A man who can swin at the rate of 2 `(km)/hr)`(in still river) crosses a river to a point exactly opposite on the other bank by swimming in a direction of `120^@` to the flow of the water in the river. The velocity of the water current in `(km)/(hr)` is

Options

A. 1
B. 3
C. `(1)/(2)`
D. `(3)/(2)`

Explanation

To solve the problem, we need to analyze the situation involving the swimmer and the river current. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We have a man swimming across a river at a speed of 2 km/hr. He swims at an angle of 120 degrees to the direction of the river current. We need to find the speed of the river current.

### Step 2: Set Up the Coordinate System
Let's assume:
- The direction of the river current is along the positive x-axis.
- The direction the man swims (120 degrees to the current) is measured counterclockwise from the positive x-axis.

### Step 3: Break Down the Swimmer's Velocity
The swimmer's velocity can be broken down into two components:
- The component in the direction of the river current (x-direction).
- The component perpendicular to the river current (y-direction).

Using trigonometry:
- The x-component (along the river current) is given by:
\[ V_{x} = V_{m} \cdot \cos(120^\circ) \]
- The y-component (across the river) is given by:
\[ V_{y} = V_{m} \cdot \sin(120^\circ) \]

Where \( V_{m} = 2 \, \text{km/hr} \).

### Step 4: Calculate the Components
Using the values of cosine and sine:
- \( \cos(120^\circ) = -\frac{1}{2} \)
- \( \sin(120^\circ) = \frac{\sqrt{3}}{2} \)

Thus:
\[ V_{x} = 2 \cdot \left(-\frac{1}{2}\right) = -1 \, \text{km/hr} \]
\[ V_{y} = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \, \text{km/hr} \]

### Step 5: Establish the Condition for Crossing the River
For the man to reach the point directly opposite on the other bank, the downstream velocity (x-component) must be equal to the velocity of the river current (V). Therefore:
\[ V_{x} = -V \]

### Step 6: Solve for the Velocity of the River Current
From the previous calculations:
\[ -1 = -V \implies V = 1 \, \text{km/hr} \]

### Conclusion
The velocity of the water current is:
\[ \boxed{1 \, \text{km/hr}} \]