(C) To ensure the particle does not return to Earth, its total mechanical energy must be at least zero at infinity.According to the law of conservatio

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(C) To ensure the particle does not return to Earth, its total mechanical energy must be at least zero at infinity.According to the law of conservation of mechanical energy:$E_{initial} = E_{final}$$\frac{1}{2}mu^2 - \frac{GMm}{R} = 0 + 0$$\frac{1}{2}mu^2 = \frac{GMm}{R}$$u^2 = \frac{2GM}{R}$$u = \sqrt{\frac{2GM}{R}}$Since $g = \frac{GM}{R^2}$, we can also write $u = \sqrt{2gR}$.

Class 11 Physics Gravitation Escape Velocity and Escape Energy

Medium

$A$ particle of mass $m$ is thrown upwards from the surface of the earth with a velocity $u$. The mass and the radius of the earth are $M$ and $R$, respectively. $G$ is the gravitational constant and $g$ is the acceleration due to gravity on the surface of the earth. The minimum value of $u$ so that the particle does not return back to earth is:

AIPMT 2011 All AIPMT

A
$(\frac{GM}{R})^{1/2}$
B
$(\frac{8GM}{R})^{1/2}$
C
$(\frac{2GM}{R})^{1/2}$
D
$(\frac{4GM}{R})^{1/2}$

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Gravitation

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Escape Velocity and Escape Energy

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