JEE MAIN - Physics (2025 - 4th April Evening Shift - No. 9)
An object is kept at rest at a distance of $3 R$ above the earth's surface where $R$ is earth's radius. The minimum speed with which it must be projected so that it does not return to earth is :
(Assume $\mathrm{M}=$ mass of earth, $\mathrm{G}=$ Universal gravitational constant)
$\sqrt{\frac{3 \mathrm{GM}}{\mathrm{R}}}$
$\sqrt{\frac{2 \mathrm{GM}}{\mathrm{R}}}$
$\sqrt{\frac{\mathrm{GM}}{2 \mathrm{R}}}$
$\sqrt{\frac{\mathrm{GM}}{\mathrm{R}}}$
Explanation
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$$\begin{aligned} &\begin{aligned} & \mathrm{P}_{\mathrm{P}}+\mathrm{k}_{\mathrm{P}}=\mathrm{P}_{\mathrm{o}}+\mathrm{k}_0 \\ & -\frac{\mathrm{GMm}}{4 \mathrm{R}}+\frac{1}{2} \mathrm{mV}_{\mathrm{P}}^2=0 \\ & \mathrm{~V}_{\mathrm{P}}=\sqrt{\frac{\mathrm{GM}}{2 \mathrm{R}}} \end{aligned}\\ &\text { Choice } 1 \end{aligned}$$
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