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 $v_0$ till mid stream of the river, then decreases to zero at the other bank. When man crosses the river in shortest time, drift is equal to :
(given $u>v_0$)

• A. $\frac{ud}{v_0}$
• B. $\frac{ud}{v_0}$
• C. $\frac{v_{0}d}{2u}$
• D. $\frac{v_{0}d}{u}$

Answer 1

The correct option is C $\frac{v_{0}d}{2u}$

Speed of river at a distance $x$ from bank to mid stream
$v=\frac{v_0}{\frac{d}{2}}x=\frac{2v_{0}x}{d}$
${\int }_0^{\frac{D}{2}}d\left(\text{drift}\right)={\int }_0^{\frac{d}{2}}\frac{dx}{u}\times \frac{2v_{0}x}{d}$
$\frac{D}{2}=\frac{2v_0}{du}\times {\left[\frac{x^2}{2}\right]}_0^{\frac{d}{2}}$
$D=\frac{4v_0}{du}\times \frac{d^2}{8}$
drift $=\frac{v_{0}d}{2u}$

Why this question? Concept: This is a classic example of velocity of river varying along the width and reinforces on the power of integration.