Person aiming to reach the exactly opposite point on the bank of a stream is swimming with a speed of `0.5ms^(-1)` at an angle of `120^(@)` with the direction of flow of water.The speed of water in the stream is

Options

A. 1.0m/s
B. 0.5m/s
C. 0.25m/s
D. 0.43m/s

Answer

To solve the problem, we need to analyze the situation using vector components. The person is swimming at an angle of 120 degrees with respect to the direction of the water flow, and we need to find the speed of the water in the stream.

### Step-by-Step Solution:

1. **Understanding the Situation**:
- Let the speed of the person swimming be \( v = 0.5 \, \text{m/s} \).
- The angle of swimming with respect to the direction of the stream is \( 120^\circ \).
- The speed of the stream is denoted as \( u \).

2. **Components of Velocity**:
- The velocity of the swimmer can be broken down into two components:
- The component in the direction opposite to the flow of the stream: \( v \cos(120^\circ) \)
- The component perpendicular to the flow of the stream: \( v \sin(120^\circ) \)

3. **Calculating the Components**:
- Calculate \( \cos(120^\circ) \) and \( \sin(120^\circ) \):
- \( \cos(120^\circ) = -\frac{1}{2} \)
- \( \sin(120^\circ) = \frac{\sqrt{3}}{2} \)
- Therefore, the components of the swimmer's velocity are:
- \( v \cos(120^\circ) = 0.5 \times -\frac{1}{2} = -0.25 \, \text{m/s} \) (opposite to the stream)
- \( v \sin(120^\circ) = 0.5 \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4} \, \text{m/s} \) (perpendicular to the stream)

4. **Balancing the Velocities**:
- For the swimmer to reach the point directly opposite, the component of the swimmer's velocity in the direction of the stream must equal the speed of the stream:
\[ -v \cos(120^\circ) = u \]
- Substituting the value of \( v \cos(120^\circ) \):
\[ -(-0.25) = u \implies u = 0.25 \, \text{m/s} \]

5. **Conclusion**:
- The speed of the water in the stream is \( u = 0.25 \, \text{m/s} \).

### Final Answer:
The speed of the water in the stream is \( 0.25 \, \text{m/s} \).