Question

A swimmer can swim in still water with a speed of $\sqrt{7}\text{m/s}$. While crossing a river, his average speed is $4\text{m/s}$. If he crosses the river in the shortest possible time, what is the speed of flow of water?

• A. $6\text{m/s}$
• B. $8\text{m/s}$
• C. $4\text{m/s}$
• D. $3\text{m/s}$

Answer 1

The correct option is D $3\text{m/s}$

To cross the river by the shortest time, he must apply his energy and effort pointing perpendicular to the river.

Let $\overrightarrow{V_{br}}$ be the velocity of the swimmer with respect to the river.

$\overrightarrow{V_b}$ and $\overrightarrow{V_r}$ are the velocities of swimmer and river w.r.t ground, respectively.


According to question,
$|\overrightarrow{V_{br}}|=\sqrt{7}\text{m/s}$ and $|\overrightarrow{V_b}|=4\text{m/s}$

We know that,

$\overrightarrow{V_{br}}=\overrightarrow{V_b}-\overrightarrow{V_r}$

$\Rightarrow \overrightarrow{V_b}=\overrightarrow{V_{br}}+\overrightarrow{V_r}$

Taking magnitude only,

$|\overrightarrow{V_b}|=\sqrt{V_{br}^2+V_r^2}$

$\Rightarrow V_r^2+7=16$

$\Rightarrow V_r=3\text{m/s}$

Hence, option $\left(a\right)$ is correct.

Alternative:

Use following equation,.

$t=\frac{AB(=d)}{V_{br}}=\frac{BC}{V_r}=\frac{AC}{V_b(=\sqrt{V_{br}^2+V_r^2)}}$

$$\begin{array}{c}\text{Why this Question?}\\ \text{To cross the river in the shortest time,}\\ \text{swimmer must apply this energy and}\\ \text{effort pointing perpendicular to the river.}\\ \text{Although he will suffer drift due to river}\\ \text{flow, but he will manage to cross the}\\ \text{river in shortest time.}\end{array}$$