The magnitude of velocity of water current in a river of width ‘d’ varies across the river as $v = \frac{u_{0} y}{d} (0 \leq y \leq d),$ where $u_{0}$ is the velocity of man in still water. The direction of water current is along + x-axis. The time taken by the man to reach from point ‘O’ to ‘B’ is

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$\frac{π d}{u_{0}}$

$\frac{d}{u_{0}}$

$\frac{d}{2 u_{0}}$

$\frac{π d}{2 u_{0}}$

answer is B.

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Detailed Solution

To reach point B, at any point $u_{0} \sin α = v_{r}$

$u_{0} \sin α = \frac{u_{0} y}{d}$

Time taken to travel a distance dy along OB is $d t = \frac{d y}{u_{0} \cos α}$

$d t = \frac{d y}{u_{0} \sqrt{1 - \frac{y^{2}}{d^{2}}}}$

Total time $t = \int_{0}^{t} d t = \frac{d}{u_{0}} \int_{0}^{d} \frac{d y}{\sqrt{d^{2} - y^{2}}}$

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