Question

A boy can jump to a height h on ground level. What should be the radius of a sphere of density d such that on jumping on it, he escapes out of the gravitational field of the sphere

• A. $\left[ \frac { 4 \pi } { 3 } \frac { G d } { g h } \right] ^ { 1 / 2 }$
• B. $\left[ \frac { 4 \pi } { 3 } \frac { g h } { G d } \right] ^ { 1 / 2 }$
• C. $\left[ \frac { 3 } { 4 \pi } \frac { g h } { G d } \right] ^ { 1 / 2 }$
• D. $\left[ \frac { 3 } { 4 \pi } \frac { G d } { g h } \right] ^ { 1 / 2 }$

Answer 1

Answer: (C)

$\left[ \frac { 3 } { 4 \pi } \frac { g h } { G d } \right] ^ { 1 / 2 }$

Answer 2

When a boy jumps from a ground level up to height $h$ then its velocity of jumping $v = \sqrt { 2 g h }$ …..(i)

and for the given condition this will become equal to escape velocity v_escape = $\sqrt { \frac { 2 G M } { R } } = \sqrt { \frac { 2 G } { R } \left( \frac { 4 } { 3 } \pi R ^ { 3 } \cdot d \right) }$ …..(ii)

Equating (i) and (ii) $R = \left[ \frac { 3 } { 4 \pi } \frac { g h } { G d } \right] ^ { 1 / 2 }$.