Question

What is the escape velocity from the surface of the earth of radius $R$ and density $\rho$ ?

• A. $2R\sqrt{\frac{2\pi \rho G}{3}}$
• B. $2\sqrt{\frac{2\pi \rho G}{3}}$
• C. $2\pi \sqrt{\frac{R}{g}}$
• D. $\sqrt{\frac{2\pi G\rho }{R^2}}$

Answer 1

The correct option is A $2R\sqrt{\frac{2\pi \rho G}{3}}$

Escape speed is given by $v_e=\sqrt{\frac{2GM}{R}}$
But $M=Volume\times density\Rightarrow M=\frac{4}{3}\pi r^3\rho$
$\Rightarrow v_e=\sqrt{2G\frac{\frac{4}{3}\pi R^3\rho }{R}}$
$\Rightarrow v_e=2R\sqrt{\frac{2\pi \rho G}{3}}$