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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, respectively, M and R. G is gravitational constant and g is 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
A
$\sqrt{2gR^2}$
B
$\sqrt{\frac{2GM}{R^2}}$
C
$\sqrt{\frac{2GM}{R}}$
D
$\sqrt{\frac{2gM}{R^2}}$
Detailed Solution
The particle does not return if it can just reach infinity, where both its kinetic and potential energies are zero. The minimum u is the escape velocity.
Conservation of energy: $\frac{1}{2}mu^2 - \frac{GMm}{R} = 0$
$u^2 = \frac{2GM}{R}$
$u = \sqrt{\frac{2GM}{R}}$
(Using $g = \frac{GM}{R^2}$ this is also $\sqrt{2gR}$.)
Conservation of energy: $\frac{1}{2}mu^2 - \frac{GMm}{R} = 0$
$u^2 = \frac{2GM}{R}$
$u = \sqrt{\frac{2GM}{R}}$
(Using $g = \frac{GM}{R^2}$ this is also $\sqrt{2gR}$.)