A particle of mass $m$ moves along a straight line on smooth horizontal plane, acted upon by a force delivering a constant power $P$ . If the initial velocity of the particle is zero, then find its displacement as a function of time $t$ .

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Solution

Power $P$ of the particle in given as the dot product of the external force $\to F$ applied on it and the velocity $\to v$ of the particle $\Rightarrow P = \to F . \to v \Rightarrow P = F v$
$\Rightarrow P = (\frac{m v d v}{d x}) v \Rightarrow v^{2} d v = \frac{P}{m} d x$
$\Rightarrow \int_{0}^{v} v^{2} d v = \int_{0}^{v} \frac{P}{m} d x \Rightarrow \frac{v^{3}}{3} = \frac{P x}{m} \Rightarrow v = \sqrt[3]{\frac{3 P}{m}} x^{1 / 3} \Rightarrow \frac{d x}{d t} = \sqrt[3]{\frac{3 P}{m}} x^{1 / 3}$
$\Rightarrow \sqrt[3]{\frac{3 P}{m}} \int_{0}^{t} d t = \int_{0}^{x} x^{- 1 / 3} d x \Rightarrow t \sqrt[3]{\frac{3 P}{m}} = \frac{+ 3}{2} x^{+ 2 / 3} \Rightarrow x^{2} = \frac{3 P}{m} t^{3} \times \frac{8}{27} \Rightarrow x = {(\frac{8 P}{9 m} t^{3})}^{1 / 2}$

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