A body is projected vertically upwards from the surface of earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be
(Take radius of earth $= 6400km$ and $g = 10{ms}^{- 2}$ )

Question

A body is projected vertically upwards from the surface of earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be
(Take radius of earth

= 6400km

and

g = 10{ms}^{- 2}

)

Quantrex Academy · Accepted Answer

Correct answer: A. 800km

Total mechanical energy of the body at the surface of earth
${M}_{i} = - \dfrac{GMm}{R} + \dfrac{1}{2}m{\left[\dfrac{1}{3}\left(\sqrt{\dfrac{2GM}{R}}\right)\right]}^{2} = - \dfrac{GMm}{R} + \dfrac{GMm}{9R} = - \dfrac{8GMm}{9R}$
Total mechanical energy of the body at the highest point
${M}_{f} = - \dfrac{GMm}{R + h}$
Using conservation of mechanical energy,
${M}_{i} = {M}_{f} ⇒ - \dfrac{GMm}{R + h} = - \dfrac{8GMm}{9R}$
$⇒ R + h = \dfrac{9R}{8}$
$⇒ h = \dfrac{R}{8} = \dfrac{6400}{8} = 800km$

A body is projected vertically upwards from the surface of earth...

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