Question

Define escape velocity. Derive an expression for it
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Answer 1

Escape velocity :-

→ It is the minimum velocity with which a body should be projected from the surface of the planet so as to reach infinity.

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→ If a body of mass m is projected with velocity v_e from the surface of a planet of mass M and radius R then by law of conservation of mechanical energy,

( K.E. + P.E. ) at surface = ( K.E. + P.E. ) at ∞

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→ $\sf \dfrac{1}{2} m v_e^2 + \Big( - \dfrac{GMm}{R} \Big) = 0 + 0$

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→ $\sf \dfrac{1}{2} m v_e^2 = \dfrac{GMm}{R}$

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→ $\sf v_e = \frac{2GM}{R}$

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∵ $\sf GM = gR^2$

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or $\sf v_e = \dfrac{2gR^2}{R}$

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→ $\sf v_e = \sqrt{2gR}$