A man desires to swim across the river in shortest time. The velocity of river water is ` 3 km h^(-1)` . He can swim in still water at ` 6 km h^(-1)` . At what angle with the velocity of flow of the river should he swim ?

Options

A. `30^(@)`
B. `60^(@)`
C. `90^(@)`
D. 120

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
We have a river with a velocity \( V_r = 3 \, \text{km/h} \) and a man who can swim in still water with a velocity \( V_m = 6 \, \text{km/h} \). We need to find the angle \( \alpha \) at which he should swim to minimize the time taken to cross the river.

**Hint:** Visualize the scenario with a diagram showing the river's flow and the swimmer's path.

### Step 2: Set Up the Velocity Components
When the man swims at an angle \( \alpha \) to the flow of the river, his velocity can be broken down into two components:
- The component perpendicular to the river flow: \( V_m \sin \alpha \)
- The component parallel to the river flow: \( V_m \cos \alpha \)

**Hint:** Remember that the swimmer's effective velocity across the river is determined by the vertical component of his swimming velocity.

### Step 3: Determine the Time to Cross the River
Let \( d \) be the width of the river. The time \( t \) taken to cross the river can be expressed as:
\[ t = \frac{d}{V_m \sin \alpha} \]
To minimize time, we need to maximize \( V_m \sin \alpha \).

**Hint:** The sine function reaches its maximum value of 1 when \( \alpha = 90^\circ \).

### Step 4: Analyze the Condition for Minimum Time
To achieve the minimum time, we want \( V_m \sin \alpha \) to be as large as possible. The maximum value of \( \sin \alpha \) is 1, which occurs when \( \alpha = 90^\circ \). This means the swimmer should swim directly across the river.

**Hint:** Think about how swimming directly across (perpendicular to the flow) affects the time taken to cross.

### Step 5: Conclusion
Thus, the angle \( \alpha \) at which the man should swim to minimize the time taken to cross the river is:
\[ \alpha = 90^\circ \]

**Final Answer:** The man should swim at an angle of \( 90^\circ \) with respect to the velocity of the river flow.