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

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

KCETKCET 1999Gravitation

Solution:

Total energy = K.E. + P.E. = $\frac{1}{2} m \left( \frac{1}{2} \sqrt{\frac{2GM}{R}}\right)^2 + (- \frac{GMm}{R})$
= $\frac{GMm}{4R} - \frac{GMm}{R} = - \frac{3GMm}{4R}$
Final energy when body comes to rest at heigh
$r = -\frac{GMm}{R} $ $\therefore \frac{-3GMm}{4R} = \frac{GMm}{r}$ i.e. $r = \frac{4R}{3}$ Max.height = r - R = $\frac{4}{3}$ R - R = $\frac{R}{3}$