A body of mass 2 kg is initially at rest. It | vedclass

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(C) Given,mass $m = 2 \ kg$ and power $P$ is constant.Work done by the source in time $t$ is $W = P 	imes t$.According to the work-energy theorem,$W

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$4$

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(C) Given,mass $m = 2 \ kg$ and power $P$ is constant.Work done by the source in time $t$ is $W = P 	imes t$.According to the work-energy theorem,$W = \Delta K = \frac{1}{2} m v^2 - 0$.So,$\frac{1}{2} m v^2 = P t \implies v = \sqrt{\frac{2 P t}{m}}$.Since $v = \frac{dx}{dt}$,we have $\frac{dx}{dt} = \sqrt{\frac{2 P}{m}} \cdot t^{1/2}$.Integrating with respect to $t$ from $0$ to $4 \ s$:$x = \int_0^4 \sqrt{\frac{2 P}{m}} \cdot t^{1/2} dt = \sqrt{\frac{2 P}{m}} \left[ \frac{t^{3/2}}{3/2} ight]_0^4$.Substituting $m = 2 \ kg$:$x = \sqrt{\frac{2 P}{2}} \cdot \frac{2}{3} \cdot (4)^{3/2} = \sqrt{P} \cdot \frac{2}{3} \cdot 8 = \frac{16}{3} \sqrt{P}$.Given displacement is $\frac{1}{3} \alpha^2 \sqrt{P}$.Comparing $\frac{16}{3} \sqrt{P} = \frac{1}{3} \alpha^2 \sqrt{P}$,we get $\alpha^2 = 16$,so $\alpha = 4$.

$A$ body of mass $2 \ kg$ is initially at rest. It starts moving unidirectionally under the influence of a source of constant power $P$. Its displacement in $4 \ s$ is $\frac{1}{3} \alpha^2 \sqrt{P} \ m$. The value of $\alpha$ will be $.............$

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