A man can swim at a speed of `3 km h^-1` in still water. He wants to cross a `500-m` wide river flowing at `2 km h^-1`. He keeps himself always at an angle to `120^@` with the river flow while swimming.
The drift of the man along the direction of flow, when he arrives at the opposite bank is.

Options

A. `(1)/(6 sqrt(3)) km`
B. ` 6 sqrt(3) cm`
C. `3 sqrt(3) km`
D. `(1)/(3 sqrt(3)) km`

Explanation

To solve the problem, we need to determine the drift of the man along the direction of the river flow when he reaches the opposite bank. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the given data
- Speed of the man in still water, \( V_m = 3 \, \text{km/h} \)
- Speed of the river, \( V_r = 2 \, \text{km/h} \)
- Width of the river, \( d = 500 \, \text{m} = 0.5 \, \text{km} \)
- Angle with the river flow, \( \theta = 120^\circ \)

### Step 2: Resolve the man's swimming speed into components
The man's swimming speed can be resolved into two components:
- **Vertical component (across the river)**:
\[ V_{m_y} = V_m \sin(120^\circ) = 3 \cdot \sin(120^\circ) = 3 \cdot \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{2} \, \text{km/h} \]
- **Horizontal component (along the river)**:
\[ V_{m_x} = V_m \cos(120^\circ) = 3 \cdot \cos(120^\circ) = 3 \cdot \left(-\frac{1}{2}\right) = -\frac{3}{2} \, \text{km/h} \]

### Step 3: Calculate the time taken to cross the river
The time taken to cross the river can be calculated using the vertical component of the man's speed:
\[ t = \frac{d}{V_{m_y}} = \frac{0.5 \, \text{km}}{\frac{3\sqrt{3}}{2} \, \text{km/h}} = \frac{0.5 \cdot 2}{3\sqrt{3}} = \frac{1}{3\sqrt{3}} \, \text{h} \]

### Step 4: Calculate the drift due to the river current
During the time \( t \), the drift caused by the river's current can be calculated as:
\[ \text{Drift} = V_r \cdot t = 2 \, \text{km/h} \cdot \frac{1}{3\sqrt{3}} \, \text{h} = \frac{2}{3\sqrt{3}} \, \text{km} \]
To convert this to meters:
\[ \text{Drift} = \frac{2}{3\sqrt{3}} \cdot 1000 \, \text{m} = \frac{2000}{3\sqrt{3}} \, \text{m} \]

### Step 5: Simplify the drift
To simplify \( \frac{2000}{3\sqrt{3}} \):
\[ \text{Drift} \approx \frac{2000}{5.196} \approx 384.9 \, \text{m} \]

### Final Answer
The drift of the man along the direction of flow when he arrives at the opposite bank is approximately \( 385 \, \text{m} \).

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