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An object is moving in a straight line under the influence of a source of constant power. If $v$ and $t$ are velocity and time respectively, then
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The correct answer is:
$v \propto t^{\frac{1}{2}}$
$\begin{aligned} & \mathrm{As} P=F V \\ & \Rightarrow \mathrm{P}=\mathrm{m} \frac{\mathrm{dV}}{\mathrm{dt}} \cdot \mathrm{V} \Rightarrow \mathrm{Pdt}=\mathrm{mV} \mathrm{dV} \\ & \Rightarrow \int_0^{\mathrm{t}} \mathrm{Pdt}=\mathrm{m} \int_0^{\mathrm{V}} \mathrm{VdV} \Rightarrow \mathrm{Pt}=\frac{1}{2} \mathrm{mV}^2 \\ & \Rightarrow \mathrm{V}=\sqrt{\frac{2 \mathrm{Pt}}{\mathrm{m}}}\end{aligned}$
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