To a stationary man, rain appears to be falling at his back at an angle `30^@` with the vertical. As he starts moving forward with a speed of `0.5 m s^-1`, he finds that the rain is falling vertically.
The speed of rain with respect to the moving man is.

Options

A. `0.5 m s^-1`
B. `1.0 m s^-1`
C. `0.5 sqrt(3) ms^-1`
D. `0.45 ms^-1`

Explanation

To solve the problem, we need to analyze the situation involving the stationary man, the rain, and the man's movement. Let's break it down step by step.

### Step 1: Understand the scenario
The stationary man observes the rain falling at an angle of \(30^\circ\) with the vertical. This means that the rain has a horizontal component of velocity due to the wind or its own motion.

### Step 2: Set up the components of the rain's velocity
Let:
- \(V_r\) be the speed of the rain.
- The vertical component of the rain's velocity is \(V_{r_y} = V_r \cos(30^\circ)\).
- The horizontal component of the rain's velocity is \(V_{r_x} = V_r \sin(30^\circ)\).

Since \(\sin(30^\circ) = \frac{1}{2}\) and \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), we can express these components as:
- \(V_{r_y} = V_r \cdot \frac{\sqrt{3}}{2}\)
- \(V_{r_x} = V_r \cdot \frac{1}{2}\)

### Step 3: Analyze the man's movement
When the man starts moving forward with a speed of \(0.5 \, \text{m/s}\), he perceives the rain to be falling vertically. This means that the horizontal component of the rain's velocity must equal the man's speed but in the opposite direction.

### Step 4: Set up the equation
Since the rain appears to fall vertically when the man is moving, we can equate the horizontal component of the rain's velocity to the man's speed:
\[ V_{r_x} = 0.5 \, \text{m/s} \]
Substituting the expression for \(V_{r_x}\):
\[ V_r \cdot \frac{1}{2} = 0.5 \]

### Step 5: Solve for the rain's speed
Multiplying both sides by 2 gives:
\[ V_r = 1 \, \text{m/s} \]

### Step 6: Find the speed of rain with respect to the moving man
Now, we need to find the speed of the rain with respect to the moving man. The vertical component of the rain's velocity is:
\[ V_{r_y} = V_r \cdot \frac{\sqrt{3}}{2} = 1 \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \, \text{m/s} \]

### Step 7: Calculate the resultant speed of rain with respect to the moving man
Using the Pythagorean theorem, the speed of the rain with respect to the moving man can be calculated as:
\[ V_{rm} = \sqrt{(V_{r_x})^2 + (V_{r_y})^2} \]
Substituting the values:
\[ V_{rm} = \sqrt{(0.5)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{0.25 + \frac{3}{4}} = \sqrt{0.25 + 0.75} = \sqrt{1} = 1 \, \text{m/s} \]

Thus, the speed of the rain with respect to the moving man is \(1 \, \text{m/s}\).

### Final Answer
The speed of rain with respect to the moving man is \(1 \, \text{m/s}\).

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