A particle of mass m is driven by a machine that delivers a constant power k watts. If the particle starts from rest the force on the particle at time t is

Given power P =constant = k (say)
If force is F and velocity is v

$\Rightarrow \mathrm{F}\cdot \mathrm{V}=\mathrm{k};\Rightarrow \frac{\mathrm{mdv}}{\mathrm{dt}}\cdot \mathrm{v}=\mathrm{k}$

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

Integrating,

$\Rightarrow \frac{{\mathrm{v}}^2}{2}=\frac{\mathrm{kt}}{\mathrm{m}}\Rightarrow \mathrm{v}=\sqrt{\frac{2\mathrm{kt}}{\mathrm{m}}}$

Differentiating the above equation,

$\frac{\mathrm{dv}}{\mathrm{dt}}=\sqrt{\frac{2\mathrm{k}}{\mathrm{m}}}\left(\frac{1}{2}\right)\left({\mathrm{t}}^{-1/2}\right)$

$\mathrm{a}=\sqrt{\frac{\mathrm{k}}{2\mathrm{m}}}\left({\mathrm{t}}^{-1/2}\right)$

$\Rightarrow \mathrm{F}=\mathrm{ma}=\mathrm{m}\left[\sqrt{\frac{\mathrm{k}}{2\mathrm{m}}}\left({\mathrm{t}}^{-1/2}\right)\right]=\sqrt{\frac{\mathrm{mk}}{2}}\left({\mathrm{t}}^{-1/2}\right)$