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 $\mathrm{m}$ )_______
(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, $x=\sqrt{\frac{2 P}{m}} \frac{t^{3 / 2}}{3 / 2}=\sqrt{\frac{2 P}{m}} \times \frac{2}{3} t^{3 / 2}$
$\Rightarrow x=\sqrt{\frac{2 \times 1}{2}} \times \frac{2}{3} \times 9^{3 / 2}=\frac{2}{3} \times 27=18$c