A body of mass travels in a straight line with a velocity $v=kx^{3/2}$ where $k$ is a constant. The work done in displacing the body from $x=0$ to $x$ is proportional to:

Given,

Velocity, $\dfrac{dx}{dt}=v=k{{x}^{3/2}}$

Acceleration,

$a=\dfrac{dv}{dt}=\dfrac{d\left( k{{x}^{3/2}} \right)}{dt}=\dfrac{1}{2}k{{x}^{1/2}}\dfrac{dx}{dt}=\dfrac{1}{2}k{{x}^{1/2}}v$

$a=\dfrac{1}{2}k{{x}^{1/2}}\left( k{{x}^{3/2}} \right)=\dfrac{1}{2}{{k}^{2}}{{x}^{2}}$

$work\,done\,=\,force\times displacemet\,=Mass\times acceleration\times displacement$

$\Rightarrow dW=m.a\,.dx=m.\dfrac{1}{2}{{k}^{2}}{{x}^{2}}dx$

On integrating

$W=m\dfrac{1}{2}{{k}^{2}}\left( \dfrac{{{x}^{3}}}{3} \right)$

Hence, Work done is proportional to ${{x}^{3}}$