Question

A man can swim with speed $u$ with respect to river. The width of river is $d$. The speed of the river is zero at the banks and increases linearly to $v_o$ till mid-stream of the river then decreases to zero to the other bank. When man crosses the river in shortest time, drift is equal to: (given $u>v_o$)

Answer 1

Time taken by man to cross the river, length $dx,t=\frac{dx}{u}$

Speed of the river from bank to mid-stream,
$v=\frac{v_o}{\frac{d}{2}}\times x$

$v=\frac{2v_o}{d}x$

So, drift = velocity $\times$ time

$=2{\int }_o^{\frac{d}{2}}\frac{2v_o}{d}x\times \frac{dx}{u}$

$=\frac{4v_0}{ud}{\int }_0^{\frac{d}{2}}xdx$

$=\frac{4v_o}{ud}\times {\left(\frac{x^2}{2}\right)}_0^{\frac{d}{2}}$

$=\frac{4v_0}{2ud}\left(\frac{d^2}{4}\right)$

$=\frac{v_{o}d}{2u}$

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