A locomotive of mass m in starts moving so that its velocity varies according to the law $v = α \sqrt{s}$ where $α$ is constant and s is the distance covered. The total work done by all forces on locomotive during first ‘t’ sec after beginning of motion is $\frac{m α^{p} t^{q}}{8}$ here $\frac{p}{q}$ value is

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Detailed Solution

$V = α s^{1 / 2}$
$a = \frac{d v}{d t} = V . \frac{d v}{d s} = \frac{1}{2} α s^{- 1 / 2} . α s^{1 / 2} = \frac{α^{2}}{2}$

$F = m a = \frac{m α^{2}}{2}$
$s = u t + \frac{1}{2} a t^{2} = \frac{α^{2} t^{2}}{4}$
Work $= \frac{m α^{2}}{2} \times \frac{α^{2} t^{2}}{4}$
$= \frac{m α^{4} t^{2}}{8}$

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