A man is trying to cross a river flowing at a speed of `5ms^(-1)` to have least possible displacement by swimming at an angle of `143^(@)` to the stream. The drift he suffers when he is crossing the same river in least possible time is ( Width of river `=1` km )

Options

A. 160 m
B. 800 m
C. 400m
D. 200m

Answer

To solve the problem, we need to find the drift suffered by a man crossing a river while swimming at an angle to the current. The river has a width of 1 km (1000 m) and flows at a speed of 5 m/s. The swimmer swims at an angle of 143° to the stream.

### Step-by-Step Solution:

1. **Understanding the Angles**:
- The angle of swimming with respect to the stream is given as 143°.
- To find the angle with respect to the perpendicular direction (the direction across the river), we calculate:
\[ \text{Angle with respect to perpendicular} = 143° - 90° = 53° \]

2. **Finding the Speed of the Swimmer**:
- Let the speed of the swimmer be \( V \).
- The vertical component of the swimmer's speed (perpendicular to the river) can be expressed using the sine function:
\[ \sin(53°) = \frac{\text{Perpendicular component}}{V} \]
- We know that \( \sin(53°) \approx 0.8 \) (or \( \frac{4}{5} \)).
- The perpendicular component of the swimmer's speed is equal to the speed of the river, which is 5 m/s:
\[ 0.8 = \frac{5}{V} \implies V = \frac{5}{0.8} = \frac{25}{4} \text{ m/s} \]

3. **Calculating the Time to Cross the River**:
- The width of the river is given as 1 km (1000 m).
- The time \( T \) taken to cross the river can be calculated using the formula:
\[ T = \frac{\text{Width of the river}}{\text{Perpendicular speed of the swimmer}} = \frac{1000 \text{ m}}{\frac{25}{4} \text{ m/s}} = 1000 \times \frac{4}{25} = 160 \text{ seconds} \]

4. **Calculating the Drift**:
- The drift \( D \) suffered while crossing the river can be calculated using the speed of the river and the time taken to cross:
\[ D = \text{Speed of river} \times T = 5 \text{ m/s} \times 160 \text{ s} = 800 \text{ m} \]

5. **Final Answer**:
- The drift suffered by the man when crossing the river in the least possible time is **800 meters**.