A person aiming to reach the exactly opposite point on the bank of a stream is swimming with a speed of `0.5(m)/(s)` at an angle of `120^@` with the direction of flow of water. The speed of water in the stream is

Options

A. `1(m)/(s)`
B. `0.5(m)/(s)`
C. 0.25(m)/(s)`
D. `0.433(m)/(s)`

Explanation

To solve the problem, we need to analyze the swimmer's motion relative to the flow of the stream. The swimmer is trying to reach a point directly opposite on the bank of the stream while swimming at an angle of 120 degrees to the direction of the stream's flow.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Speed of the swimmer, \( v_s = 0.5 \, \text{m/s} \)
- Angle with the direction of the stream, \( \theta = 120^\circ \)

2. **Break Down the Swimmer's Velocity:**
The swimmer's velocity can be resolved into two components:
- **Horizontal Component (along the stream):**
\[ v_{sx} = v_s \cdot \cos(120^\circ) = 0.5 \cdot \cos(120^\circ) = 0.5 \cdot (-0.5) = -0.25 \, \text{m/s} \]
- **Vertical Component (across the stream):**
\[ v_{sy} = v_s \cdot \sin(120^\circ) = 0.5 \cdot \sin(120^\circ) = 0.5 \cdot \left(\frac{\sqrt{3}}{2}\right) = \frac{\sqrt{3}}{4} \, \text{m/s} \]

3. **Determine the Speed of the Stream:**
To reach the point directly opposite, the horizontal component of the swimmer's velocity must cancel out the speed of the stream. Let \( u \) be the speed of the stream. Therefore, we have:
\[ -0.25 \, \text{m/s} = -u \implies u = 0.25 \, \text{m/s} \]

4. **Final Result:**
The speed of the water in the stream is:
\[ u = 0.25 \, \text{m/s} \]