A body is projected vertically upwards from the surface of the Earth with a velocity equal to half the escape velocity. The maximum height reached by the body is:

A. R/3 — Using conservation of energy, the kinetic energy at the surface equals the change in gravitational potential energy at height h.

JEE Advanced · PhysicsMedium

$A body is projected vertically upwards from the surface of the Earth with a velocity v = v_{e} /2, where v_{e} is the escape velocity. The maximum height h reached by the body is:$

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Correct answer: A — $R /3$

$Initial energy is E_{i} = (1/2) m v^{2} - GM m / R .$
$Since v = v_{e} /2 = (1/2) 2 GM / R, v^{2} = GM / (2 R) .$
$At maximum height h, E_{f} = - GM m / (R + h) .$
$Equating E_{i} = E_{f}, we get m (GM /4 R) - GM m / R = - GM m / (R + h) .$
$Solving for h, we find - 3/4 R = - 1/ (R + h), so R + h = 4 R /3, which gives h = R /3 .$
$Hence the answer is (A).$