First slide

Gravitaional potential energy

Question

A body is projected vertically upwards from the surface of a planet of radius fl with a velocity equal to half the escape velocity for that planet. The maximum height attained by the body is

Moderate

A

R/3
B

R/2
C

R/4
D

R/5

Solution

When the body goes up, then
Decrease in K.E, = Increase in P.E.
$∴ \frac{1}{2} {mv}^{2} - 0 = m [\frac{g}{1 + (h / R)}] h$
or $\frac{1}{2} m {[\frac{\sqrt{2 gR}}{2}]}^{2} = m [\frac{g}{1 + (h / R)}] h (∵ v_{e} = \sqrt{2 gR)}$
$\frac{mgR}{4} = \frac{mgh}{1 + (h / R)}$
or $R [1 + \frac{h}{R}] = 4 h or R + h = 4 h or R = 3 h$
$∴ h = \frac{R}{3}$

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