Question

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

• A. $20m/min$
• B. $12m/min$
• C. $10m/min$
• D. $8m/min$

Answer 1

The correct option is A $20m/min$

Let $L$ be the width of the river $u$ be the speed of river and $v$ be the speed of man w.r.t water.
For shortest time path, $t=10=\frac{L}{v}...\left(i\right)$
For shortest distance path,



$t=12.5=\frac{L}{\sqrt{v^2-u^2}}=\frac{L}{v\sqrt{1-u^2/v^2}}...\left(ii\right)$

Give Drift $=120m$ in $12min$. So, river velocity $u=\frac{120}{12}=10m/min$

Dividing $\left(i\right)$ by $\left(ii\right)$,
$\frac{10}{12.5}=\frac{L}{v}\times \frac{v\sqrt{1-u^2/v^2}}{L}$
$\Rightarrow \frac{4}{5}=\sqrt{1-\frac{{12}^2}{v^2}}$
$\Rightarrow \frac{16}{25}=1-\frac{{12}^2}{v^2}$
$\Rightarrow \frac{{12}^2}{v^2}=1-\frac{16}{25}=\frac{9}{25}$
$\Rightarrow \frac{12}{v}=\frac{3}{5}$
$\Rightarrow v=\frac{12\times 5}{3}=20m/min$