(A) To ensure the particle does not return,its total mechanical energy at infinity must be at least zero.Let $v$ be the projection speed at height $h

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(A) To ensure the particle does not return,its total mechanical energy at infinity must be at least zero.Let $v$ be the projection speed at height $h = 3R$ from the surface.The distance from the center of the Earth is $r = R + 3R = 4R$.Using the law of conservation of energy: $K_i + U_i = K_f + U_f$.Here,$K_i = \frac{1}{2}mv^2$,$U_i = -\frac{GMm}{4R}$,$K_f = 0$,and $U_f = 0$.$\frac{1}{2}mv^2 - \frac{GMm}{4R} = 0$.$\frac{1}{2}v^2 = \frac{GM}{4R}$.$v^2 = \frac{GM}{2R}$.$v = [\frac{GM}{2R}]^{1/2}$.

Class 11 Physics Gravitation Escape Velocity and Escape Energy

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$A$ particle of mass $m$ is kept at rest at a height $3R$ from the surface of the Earth,where $R$ is the radius of the Earth and $M$ is the mass of the Earth. The minimum speed with which it should be projected upward so that it does not return back is ($g$ = acceleration due to gravity on the Earth's surface).

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A
$[\frac{GM}{2R}]^{1/2}$
B
$[\frac{gR}{4}]^{1/2}$
C
$[\frac{2g}{R}]^{1/2}$
D
$[\frac{GM}{R}]^{1/2}$

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