Question

A locomotive of mass $m$ starts moving so that its velocity varies according to the law $v=\alpha \sqrt{s}$, where $\alpha$ is constant and $s$ is the distance covered. Find the total work done by all the forces acting on the locomotive during the first $t$ seconds after the beginning of motion.

Answer 1

The velocity is given as,

$v=\alpha \sqrt{s}$

$\frac{ds}{dt}=\alpha \sqrt{s}$

${\int }_0^x\frac{ds}{\alpha \sqrt{s}}={\int }_0^{t}dt$

$\frac{2\sqrt{s}}{\alpha }=t$

$s=\frac{{\alpha }^2{t}^2}{4}$

The acceleration is given as,

$a=v\frac{dv}{ds}$

$=\alpha \sqrt{s}\times \frac{\alpha }{2\sqrt{s}}$

$=\frac{{\alpha }^2}{2}$

The total work done is given as,

$W=m\times a\times s$

$=m\times \frac{{\alpha }^2}{2}\times \frac{{\alpha }^2{t}^2}{4}$

$=\frac{m{\alpha }^4{t}^2}{8}$

Thus, the total work done by all forces acting on the locomotive is $\frac{m{\alpha }^4{t}^2}{8}$.