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A locomotive of mass m starts moving so that its velocity varies as $v = K \sqrt{s}$ , where K is a constant and 's' is the distance traversed. The total work done by all the forces acting on the locomotive during the first t second after the start of motion is

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$\frac{1}{16} {mK}^{4} t^{2}$

$\frac{1}{8} {mK}^{4} t^{2}$

$\frac{1}{2} {mK}^{4} t^{2}$

$\frac{1}{4} {mK}^{4} t^{2}$

answer is C.

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Detailed Solution

$a = \frac{dv}{dt} = \frac{K}{2 \sqrt{s}} \frac{ds}{dt} = \frac{K}{2 \sqrt{s}} K \sqrt{s} = \frac{K^{2}}{2}$

So the acceleration is constant.

Gain in kinetic energy is equal to $the work done, W = Fs = \frac{{mK}^{2}}{2} \frac{(K^{2} t^{2})}{4} = \frac{{mK}^{4} t^{2}}{8}$

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