Question

A boy can jump to a height $h$ from ground on earth . What should be the radius of a sphere of density $\delta$ such that on jumping on it, he escapes out of the gravitational field of the sphere?

• A. $\sqrt{\frac{4\pi G\delta }{3gh}}$
• B. $\sqrt{\frac{4\pi gh}{3G\delta }}$
• C. $\sqrt{\frac{3gh}{4\pi G\delta }}$
• D. $\sqrt{\frac{3G\delta }{4\pi gh}}$

Answer 1

The correct option is C $\sqrt{\frac{3gh}{4\pi G\delta }}$

While jumping he converts his initial $kinetic$ $energy$ to $potential$ $energy$

$\frac{m{v}^2}{2}=mgh$

$v=\sqrt{2gh}$

Now this $velocity$ has to be equal to $escape$ $velocity$ on the sphere

$Escape$ $velocity=\sqrt{\frac{2G{M}_s}{R}}=\sqrt{\frac{8\pi G{R}_s^2\delta }{3}}$

By comparing both the velocities we get $Radius\left(R_s\right)=\sqrt{\frac{3gh}{4\pi G\delta }}$