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A body is moved in straight line by machine with constant power. The distance travelled by it in time duration $t$ is proportional to
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Verified Answer
The correct answer is:
$t^{3 / 2}$
Here, $P=$ constant, or, $\frac{\Delta W}{\Delta t}=P$ $\Delta W=P \Delta t$ or, $W=P t$ or $\frac{1}{2} m v^2=P t \quad$ [Using work-energy theorem] or $v^2=\frac{2 P}{m} t$ or $v=\sqrt{\frac{2 P}{m}} t^{1 / 2}$ or $d x=\sqrt{\frac{2 P}{m}} t^{1 / 2} d t$ Integrating both sides, $\int_0^x d x=\sqrt{\frac{2 P}{m}} \int_0^t t^{1 / 2} d t$
$$
\begin{aligned}
& x=\sqrt{\frac{2 P}{m}}\left(\frac{2}{3}\right) t^{3 / 2} \\
& \therefore \quad x \propto t^{3 / 2} \\
& \text { or displacement } \propto(\text { time })^{3 / 2}
\end{aligned}
$$
$$
\begin{aligned}
& x=\sqrt{\frac{2 P}{m}}\left(\frac{2}{3}\right) t^{3 / 2} \\
& \therefore \quad x \propto t^{3 / 2} \\
& \text { or displacement } \propto(\text { time })^{3 / 2}
\end{aligned}
$$
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