A river is flowing from east to west at a speed of `5m//min`. A man on south bank of river, capable of swimming `10m//min` in still water, wants to swim across the river in shortest time. He should swim

Options

A. Due north
B. Due north -east
C. Due north-east with double the speed of river
D. None of these

Answer

To solve the problem, we need to determine the direction in which the man should swim to cross the river in the shortest time. Here’s a step-by-step solution:

### Step 1: Understand the Problem
The river flows from east to west with a speed of 5 m/min. The man can swim at a speed of 10 m/min in still water. We need to find the direction he should swim to cross the river in the shortest time.

### Step 2: Set Up the Coordinate System
Let's set up a coordinate system:
- Let the south bank of the river be at point A (where the man starts).
- Let the north bank of the river be at point B (where the man wants to reach).
- The river flows horizontally from east (point C) to west (point D).

### Step 3: Analyze the Swimming Direction
To minimize the time taken to cross the river, the man should swim in such a way that he compensates for the downstream current of the river. If he swims directly north, he will be carried downstream by the river's current.

### Step 4: Determine the Components of Velocity
Let’s denote:
- The speed of the river (u) = 5 m/min (westward).
- The speed of the man in still water (V) = 10 m/min.

If the man swims at an angle θ to the north, his velocity can be broken down into two components:
- Vertical component (towards north): \( V \cos(\theta) \)
- Horizontal component (towards east): \( V \sin(\theta) \)

### Step 5: Calculate the Time to Cross the River
The time taken to cross the river can be expressed as:
\[ t = \frac{d}{V \cos(\theta)} \]
where \( d \) is the width of the river.

### Step 6: Minimize the Time
To minimize the time \( t \), we need to maximize \( V \cos(\theta) \). The maximum value of \( \cos(\theta) \) is 1, which occurs when \( \theta = 0 \). This means the man should swim directly north.

### Step 7: Conclusion
Thus, the man should swim directly north (0 degrees to the vertical) to cross the river in the shortest time.

### Final Answer
The man should swim **directly north**.

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