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A river is flowing with a variable flow rate given by $v= (v_{0} +αx)$ . A swimmer starts from origin having speed v with respect to water wants to cross the river in minimum time. Then drift of swimmer along x – axis would be
(width of river is L)

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$(\frac{v_{0} Lα}{V})$

$\frac{v_{0}}{α} e^{\frac{Lα}{v}}$

$\frac{v_{0}}{α} [e^{\frac{Lα}{v}} -1]$

$In (\frac{v_{0}}{α} \frac{L}{V})$

answer is A.

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Detailed Solution

$\vec{V} s n = (V_{0} + α x) \hat{i} + v j$
time taken, $t= \frac{L}{V}$
Now, $\frac{d x}{d t} = (v_{0} + α x) \Rightarrow \int_{0}^{x} \frac{d x}{v_{0} + α x} = \int_{0}^{t} d t$
$x = \frac{v_{0}}{α} [e^{\frac{L α}{v}} - 1]$

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