A particle of mass $m$ is driven by a machine that delivers a constant power $k$ watts. If the parti

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

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(c) : Constant power acting on the particle of mass $m$ is $k$ watt.\ or $\quad P=k ; \frac{d W}{d t}=k ; \quad d W=k d t$\ Integrating both sides, $\int_{0}^{W} d W=\int_{0}^{t} k d t$\ $\Rightarrow \quad W=k t$\ Using work energy theorem, $$\begin{align*} & W=\frac{1}{2} m v^{2}-\frac{1}{2} m(0)^{2} \Rightarrow k t=\frac{1}{2} m v^{2}  	ag{i}\ & v=\sqrt{\frac{2 k t}{m}} \end{align*}$$ [Using (i)] Acceleration of the particle, $a=\frac{d v}{d t}$ $$ a=\frac{1}{2} \sqrt{\frac{2 k}{m}} \frac{1}{\sqrt{t}}=\sqrt{\frac{k}{2 m t}} $$ Force on the particle, $F=m a=\sqrt{\frac{m k}{2 t}}=\sqrt{\frac{m k}{2}} t^{-1 / 2}$\