A body is moved in straight line by machine with constant power....

Here, $P=$ constant, or, $\frac{\Delta W}{\Delta t}=P$ $\Delta W=P \Delta t$ or, $W=P t$ or $\frac{1}{2} m v^2=P t \quad$ [Using work-energy theorem] or $v^2=\frac{2 P}{m} t$ or $v=\sqrt{\frac{2 P}{m}} t^{1 / 2}$ or $d x=\sqrt{\frac{2 P}{m}} t^{1 / 2} d t$ Integrating both sides, $\int_0^x d x=\sqrt{\frac{2 P}{m}} \int_0^t t^{1 / 2} d t$ \begin{aligned} & x=\sqrt{\frac{2 P}{m}}\left(\frac{2}{3}\right) t^{3 / 2} \\ & \therefore \quad x \propto t^{3 / 2} \\ & \text{ or displacement } \propto(\text{ time })^{3 / 2} \end{aligned}