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# A body is moving unidirectionally under the influence of a source of constant power. Its displacement in time t is proportional to

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## Step 1: Given dataA body is moving unidirectionally under the influence of a source of constant power.Step 2: Formula used$P=\frac{W}{t}\left[whereP=power,W=workdone,t=time\right]$Step 3: Calculating displacementNow, the total work done by this force is equal to the product of the magnitude of applied force and the distance travelled by the body.$W=Fd$Substituting the value of work done in the power formula$P=\frac{Fd}{t}$Now as the velocity is given by $v=\frac{d}{t}\left[d=dis\mathrm{tan}ce,t=time\right]$ and the force is given by $F=ma\left[m=mass,a=acceleration\right]$The power can be expressed as $P=mav$Finally, we substitute $v=at$ to obtain $P=m{a}^{2}t$$a=\sqrt{\frac{P}{mt}}......\left(i\right)$Now Newton’s first equation of motion gives the displacement of the body as $d=ut+\frac{1}{2}a{t}^{2}............\left(ii\right)$We assume that the body was initially at rest, i.e., $u=0$.So equation (ii) becomes $d=\frac{1}{2}a{t}^{2}.......\left(iii\right)$Substituting equation (i) and (iii) we get, $d=\frac{1}{2}\left(\sqrt{\frac{P}{mt}}\right){t}^{2}$On simplifying we get, $d=\frac{1}{2}\sqrt{\frac{P}{m}}×{t}^{\left[2-\frac{1}{2}\right]}\phantom{\rule{0ex}{0ex}}d=\frac{1}{2}\sqrt{\frac{P}{m}}×{t}^{3/2}\phantom{\rule{0ex}{0ex}}d\propto {t}^{3/2}$Hence, its displacement is directly proportional to time ${t}^{3/2}$.

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