocity of the rain \( V_r \) can be broken down into two components: a vertical component (downward) and a horizontal component (which we need to find).

3. **Using the Angle of the Umbrella:**
- When the man stops, the umbrella makes an angle of \( 30^\circ \) with the vertical.
- This means that the rain appears to be coming at an angle of \( 30^\circ \) from the vertical.

4. **Applying Trigonometry:**
- The vertical component of the rain's velocity \( V_{ry} \) can be expressed as:
\[ V_{ry} = V_r \cos(30^\circ) \]
- The horizontal component of the rain's velocity \( V_{rx} \) can be expressed as:
\[ V_{rx} = V_r \sin(30^\circ) \]

5. **Relating the Components:**
- Since the man is moving west with a velocity of \( 1.5 \, \text{m/s} \), when he stops, the horizontal component of the rain's velocity must equal the man's velocity:
\[ V_{rx} = V_m = 1.5 \, \text{m/s} \]
- Therefore, we have:
\[ V_r \sin(30^\circ) = 1.5 \]

6. **Solving for \( V_r \):**
- We know that \( \sin(30^\circ) = \frac{1}{2} \):
\[ V_r \cdot \frac{1}{2} = 1.5 \]
- Solving for \( V_r \):
\[ V_r = 1.5 \cdot 2 = 3 \, \text{m/s} \]

7. **Conclusion:**
- The velocity of the rain is \( V_r = 3 \, \text{m/s} \).