Animated Solution for Physics - Kinematics: A river is flowing from West to East at a speed of $5\text{ m/min}$. A man on the South bank of the river, capable of swimming at $10\text{ m/min}$ in still water, wants to swim across the river in the shortest time. He should swim in a direction

The problem of crossing a river in the shortest possible time is a classic test of your understanding of vector independence. It often tricks students into overthinking the effect of the river's current. Let's break down the physics step-by-step.

Imagine you are standing on the South bank of a river, looking across to the North bank. The river is flowing steadily from West to East with a velocity of $5\text{ m/min}$. You are a capable swimmer, able to maintain a speed of $10\text{ m/min}$ in perfectly still water.

Your objective is simple: reach the opposite bank in the absolute minimum amount of time.

The most crucial conceptual leap in this problem is realizing that motion in perpendicular directions is completely independent.

The river's current only flows horizontally (West to East). It has absolutely zero component in the vertical direction (South to North). Therefore, the river's flow cannot help you cross the river faster, nor can it slow down your crossing time. It can only push you downstream.

To understand how to minimize your time, let's assume you dive into the water at an arbitrary angle $\theta$ with respect to the due North direction.

Your swimming velocity, $v_m$, can be resolved into two perpendicular components: 1. The component parallel to the river flow: $v_m\sin \theta$ 2. The component perpendicular to the river flow: $v_m\cos \theta$

Only the perpendicular component, $v_m\cos \theta$, is responsible for closing the distance between the South and North banks.

If we let the width of the river be $d$, the time $t$ taken to cross is simply the distance divided by the velocity component in that direction:

Now, we look at our time equation. The width of the river $d$ is a constant, and your swimming speed $v_m$ is also a constant. The only variable you control is the angle $\theta$.

To make the time $t$ as small as possible, we must make the denominator of our fraction as large as possible.

We need to maximize the value of $\cos \theta$. From trigonometry, we know that the maximum possible value of the cosine function is $1$, and this occurs when the angle is $0^{\circ }$.

Setting $\theta =0^{\circ }$ means you should not angle your swim at all. You must aim your body directly across the river, due North.

By doing so, you dedicate 100% of your swimming effort ($10\text{ m/min}$) to crossing the river, ensuring you reach the other side in the shortest possible time. While the river will certainly carry you downstream (you will drift East), your time spent in the water will be minimized.