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. `1ms^(-1)`
B. `0.25ms^(-1)`
C. `0.67ms^(-1)`
D. `3ms^(-1)`

Explanation

To solve the problem, we need to analyze the motion of the swimmer with respect to the water and the stream. Here is a step-by-step solution:

### Step 1: Understand the problem
The swimmer is trying to reach a point directly opposite on the bank of the stream. They swim at a speed of \(0.5 \, \text{m/s}\) at an angle of \(120^\circ\) with respect to the direction of the water flow.

### Step 2: Break down the swimmer's velocity
The swimmer's velocity can be broken down into two components:
- **Horizontal Component (along the direction of the stream)**: \(V_{sx} = V_s \cos(120^\circ)\)
- **Vertical Component (against the current)**: \(V_{sy} = V_s \sin(120^\circ)\)

Where \(V_s = 0.5 \, \text{m/s}\).

### Step 3: Calculate the components
Using the trigonometric values:
- \(\cos(120^\circ) = -\frac{1}{2}\)
- \(\sin(120^\circ) = \frac{\sqrt{3}}{2}\)

Now we can calculate:
- \(V_{sx} = 0.5 \cos(120^\circ) = 0.5 \times -\frac{1}{2} = -0.25 \, \text{m/s}\)
- \(V_{sy} = 0.5 \sin(120^\circ) = 0.5 \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4} \, \text{m/s}\)

### Step 4: Set up the equation for horizontal motion
Since the swimmer is aiming to reach the point directly opposite, the horizontal component of their velocity must cancel out the velocity of the stream. This means:
\[ V_{sx} + V = 0 \]
Where \(V\) is the speed of the stream.

### Step 5: Solve for the speed of the stream
From the equation:
\[ -0.25 + V = 0 \]
We can solve for \(V\):
\[ V = 0.25 \, \text{m/s} \]

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

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