Question

A swimmer wants to cross a river which is flowing at a speed 'v'. If the swimmer can swim in still water at speed 'V', the direction he should swim to cross the river in least time is

• A. Along the flow of river
• B. Opposite to the flow of river
• C. Perpendicular to the flow of river
• D. $45^\circ$ to the flow of river

Hint:

River Boat Strategy:

• Least time $\Rightarrow$ swim perpendicular to flow.
• Shortest path (zero drift) $\Rightarrow$ swim at angle $\theta$, such that $V\sin\theta = v$.

Answer 1

The Correct Option is C

To minimize crossing time, swimmer must maximize velocity component perpendicular to river: $V_\perp = V$.
This happens when swimming directly across, i.e., perpendicular to river flow.
Time to cross: $t = \frac{W}{V}$ (minimum).
Downstream drift will occur, but crossing time is minimized.

Answer 2

Step 1: Understand the situation
The river flows with speed \(v\) and the swimmer can swim with speed \(V\) in still water.
The swimmer wants to cross the river in the least time possible.

Step 2: Analyze swimming directions
- If the swimmer swims directly across the river (perpendicular to the flow), the velocity component across the river is maximum.
- Swimming at any angle other than perpendicular increases the path length, thereby increasing the crossing time.

Step 3: Reasoning for least time
The time taken to cross the river depends on the component of the swimmer's velocity perpendicular to the river flow.
- Time \(t = \frac{\text{width of river}}{\text{velocity component perpendicular to flow}}\).
- To minimize time, the swimmer should maximize this perpendicular component.

Step 4: Conclusion
Hence, the swimmer should swim perpendicular to the flow of the river to cross in the least time.
Swimming at any other angle will increase the time taken due to reduced effective speed across the river.