A man desires to swim across the river in shortest time. The velcoity 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 determining the angle at which the man should swim to cross the river in the shortest time, we can follow these steps:

### Step 1: Understand the Problem
The man wants to swim across a river that has a current. The velocity of the river (Vr) is given as 3 km/h, and the man's swimming speed in still water (Vm) is 6 km/h. We need to find the angle (θ) at which he should swim relative to the flow of the river to minimize his crossing time.

**Hint:** Visualize the river flow and the swimmer's path to understand the components of velocity.

### Step 2: Set Up the Velocity Components
When the man swims at an angle θ with respect to the direction of the river flow, his velocity can be broken down into two components:
- The component of his swimming velocity in the direction across the river: \( V_{m \perp} = V_m \sin(\theta) \)
- The component of his swimming velocity in the direction of the river flow: \( V_{m \parallel} = V_m \cos(\theta) \)

**Hint:** Remember that the total velocity in the direction of the river flow is the sum of the swimmer's velocity component and the river's velocity.

### Step 3: Determine the Effective Velocity Across the River
To cross the river, the effective velocity across the river (perpendicular to the flow) is given by:
\[ V_{effective} = V_m \sin(\theta) \]

**Hint:** The time taken to cross the river depends on how fast he can swim across, which is determined by this effective velocity.

### Step 4: Calculate the Time to Cross the River
The time taken (T) to cross the river of width d is given by:
\[ T = \frac{d}{V_{effective}} = \frac{d}{V_m \sin(\theta)} \]

**Hint:** To minimize the time, we need to maximize \( V_{effective} \).

### Step 5: Maximize the Effective Velocity
To minimize time, \( V_{effective} \) must be maximized. The maximum value of \( \sin(\theta) \) is 1, which occurs when \( \theta = 90^\circ \). Thus, the swimmer should swim directly across the river.

**Hint:** Consider the implications of swimming at 90 degrees; he will not be affected by the current in terms of crossing time.

### Step 6: Conclusion
The angle θ at which the man should swim to cross the river in the shortest time is:
\[ \theta = 90^\circ \]

**Final Answer:** The required angle is \( 90^\circ \).

### Summary of Steps:
1. Understand the problem and the velocities involved.
2. Break down the swimmer's velocity into components.
3. Determine the effective velocity across the river.
4. Calculate the time to cross the river.
5. Maximize the effective velocity to minimize time.
6. Conclude that the angle should be \( 90^\circ \).