Question

A locomotive of mass $m$ starts moving so that its velocity varies according to the law $v=a\sqrt{s}$, where $a$ is a constant, and $s$ is the distance covered. The total work performed by all the forces which are acting on the locomotive during the first $t$ seconds after the beginning of motion is $W=\frac{m{a}^4{t}^2}{x}$. Find $x$.

Answer 1

The acceleration $w$ is obtained by differentiating $v$ with respect to time.
$w=\frac{dv}{dt}=\frac{a}{2\sqrt{s}}\frac{ds}{dt}=\frac{a}{2\sqrt{s}}a\sqrt{s}=\frac{a^2}{2}$ (As locomotive is in unidirectional motion)
Hence force acting on the locomotive $F=mw=\frac{m{a}^2}{2}$
Let, $v=0$ at $t=0$ then the distance covered during the first $t$ seconds
$s=\frac{1}{2}w{t}^2=\frac{1}{2}\frac{a^2}{2}{t}^2=\frac{a^2}{4}{t}^2$
Hence the work, $W=Fs=\frac{m{a}^2}{2}\frac{\left(a^2{t}^2\right)}{4}=\frac{m{a}^4{t}^2}{8}$

So, $x=8$