Question

A mass of 6 × 10²⁴ kg (equal to the mass of the earth) is to be compressed in a sphere in such a way that the escape velocity from its surface is 3 × 10⁸ m s⁻¹. What should be the radius of the sphere?

Answer 1

Mass of the sphere = 6 × 10²⁴ kg
Escape velocity = 3 × 10⁸ m/s

Escape velocity is given by
$$\begin{gathered} v_e=\frac{\mathit{2}GM}{R} \\ \Rightarrow R=\frac{\mathit{2}GM}{{v_e}^{\mathit{2}}} \\ =\frac{2\times 6.67\times {10}^{-11}\times 6\times {10}^{24}}{{\left(3\times {10}^8\right)}^2} \\ =\frac{2\times 40.02\times {10}^{13}}{9\times {10}^{16}} \\ =\frac{80.02}{9}\times {10}^{-3}\mathrm{m} \\ =8.89\times {10}^{-3}\mathrm{m} \\ =9\mathrm{mm} \end{gathered}$$