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. E
B. `E//sqrt(2)`
C. `E//2`
D. zero

Explanation

To find the angle at which the man should swim to reach a point directly opposite his starting point across a river, we can follow these steps:

### Step 1: Understand the Problem
The man is swimming 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 he should swim relative to the flow of the river to reach the point directly opposite.

### Step 2: Set Up the Vector Diagram
1. **Velocity of the man in still water (V_m)**: 2 km/h (this is the hypotenuse of the triangle).
2. **Velocity of the river (V_r)**: 1 km/h (this is the base of the triangle).
3. **Resultant velocity (V_g)**: The velocity of the man with respect to the ground, which should be directed straight across the river.

### Step 3: Use the Right Triangle
Using the right triangle formed by the velocities:
- The vertical component (across the river) is \( V_g \).
- The horizontal component (along the river) is \( V_r \).

### Step 4: Apply the Pythagorean Theorem
Since the man wants to swim directly across, we can use the sine function to find the angle \( \theta \) he should swim at:
\[ \sin(\theta) = \frac{V_r}{V_m} \]
Where:
- \( V_r = 1 \) km/h (velocity of the river)
- \( V_m = 2 \) km/h (velocity of the man)

### Step 5: Calculate the Sine of the Angle
Substituting the values:
\[ \sin(\theta) = \frac{1}{2} \]

### Step 6: Find the Angle
Now, we can find \( \theta \):
\[ \theta = \sin^{-1}\left(\frac{1}{2}\right) \]
This gives:
\[ \theta = 30^\circ \]

### Step 7: Determine the Angle with Respect to the Flow of the River
Since the angle \( \theta \) is measured from the direction directly across the river, the angle with respect to the flow of the river will be:
\[ 90^\circ + 30^\circ = 120^\circ \]

### Final Answer
The angle at which the man should swim to reach a point directly opposite his starting point is **120 degrees** with respect to the flow of the river.

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