Question

A man wishes to cross a river in a boat. If he crosses the river in minimum time he takes $10\text{minutes}$ with a drift of $120\text{m}$. If he crosses the river taking the shortest route, he takes $12.5\text{minutes}$, find the velocity of the boat with respect to water.

• A. $12m/min$
• B. $20m/min$
• C. $5m/min$
• D. $31m/min$

Answer 1

The correct option is B

$20\text{m/min}$

Case $-1:$ If a man crosses the river in minimum time he should move perpendicular to the bank or normal to the direction of water flow.

Man crossing river taking minimum time.

Let $|{\overrightarrow{v}}_{mw}|=v;|{\overrightarrow{v}}_w|=u$

Time to cross the river

$t_1=10=\frac{d}{v}\Rightarrow d=10v$

Drift,
$x=u{t}_1\Rightarrow 120=u\times 10$

$\Rightarrow u=12\text{m/min}$

Case $-2:$
If a man crosses the river taking the shortest route, the drift should be zero.

Man crossing river taking shortest route

Time to cross river $t_2=12.5=\frac{d}{\sqrt{v^2-u^2}}$

From case$-1$:

$d=10v$
and $u=12\text{m/min}$;

$12.5=\frac{10v}{\sqrt{v^2-{12}^2}}\Rightarrow v=20\text{m/min}$

Hence, option $\left(\text{b}\right)$ is correct.