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}.

(a) $\displaystyle t_1 = \sqrt{t_2 t_3} $

(b) $ \displaystyle t_1 = \sqrt{t_2 / t_3} $

(c) $ \displaystyle t_1 = \sqrt{t_3 / t_2} $

(d) $ \displaystyle t_1 = t_3 t_2 $

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Sol: Let u = velocity of swimmer in still water

v = velocity of river flow

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

$ \displaystyle t_2 = \frac{d}{u+v}+\frac{d}{u-v} $ …(ii)

$\displaystyle t_3 = \frac{2 d}{u} $ …(iii)

On multiplying (ii) and (iii) we get

$ \displaystyle t_2 \times t_3 = t_1^2 $

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