A person aiming to reach exactly opposite point on the bank of a stream is swimming with a speed of 0.6 m/s at an angle of `120^(@)` with the direction of flow of water . The speed of water in the stream is

Options

A. 0.3 m/s
B. 0.5 m/s
C. 1.0 m/s
D. 0.433 m/s

Answer

To solve the problem of a person swimming across a stream at an angle while trying to reach a point directly opposite on the bank, we can break down the situation step by step.

### Step-by-Step Solution:

1. **Understand the Problem**:
The swimmer is swimming at a speed of \(0.6 \, \text{m/s}\) at an angle of \(120^\circ\) with respect to the direction of the flow of the water. We need to find the speed of the stream, denoted as \(U\).

2. **Identify the Components of Velocity**:
The swimmer's velocity can be broken down into two components:
- **Horizontal Component** (along the direction of the stream): \(V_x = V \cos(120^\circ)\)
- **Vertical Component** (across the stream): \(V_y = V \sin(120^\circ)\)

Where \(V = 0.6 \, \text{m/s}\).

3. **Calculate the Components**:
- The angle \(120^\circ\) can be converted to its reference angle \(30^\circ\) for calculations:
- \(V_x = 0.6 \cos(120^\circ) = 0.6 \cos(30^\circ + 90^\circ) = -0.6 \cos(30^\circ) = -0.6 \times \left(-\frac{\sqrt{3}}{2}\right) = -0.3\sqrt{3} \, \text{m/s}\)
- \(V_y = 0.6 \sin(120^\circ) = 0.6 \sin(30^\circ) = 0.6 \times \frac{1}{2} = 0.3 \, \text{m/s}\)

4. **Condition for Reaching the Opposite Point**:
For the swimmer to reach the point directly opposite, the horizontal component of the swimmer's velocity must be equal to the speed of the stream:
\[ U = V_x \]
Therefore, we know:
\[ U = 0.6 \sin(30^\circ) \]

5. **Calculate the Speed of the Stream**:
Since \(\sin(30^\circ) = \frac{1}{2}\):
\[ U = 0.6 \times \frac{1}{2} = 0.3 \, \text{m/s} \]

### Final Answer:
The speed of the stream is \(0.3 \, \text{m/s}\).