The deceleration experienced by a moving motorboat, after its engine is cut – off is given by $\frac{d υ}{d t} = - k υ^{3}$ , where $k$ is constant. If $υ_{0}$ is the magnitude of velocity at cut – off, the magnitude of velocity at time $' t'$ after the cut – off is:

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$\frac{υ_{0}}{2}$

$\frac{υ_{0}}{\sqrt{2 υ_{0}^{2} k t + 1}}$

$υ_{0} e^{- k t}$

$υ_{0}$

answer is D.

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Detailed Solution

$\frac{d υ}{d t} = - k υ^{3}$

$\Rightarrow$ $\int_{υ_{0}}^{υ} \frac{d υ}{υ^{3}} = - \int_{0}^{t} k d t$

$- \frac{1}{2} [\frac{1}{υ^{2}} - \frac{1}{υ_{0}^{2}}] = - k t$

$\Rightarrow$ $υ = \frac{υ_{0}}{\sqrt{2 υ_{0}^{2} k t + 1}}$

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