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Apparent Depth

A swimmer cro...

Question

A swimmer crosses a flowing stream of width $^{'} ω^{'}$ to and fro in time $t_{1}$ . The time taken to cover the same distance up & down the stream is $t_{2}$ . if $t_{3}$ is the time swimmer would take to swim a distance $2 ω$ in still water, then:

A

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

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B

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

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C

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

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D

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

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Solution

The correct option is A $t_{1}^{2} = t_{2} t_{3}$
Let 'u' be the velocity of swimmer in still water and 'v' be velocity of river.

in case 1
Since v and u are perpendicular to each other.
$t_{1} = 2 (\frac{w}{\sqrt{u^{2} - v^{2}}})$

In case 2nd
The swimmer goes upstream and downstream.
$t_{2} = \frac{w}{u + t} + \frac{w}{u - t} = \frac{2 u w}{u^{2} - v^{2}}$

In case 3rd
Still water
$t_{3} = \frac{2 w}{4}$
We see $t_{1}^{2} = t_{2} t_{3}$

Hence option (A) is correct

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