A body is projected vertically upwards from the surface of earth with… | JEE Main 2021 (22 Jul, Shift 2)

Q52JEE Main 2021 · 22 JUL · SHIFT 2Difficult

Escape Velocity

A body is projected vertically upwards from the surface of earth with a velocity sufficient enough to carry it to infinity. The time taken by it to reach height h is ___________ s.

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Answer: (d)

Projected with escape speed, so at every distance $r$ : $v = \sqrt{\dfrac{2GM}{r}} = R_e\sqrt{\dfrac{2g}{r}}$ .
$\dfrac{dr}{dt} = R_e\sqrt{2g}\, r^{-1/2} \Rightarrow \int_{R_e}^{R_e+h} r^{1/2}\,dr = R_e\sqrt{2g}\; t$
$\tfrac23\left[(R_e+h)^{3/2} - R_e^{3/2}\right] = R_e\sqrt{2g}\; t$
$t = \dfrac{2}{3}\sqrt{\dfrac{R_e}{2g}}\left[\left(1+\dfrac{h}{R_e}\right)^{3/2}-1\right] = \dfrac13\sqrt{\dfrac{2R_e}{g}}\left[\left(1+\dfrac{h}{R_e}\right)^{3/2}-1\right]$
So the answer is (d).

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Escape Velocity question from JEE Main 2021 (22 Jul, Shift 2)

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