A man can swim in still water with speed u. A river of width d flows with speed v (<1) per cent relative to the ground. A man is at a point P on one bank of the river and he swims to a point opposite to the direction of the river on the other bank of the river and swims back to point P. If the banks are straight and parallel and PQ forms an angle a (0 < a < 1/2) with the direction opposite to the direction of the river, then the velocity triangles of the relative velocities are in the same diagram.

Draw the total time it takes the man to swim to ) and return to P

{2 d√(u² cosec² α -v^2)}/u^2 - v^2

show Declare (i.e. the value of ) the total time to be minimum when ) is located on the bank of the point

To solve this problem, let's break down the scenario and derive the total time it takes for the man to swim from point \( P \) to point \( Q \) on the opposite bank and then return to point \( P \).

### Problem Breakdown

1. **Given:**
   - Speed of the man in still water: \( u \)
   - Speed of the river: \( v \) where \( v \) is a small percentage compared to the ground (\(< 1\%\)).
   - Width of the river: \( d \)
   - Angle between the direction opposite to the flow of the river and line \( PQ \): \( \alpha \) where \( 0 < \alpha < \frac{1}{2} \).

2. **Objective:**
   - Determine the time taken to swim from \( P \) to \( Q \) and back to \( P \).
   - Show that the total time is minimized when \( Q \) is directly across from \( P \) (i.e., when \( \alpha = 0 \)).

### Time Calculation

#### 1. Time from \( P \) to \( Q \):

The velocity of the man relative to the ground will have two components:
- **Across the river (perpendicular to the river flow):**
  - \( u \cos(\alpha) \)
- **Along the river (parallel to the river flow):**
  - \( u \sin(\alpha) - v \) (where \( v \) opposes the man's velocity along the river).

The time to cross from \( P \) to \( Q \) is given by the distance divided by the effective speed:
\[ t_1 = \frac{d}{\sqrt{u^2 \cos^2(\alpha) - v^2}} \]

#### 2. Time from \( Q \) to \( P \):

For the return journey from \( Q \) to \( P \), the man's effective velocity will change because the direction of the river's current is now aligned with the man's swimming direction. The time is:
\[ t_2 = \frac{d}{\sqrt{u^2 \cos^2(\alpha) - v^2}} \]

#### 3. Total Time:

Thus, the total time \( T \) for the round trip is:
\[ T = t_1 + t_2 = \frac{2d}{\sqrt{u^2 \cos^2(\alpha) - v^2}} \]

### Minimizing the Total Time

To minimize the total time, we need to find the value of \( \alpha \) that minimizes \( T \). Let's analyze the expression for \( T \):

\[ T = \frac{2d}{\sqrt{u^2 \cos^2(\alpha) - v^2}} \]

The minimum time occurs when \( \alpha = 0 \) (since \( \cos(\alpha) \) is maximum when \( \alpha = 0 \)). At this point, the swimmer heads directly perpendicular to the current, and the time \( T \) is minimized.

### Final Expression

When \( \alpha = 0 \):
\[ T_{\text{min}} = \frac{2d}{\sqrt{u^2 - v^2}} \]

This represents the minimum time required to complete the round trip from \( P \) to \( Q \) and back to \( P \) when \( Q \) is directly opposite \( P \) on the other bank.

### Conclusion

The total time for the round trip is minimized when the man swims directly across the river and back, i.e., when \( \alpha = 0 \).

Would you like to go through the derivation step-by-step, or do you have any specific questions?

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Here are 5 related questions:

1. How would the total time change if the river's speed \( v \) was more significant?
2. What if the swimmer aimed at an angle other than \( \alpha = 0 \)? How does the trajectory affect the time?
3. Can you determine the swimmer's drift downstream during the crossing?
4. What is the physical significance of the angle \( \alpha \) in minimizing time?
5. How does the formula change if the river's speed \( v \) is greater than \( u \)?

**Tip:** When solving problems involving relative velocities, breaking down the velocities into components and carefully analyzing their contributions is crucial for accuracy.

Explore how to calculate the optimal swim time across a river using relative velocities and trigonometric principles. Understand how the angle \( \alpha \) affects the total time taken. Suitable for advanced high school or college-level mathematics.

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