Rain, driven by the wind, falls on a railway compartment with a velocity of `20 ms^-1`, at an angle of `30^@` to the vertical. The train moves, along the direction of wind flow, at a speed of `108 km h^-1`. Determine the apparent velocity of rain for a person sitting in the train.

Options

A. `20 sqrt(7) ms^-1`
B. `10 sqrt(7) ms^-1`
C. `15 sqrt(7) ms^-1`
D. `10 sqrt(7) km h^-1`

Explanation

To determine the apparent velocity of rain for a person sitting in a train, we need to analyze the situation using vector components. Here’s a step-by-step solution:

### Step 1: Convert the speed of the train
The speed of the train is given as 108 km/h. We need to convert this to meters per second (m/s).

\[ \text{Speed of train} = 108 \, \text{km/h} = \frac{108 \times 1000}{3600} \, \text{m/s} = 30 \, \text{m/s} \]

### Step 2: Resolve the velocity of rain into components
The velocity of rain is given as 20 m/s at an angle of 30° to the vertical. We need to resolve this into vertical and horizontal components.

- Vertical component (\(V_{ry}\)):
\[ V_{ry} = 20 \cos(30^\circ) = 20 \times \frac{\sqrt{3}}{2} = 10\sqrt{3} \, \text{m/s} \]

- Horizontal component (\(V_{rx}\)):
\[ V_{rx} = 20 \sin(30^\circ) = 20 \times \frac{1}{2} = 10 \, \text{m/s} \]

### Step 3: Determine the apparent velocity of rain
For a person sitting in the train, the apparent velocity of rain is the vector sum of the rain's velocity and the negative of the train's velocity (since the train is moving in the same direction as the rain's horizontal component).

- The horizontal component of the apparent velocity (\(V_{a_x}\)):
\[ V_{a_x} = V_{rx} - V_{train} = 10 - 30 = -20 \, \text{m/s} \]

- The vertical component of the apparent velocity (\(V_{a_y}\)):
\[ V_{a_y} = V_{ry} = 10\sqrt{3} \, \text{m/s} \]

### Step 4: Calculate the magnitude of the apparent velocity
Now we can find the magnitude of the apparent velocity using the Pythagorean theorem:

\[ V_a = \sqrt{(V_{a_x})^2 + (V_{a_y})^2} = \sqrt{(-20)^2 + (10\sqrt{3})^2} \]

Calculating each term:

\[ (-20)^2 = 400 \]
\[ (10\sqrt{3})^2 = 100 \times 3 = 300 \]

Adding these:

\[ V_a = \sqrt{400 + 300} = \sqrt{700} \, \text{m/s} \]

### Step 5: Final answer
Thus, the apparent velocity of rain for a person sitting in the train is:

\[ V_a = \sqrt{700} \approx 26.46 \, \text{m/s} \]