First slide

Gravitaional potential energy

Question

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

Easy

A

${(\frac{G M}{2 R})}^{1 / 2}$
B

${(\frac{g R}{4})}^{1 / 2}$
C

${(\frac{2 g}{R})}^{1 / 2}$
D

${(\frac{G M}{R})}^{1 / 2}$

Solution

The minimum speed with which the particle should be projected from the surface of the earth so that it does not return back is known as escape speed and it is given by
$v_{e} = \sqrt{\frac{2 G M}{(R + h)}} Here, h = 3 R ∴ v_{e} = \sqrt{\frac{2 G M}{(R + 3 R)}} = \sqrt{\frac{2 G M}{4 R}} = \sqrt{\frac{G M}{2 R}} = \sqrt{\frac{g R}{2}} (∵ g = \frac{G M}{R^{2}})$

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