Question

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

• A. $\frac {R} {3}$
• B. $\frac {R} {2}$
• C. $\frac {R} {4}$
• D. $\frac {R} {5}$

Answer 1

Answer: (A)

$\frac {R} {3}$

Answer 2

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 = $\frac{R}{3}$