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 $W=\frac{1}{8}m{k}^4{t}^2$

W.D by all forces $=$ change in KE.
Since $v=k\sqrt{S}$
Hence $v^2=k^{2}S$
$2v\frac{dv}{ds}=k^2$
$\frac{dv}{dt}=\frac{k^2}{2}\Rightarrow dv=\frac{k^2}{2}dt$
$\Rightarrow v=\frac{k^2}{2}t$
W.D. by all forces $=\Delta KE=\frac{1}{2}m{v}^2$
$=\frac{1}{2}m{\left(\frac{k^2}{2}t\right)}^2=\frac{1}{8}m{k}^4{t}^2$