A man can swim at 4 m/s in a still water swimming pool. He enters a 200 m wide river at one bank and swims ( w.r.t water) at an angle of `60^(@)` to the river flow velocity. The river flow velocity is 5 m/s . In how much -time does he cross the river ? Calculate his drift.

Options

A. ` 200/sqrt 3 s, 500/sqrt3 m`
B. ` 100/sqrt3 s, 700/sqrt3 m`
C. 50s, 1000 m
D. `1000/sqrt3 s, 200/sqrt3 m`

Answer

To solve the problem, we need to analyze the swimmer's motion in the river, taking into account both his swimming speed and the river's flow speed. Here’s a step-by-step solution:

### Step 1: Understand the components of motion
The swimmer swims at an angle of 60° to the direction of the river flow. We need to break down his swimming speed into two components:
- One component perpendicular to the river flow (across the river).
- One component parallel to the river flow (downstream).

### Step 2: Calculate the swimmer's speed components
The swimmer's speed in still water is 4 m/s. We can find the components of his swimming speed:
- Perpendicular component (across the river):
\[ V_{y} = V \cdot \sin(60°) = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3} \, \text{m/s} \]
- Parallel component (downstream):
\[ V_{x} = V \cdot \cos(60°) = 4 \cdot \frac{1}{2} = 2 \, \text{m/s} \]

### Step 3: Determine the time to cross the river
The width of the river is 200 m. To find the time taken to cross the river, we use the perpendicular component of the swimmer's speed:
\[ \text{Time} = \frac{\text{Width of the river}}{\text{Perpendicular speed}} = \frac{200 \, \text{m}}{2\sqrt{3} \, \text{m/s}} = \frac{200}{2\sqrt{3}} \approx 57.74 \, \text{s} \]

### Step 4: Calculate the drift
While the swimmer is crossing the river, he is also being carried downstream by the river's flow. The river's flow speed is 5 m/s. The distance drifted downstream can be calculated as:
\[ \text{Drift} = \text{River speed} \times \text{Time} = 5 \, \text{m/s} \times 57.74 \, \text{s} \approx 288.7 \, \text{m} \]

### Final Answers
- Time taken to cross the river: **approximately 57.74 seconds**
- Drift downstream: **approximately 288.7 meters**