Q: A swimmer crosses a flowing stream of width ‘d’ to and fro normal to the flow of the river in time t_{1}. The time taken to cover the same distance up and down the stream is t_{2} . If t_{3} is the time the swimmer would take to swim a distance 2d in still water, then relation between t_{1} , t_{2} & t_{3} .

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Sol. Let v be the river velocity and u be the velocity of swimmer in still water. Then

$\large t_1 = 2 (\frac{d}{\sqrt{u^2 – v^2}}) $ …(i)

$\large t_2 = \frac{d}{u+v} + \frac{d}{u-v} = \frac{2ud}{u^2 -v^2}$ …(ii)

and , $\large t_3 = \frac{2d}{u}$ …(iii)

From equation (i), (ii) and (iii)

$\large t_1^2 = t_2 t_3$

$\large t_1 = \sqrt{t_2 t_3}$