Q: A swimmer crosses a flowing stream of width ‘d’ to and fro normal to the flow of the river in time t1. The time taken to cover the same distance up and down the stream is t2 . If t3 is the time the swimmer would take to swim a distance 2d in still water, then relation between t1 , t2 & t3 .
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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)
$\large t_1^2 = t_2 t_3$
$\large t_1 = \sqrt{t_2 t_3}$