A river is flowing with a velocity of 2m/s. If the width of river in 100 m and swimmer wants to cross river is shortest time, what time (in sec) would he take if velocity of swimmer in still water is 4 m/s ?

Explanation

To solve the problem, we need to determine the time it takes for a swimmer to cross a river of width 100 meters while considering the river's current. The swimmer's velocity in still water is 4 m/s, and the river's current flows at 2 m/s.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The river flows with a velocity \( v_r = 2 \, \text{m/s} \).
- The swimmer's velocity in still water is \( v_s = 4 \, \text{m/s} \).
- The width of the river is \( W = 100 \, \text{m} \).

2. **Determine the Direction of Swimming**:
- To cross the river in the shortest time, the swimmer should swim at an angle such that his velocity component perpendicular to the river's flow is maximized. This means he should swim directly across the river while allowing the current to carry him downstream.

3. **Using Vector Components**:
- The swimmer's velocity can be broken down into two components:
- \( v_{s_y} \): The component of the swimmer's velocity in the direction across the river (perpendicular to the flow).
- \( v_{s_x} \): The component of the swimmer's velocity in the direction of the river's flow (parallel to the flow).
- For the swimmer to cross directly, he must swim at an angle such that \( v_{s_y} = v_s \sin(\theta) \) and \( v_{s_x} = v_s \cos(\theta) \).

4. **Finding the Time to Cross**:
- The time taken to cross the river can be calculated using the formula:
\[ t = \frac{W}{v_{s_y}} \]
- Since \( v_{s_y} \) must equal the swimmer's speed in the perpendicular direction, we can set \( v_{s_y} = 4 \, \text{m/s} \) (the full speed of the swimmer since he is swimming directly across).

5. **Calculating the Time**:
- Plugging in the values:
\[ t = \frac{100 \, \text{m}}{4 \, \text{m/s}} = 25 \, \text{s} \]

### Final Answer:
The time taken for the swimmer to cross the river is **25 seconds**.

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