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Q.

A swimmer crosses a flowing stream of width w to and fro in time t₁.The time taken to cover the same distance up and down the stream is t₂. If t₃ is the timethe swimmer would take to swim a distance 2w in still water, then

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a

$t_{2}^{2} = t_{1} t_{3}$

b

$t_{1}^{2} = t_{2} t_{3}$

c

$t_{3} = t_{1} + t_{2}$

d

$t_{3}^{2} = t_{1} t_{2}$

answer is A.

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Detailed Solution

Let v be the river velocity and u the velocity of swimmer in still water.
Then $\begin{array}{l} t_{1} = 2 (\frac{W}{\sqrt{u^{2} - v^{2}}}) \\ t_{2} = \frac{W}{u + v} + \frac{W}{u - v} = \frac{2 u W}{u^{2} - v^{2}} \\ a n d t_{3} = \frac{2 W}{u} \end{array}$
Now we can seen that $t_{1}^{2} = t_{2} t_{3}$

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