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 determine two things: the time taken for the man to cross the river and the drift caused by the river's current.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Swimming speed of the man in still water, \( V_m = 4 \, \text{m/s} \)
- Width of the river, \( d = 200 \, \text{m} \)
- River flow velocity, \( V_r = 5 \, \text{m/s} \)
- Angle with respect to the river flow, \( \theta = 60^\circ \)

2. **Calculate the Component of the Man's Swimming Velocity:**
- The man swims at an angle of \( 60^\circ \) to the river flow.
- The vertical component of his swimming velocity (across the river) is given by:
\[ V_{m_y} = V_m \sin \theta = 4 \sin 60^\circ = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3} \, \text{m/s} \]

3. **Calculate the Time Taken to Cross the River:**
- The time taken \( t \) to cross the river can be calculated using the formula:
\[ t = \frac{d}{V_{m_y}} = \frac{200 \, \text{m}}{2\sqrt{3} \, \text{m/s}} = \frac{100}{\sqrt{3}} \, \text{s} \]

4. **Calculate the Drift:**
- The horizontal component of the man's swimming velocity (along the river) is:
\[ V_{m_x} = V_m \cos \theta = 4 \cos 60^\circ = 4 \cdot \frac{1}{2} = 2 \, \text{m/s} \]
- The total effective velocity in the direction of the river flow is:
\[ V_{\text{effective}} = V_r + V_{m_x} = 5 \, \text{m/s} + 2 \, \text{m/s} = 7 \, \text{m/s} \]
- The drift \( d_{\text{drift}} \) can be calculated as:
\[ d_{\text{drift}} = V_{\text{effective}} \cdot t = 7 \, \text{m/s} \cdot \frac{100}{\sqrt{3}} \, \text{s} = \frac{700}{\sqrt{3}} \, \text{m} \]

### Final Answers:
- Time taken to cross the river: \( \frac{100}{\sqrt{3}} \, \text{s} \)
- Drift: \( \frac{700}{\sqrt{3}} \, \text{m} \)