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Question

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 the earth is

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Solution

The correct option is **C** √2GMR

Particle will not return if it is thrown upwards with escape velocity.

Conserving mechanical energy between the instants the particle is at the surface of the earth and at a very large distance

Ui+Ki=Uf+Kf

At a very large distance, Uf=0 and for the particle to just escape gravity. its final velocity will be 0, so Kf=0.

−GMmR+12mv2e=0+0

⇒ve=√2GMR

Hence, option (c) is correct.

Why this question ?Tip: Escape velocity can be calculated for any orbit not just the surface of the planet. For a particle to escape, its total mechanical energy should be equal to zero.

Particle will not return if it is thrown upwards with escape velocity.

Conserving mechanical energy between the instants the particle is at the surface of the earth and at a very large distance

Ui+Ki=Uf+Kf

At a very large distance, Uf=0 and for the particle to just escape gravity. its final velocity will be 0, so Kf=0.

−GMmR+12mv2e=0+0

⇒ve=√2GMR

Hence, option (c) is correct.

Why this question ?Tip: Escape velocity can be calculated for any orbit not just the surface of the planet. For a particle to escape, its total mechanical energy should be equal to zero.

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