Question

A locomotive of mass $m$ starts moving so that its velocity varies as $v=\alpha s^{2/3}$, where $\alpha$ is a constant and $s$ is the distance traversed. The total work done by all the forces acting on the locomotive during the first $t$ second after the start of motion is

• A. $\frac{1}{8}m{\alpha }^4{t}^2$
• B. $\frac{m{\alpha }^6{t}^4}{162}$
• C. $\frac{m{\alpha }^6{t}^4}{81}$
• D. $\frac{m{\alpha }^4{t}^2}{162}$

Answer 1

The correct option is B $\frac{m{\alpha }^6{t}^4}{162}$

$v=\alpha s^{2/3}$
$\frac{ds}{dt}=\alpha s^{2/3}$
$\underset{0}{\overset{s}{\int }}\frac{ds}{s^{2/3}}=\underset{0}{\overset{t}{\int }}\alpha dt\Rightarrow 3s^{1/3}=\alpha t$
$s=\frac{{\alpha }^3}{27}{t}^3\Rightarrow ds=\frac{{\alpha }^3}{9}{t}^{2}dt$
$v=\frac{{\alpha }^3}{9}{t}^2$
$W=\frac{1}{2}m{v}^2=\frac{1}{2}m{\left(\frac{{\alpha }^3{t}^2}{9}\right)}^2=\frac{m{\alpha }^6{t}^4}{162}$