A body is moved along a straight line by a machine delivering constant power the distance moved by the body in time “t” proportional to

Power $\mathrm{p}=\mathrm{constant}$  

$\Rightarrow \mathrm{F}\times \mathrm{V}=\mathrm{constant}$

$m\frac{dv}{dt}v=k$

$\mathrm{vdv}=\frac{\mathrm{k}}{\mathrm{m}}\mathrm{dt}$   

Integrating on both sides

 $\int \mathrm{vdv}=\frac{\mathrm{k}}{\mathrm{m}}\int \mathrm{dt}$  

$\frac{{\mathrm{v}}^2}{2}=\frac{\mathrm{k}}{\mathrm{m}}\mathrm{t}$

 ${\mathrm{v}}^2=\frac{2\mathrm{k}}{\mathrm{m}}\mathrm{t}$ 

 $\mathrm{v}={\left(\frac{2\mathrm{k}}{\mathrm{m}}\right)}^{\frac{1}{2}}{\mathrm{t}}^{\frac{1}{2}}$

$\frac{\mathrm{dx}}{\mathrm{dt}}={\left(\frac{2\mathrm{k}}{\mathrm{m}}\right)}^{\frac{1}{2}}{\mathrm{t}}^{\frac{1}{2}}$

$\mathrm{dx}={\left(\frac{2\mathrm{k}}{\mathrm{m}}\right)}^{\frac{1}{2}}{\mathrm{t}}^{\frac{1}{2}}\mathrm{dt}$
       
Integrating on both sides

$\int \mathrm{dx}={\left(\frac{2\mathrm{k}}{\mathrm{m}}\right)}^{\frac{1}{2}}\int {\mathrm{t}}^{\frac{1}{2}}\mathrm{dt}$

 $\mathrm{x}={\left(\frac{2\mathrm{k}}{\mathrm{m}}\right)}^{\frac{1}{2}}\frac{{\mathrm{t}}^{\frac{3}{2}}}{\frac{3}{2}}$   

$\mathrm{x}=\frac{2}{3}{\left(\frac{2\mathrm{k}}{\mathrm{m}}\right)}^{\frac{1}{2}}{\mathrm{t}}^{\frac{3}{2}}$

$\mathrm{x}\propto {\mathrm{t}}^{\frac{3}{2}}$