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A constant power P is applied on a particle of mass m. find kintic energy, velocity and displacement of particle as function of time t.

Answer

As is known, \({P}={F}{v}={\left({m}{a}\right)}{v}={m}{\left(\frac{{{d}{v}}}{{{\left.{d}{t}\right.}}}\right)}{v}.\)
\(\therefore{v}{d}{v}=\frac{{{P}}}{{{m}}}{\left.{d}{t}\right.}\)
Integrating both sides, we get
\(\frac{{{v}^{{{2}}}}}{{{2}}}=\frac{{{P}}}{{{m}}}{t},{v}=\sqrt{{\frac{{{2}{P}}}{{{m}}}}}{t}^{{{1}/{2}}}\)
\({v}=\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}=\sqrt{{\frac{{{2}{P}}}{{{m}}}}}{t}^{{{1}/{2}}}{\quad\text{or}\quad}{\left.{d}{x}\right.}=\sqrt{{\frac{{{2}{P}}}{{{m}}}}}{t}^{{{1}/{2}}}{\left.{d}{t}\right.}\)
Integrating both sides, we get
\({x}=\sqrt{{\frac{{{2}{P}}}{{{m}}}}}\frac{{{t}^{{{3}/{2}}}}}{{{3}/{2}}}=\frac{{{2}}}{{{3}}}\sqrt{{\frac{{{2}{P}}}{{{m}}}}}{t}^{{{3}/{2}}}\)
This is the required expression.
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