A man wishes to swim across a river 0.5km. wide if he can swim at the rate of 2 km/h. in still water and the river flows at the rate of 1km/h. The angle (w.r.t. the flow of the river) along which he should swin so as to reach a point exactly oppposite his starting point, should be:-

Options

A. `60^(@)`
B. `120^(@)`
C. `145^(@)`
D. `90^(@)`

Answer

To solve the problem of the man swimming across the river, we can break it down into several steps:

### Step 1: Understand the Problem
The man wants to swim across a river that is 0.5 km wide. He can swim at a speed of 2 km/h in still water, while the river flows at a speed of 1 km/h. We need to find the angle at which he should swim to reach a point directly opposite his starting point.

### Step 2: Set Up the Coordinate System
Let's set up a coordinate system where:
- The width of the river (0.5 km) is along the y-axis.
- The flow of the river (1 km/h) is along the x-axis.

### Step 3: Determine the Components of Velocity
When the man swims at an angle θ with respect to the flow of the river, his swimming velocity can be broken down into two components:
- The component of his swimming velocity in the direction across the river (y-direction): \( V_{y} = V_{s} \cdot \cos(\theta) \)
- The component of his swimming velocity in the direction of the river flow (x-direction): \( V_{x} = V_{s} \cdot \sin(\theta) \)

Where:
- \( V_{s} = 2 \) km/h (swimming speed)
- The river speed \( V_{r} = 1 \) km/h.

### Step 4: Set Up the Equation for the x-direction
To reach a point directly opposite his starting point, the downstream drift caused by the river must be equal to the upstream component of his swimming. This gives us the equation:
\[ V_{x} = V_{r} \]
Substituting the values:
\[ V_{s} \cdot \sin(\theta) = V_{r} \]
\[ 2 \cdot \sin(\theta) = 1 \]

### Step 5: Solve for sin(θ)
Now, we can solve for \( \sin(\theta) \):
\[ \sin(\theta) = \frac{1}{2} \]

### Step 6: Find the Angle θ
To find the angle θ, we take the inverse sine:
\[ \theta = \sin^{-1}\left(\frac{1}{2}\right) \]
This gives:
\[ \theta = 30^\circ \]

### Conclusion
The angle at which the man should swim with respect to the flow of the river to reach a point directly opposite his starting point is **30 degrees**.

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