Question

A locomotive of mass $m$ starts moving so that its velocity varies according to the law $v=k\sqrt{s}$ where $k$ is 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.

• A. $W=\frac{1}{8}m{k}^4{t}^2$
• B. $W=\frac{1}{4}{m}^2{k}^4{t}^2$
• C. $W=\frac{1}{4}m{k}^4{t}^4$
• D. $W=\frac{1}{8}m{k}^4{t}^4$

Answer 1

The correct option is A $W=\frac{1}{8}m{k}^4{t}^2$

Given $v=k\sqrt{s}$
differentiating above equation
$\frac{dv}{dt}=\frac{k}{2\sqrt{s}}.\frac{ds}{dt}$
$a=\frac{k^2}{2}$
$F=ma=\frac{m{k}^2}{2}$

(Distanced covered in $t$ seconds)
$\frac{ds}{dt}=v,{\int }_0^x\frac{1}{v}.ds={\int }_0^{t}dt{\int }_0^x\frac{1}{k\sqrt{s}}.ds={\int }_0^{t}dt\frac{2\sqrt{x}}{k}=t\Rightarrow x=\frac{t^2{k}^2}{4}$

Workdone during $t$ seconds is
$\int dW={\int }_0^{x}F.ds=\frac{m{k}^2}{2}{\int }_0^{\frac{t^2{k}^2}{4}}ds=\frac{m{k}^4{t}^2}{8}$