A body of mass 2kg is driven by

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Question:

A body of mass $2 \mathrm{~kg}$ is driven by an engine delivering a constant power of $1 \mathrm{~J} / \mathrm{s}$. The body starts from rest and moves in a straight line. After 9 seconds, the body has moved a distance (in $\mathrm{m}$ )_______

Solution:

(18)

Given, Mass of the body, $m=2 \mathrm{~kg}$

Power delivered by engine, $P=1 \mathrm{~J} / \mathrm{s}$

Time, $t=9$ seconds

Power, $P=F v$

$\Rightarrow P=m a v$                             $[\because F=m a]$

$\Rightarrow m \frac{d v}{d t} v=P$         $\left(\because a=\frac{d v}{d t}\right)$

$\Rightarrow v d v=\frac{P}{m} d t$

Integrating both sides we get

$\Rightarrow \int_{0}^{v} v d v=\frac{P}{m} \int_{0}^{t} d t$

$\Rightarrow \frac{v^{2}}{2}=\frac{P t}{m} \Rightarrow v=\left(\frac{2 P t}{m}\right)^{1 / 2}$

$\Rightarrow \frac{d x}{d t}=\sqrt{\frac{2 P}{m}} t^{1 / 2}$         $\left(\because v=\frac{d x}{d t}\right)$

$\Rightarrow \int_{0}^{x} d x=\sqrt{\frac{2 P}{m}} \int_{0}^{t} t^{1 / 2} d t$

$\therefore$ Distance, $x=\sqrt{\frac{2 P}{m}} \frac{t^{3 / 2}}{3 / 2}=\sqrt{\frac{2 P}{m}} \times \frac{2}{3} t^{3 / 2}$

$\Rightarrow x=\sqrt{\frac{2 \times 1}{2}} \times \frac{2}{3} \times 9^{3 / 2}=\frac{2}{3} \times 27=18$

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