Question

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

Answer 1

Case 1: If the man crosses the river in minimum time he should move perpendicular to bank or normal to the direction of water flow.
Let $|{\overrightarrow{v}}_{m.w}|=v$ and $|{\overrightarrow{v}}_w|=u$
Time to cross river, $t_1=10=\frac{d}{v}\Rightarrow d=10v,$
$x=u{t}_1\Rightarrow 120u=u\times 10\Rightarrow u=12mmi{n}^{-1}$
Case 2: If the man crosses river taking the shortest route, the drift should be zero. Time to cross river,
$t_2=12.5=\frac{d}{\sqrt{v^2-u^2}}$
From case 1, $d=10v$ and $v=12m{s}^{-1}$
$12.5=\frac{10v}{\sqrt{v^2-{12}^2}}\Rightarrow v=20m{s}^{-1}$