A man wants to reach point B on the opposite bank of a river flowing at a speed as shown in figure. What minimum speed relative to water should the man have so that he can reach point B? In which direction should he swim?

Explanation

To solve the problem of a man wanting to reach point B on the opposite bank of a river flowing at a certain speed, we can break down the solution into several steps.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- The man is swimming across a river that has a current. We need to determine the minimum speed he must swim relative to the water to reach point B directly across from his starting point.

2. **Define Variables:**
- Let \( V \) be the speed of the river (current).
- Let \( U \) be the speed of the man relative to the water.
- The man needs to swim at an angle \( \theta \) to counteract the current.

3. **Break Down the Velocities:**
- The man's velocity can be broken down into two components:
- \( U \cos \theta \) (horizontal component, against the current)
- \( U \sin \theta \) (vertical component, across the river)

4. **Set Up the Equation:**
- To reach point B directly, the horizontal component of the man's velocity must equal the speed of the river:
\[ U \cos \theta = V \]
- The vertical component must allow him to swim across the width of the river:
\[ U \sin \theta = \text{Width of the river} / t \]
- Here, \( t \) is the time taken to cross the river.

5. **Relate Time to Velocity:**
- The time \( t \) it takes to cross the river can be expressed as:
\[ t = \frac{\text{Width of the river}}{U \sin \theta} \]

6. **Combine the Equations:**
- From the first equation, we can express \( U \):
\[ U = \frac{V}{\cos \theta} \]
- Substitute this into the second equation:
\[ \frac{V}{\cos \theta} \sin \theta = \text{Width of the river} / t \]
- Rearranging gives:
\[ t = \frac{\text{Width of the river} \cos \theta}{V \sin \theta} \]

7. **Minimize the Speed:**
- To find the minimum speed \( U \), we need to maximize \( \sin \theta + \cos \theta \). The maximum value occurs when \( \theta = 45^\circ \):
\[ \sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}} \]
- Therefore, the minimum speed \( U \) can be expressed as:
\[ U = \frac{V}{\cos 45^\circ} = V \sqrt{2} \]

8. **Conclusion:**
- The minimum speed relative to water that the man should have to reach point B is \( U = V \sqrt{2} \).
- The direction in which he should swim is at an angle of \( 45^\circ \) to the bank of the river.

### Final Answer:
- Minimum speed \( U = V \sqrt{2} \)
- Direction: \( 45^\circ \) to the bank of the river.