A mass of $6 imes 10^{24}~kg$ (equal to the mass of the earth) is to be compressed into a sphere i

Question

A mass of $6 	imes 10^{24}~kg$ (equal to the mass of the earth) is to be compressed into a sphere in such a way that the escape velocity from its surface is $3 	imes 10^8~m~s^{-1}$. What should be the approximate radius of the sphere?

Professor PC Thomas Classes · Answer

Escape velocity is given by $v_e = \sqrt{\frac{2GM}{R}}$. Rearranging for $R$, we get $R = \frac{2GM}{v_e^2} = \frac{2 	imes 6.67 	imes 10^{-11} 	imes 6 	imes 10^{24}}{(3 	imes 10^8)^2} \approx 8.89 	imes 10^{-3}~m \approx 9~mm$.