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A particle of mass ' $m$ ' is kept at rest at a height $3 R$ from the surface of earth, where ' $R$ ' is radius of earth and ' $M$ ' is mass of earth. The minimum speed with which is should be projected, so that it does not return back, is $(g$ is acceleration due to gravity on the surface of earth)
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Verified Answer
The correct answer is:
$\left(\frac{G M}{2 R}\right)^{1 / 2}$
Escape velocity $V_e=\sqrt{\frac{2 \mathrm{GM}}{(\mathrm{R}+h)}}$ from the question $h=3 R$
$$
\begin{aligned}
& \therefore V_e^{\prime}=\sqrt{\frac{2 \mathrm{GM}}{\mathrm{R}+3 \mathrm{R}}} \\
& =\sqrt{\frac{2 G M}{4 R}}=\sqrt{\frac{G M}{2 R}}=\sqrt{\frac{g R}{2}} \\
& \text { where } g=\frac{G M}{R^2}
\end{aligned}
$$
$$
\begin{aligned}
& \therefore V_e^{\prime}=\sqrt{\frac{2 \mathrm{GM}}{\mathrm{R}+3 \mathrm{R}}} \\
& =\sqrt{\frac{2 G M}{4 R}}=\sqrt{\frac{G M}{2 R}}=\sqrt{\frac{g R}{2}} \\
& \text { where } g=\frac{G M}{R^2}
\end{aligned}
$$
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