A river is flowing from east to west at a speed of `5(m)/(min)`. A man on south bank of river, capable of swimming `(10m)/(min)` ini 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. due north-west

Answer

To solve the problem of the man swimming across the river in the shortest time, we can follow these steps:

### Step 1: Understand the Problem
The river is flowing from east to west at a speed of 5 m/min. The man can swim at a speed of 10 m/min in still water. We need to determine the direction in which 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 the origin (0, 0).
- The river flows along the x-axis (east to west), and the man needs to swim along the y-axis (north).

### Step 3: Determine the Velocity Components
The man's swimming velocity can be broken down into two components:
- \( V_{mx} \): the component of the man's velocity in the x-direction (along the river).
- \( V_{my} \): the component of the man's velocity in the y-direction (across the river).

Given that the man's speed in still water is 10 m/min, we can express these components as:
- \( V_{mx} = 10 \sin(\theta) \)
- \( V_{my} = 10 \cos(\theta) \)

Where \( \theta \) is the angle at which the man swims relative to the north direction.

### Step 4: Find the Effective Velocity Across the River
To cross the river in the shortest time, the man needs to maximize his effective velocity in the y-direction (across the river). The effective velocity in the y-direction is given by:
\[ V_{effective} = V_{my} = 10 \cos(\theta) \]

### Step 5: Analyze the River's Current
The river's current affects the man's position in the x-direction. The river flows at 5 m/min, so the man's total velocity in the x-direction is:
\[ V_{total,x} = V_{mx} - 5 = 10 \sin(\theta) - 5 \]

### Step 6: Determine the Conditions for Shortest Time
To swim directly across the river (shortest time), the man should not be carried downstream. Thus, we set the x-component of his swimming velocity equal to the river's current:
\[ 10 \sin(\theta) = 5 \]
This gives:
\[ \sin(\theta) = \frac{5}{10} = 0.5 \]
Thus, \( \theta = 30^\circ \).

### Step 7: Conclusion
To swim across the river in the shortest time, the man should swim at an angle of \( 30^\circ \) upstream relative to the north direction.

### Final Answer
The man should swim at an angle of \( 30^\circ \) upstream to cross the river in the shortest time.