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 : 

\([\frac{3 g h}{4 \pi G d}]^{1/2}\)

When a boy jumps from a ground level up to height h then its velocity of jumping v = √2gh ... (i)

and for the given condition this will become equal to escape velocity v_escape : 

  = \(\sqrt{\frac{2GM}{R}} \)

  = \(\sqrt{\frac{2 G}{R}\frac{4}{3} \pi R^3 d}\) ... (ii)

From (i) and (ii) :

\(\sqrt{2gh} = R \sqrt{\frac{2}{3} \pi G d}\)

\(\Rightarrow R = [\frac{3}{4 \pi} \frac{gh}{dG}]^{1/2}\)