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. Find the total work performed by all the forces which are acting on the locomotive during the first $t$ seconds after the beginning of motion.

Answer 1

hence force acting on the locomotive $F=mw=\frac{m{a}^2}{2}$
Let $at=\nu =0$ 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 sought work, $A=Fs=\frac{m{a}^2}{2}\frac{\left(a^2{t}^2\right)}{4}=\frac{m{a}^3{t}^2}{8}$