Question

The escape velocity from the surface of the earth of radius R and density $\rho$ is given by

• A. $\sqrt{\frac{8G\pi R^2\rho }{3}}$
• B. $\sqrt{\frac{8G\pi R^2\rho }{9}}$
• C. $\sqrt{\frac{8G\pi R^2\rho }{7}}$
• D. $\sqrt{\frac{8G\pi R^2\rho }{5}}$

Answer 1

The correct option is A $\sqrt{\frac{8G\pi R^2\rho }{3}}$

The escape velocity from a surface of a planet of mass $M$ and radius $R$ is given by,$$v_e=\sqrt{\frac{2GM}{R}}$$If, $\rho$ is the density of the planet, then $M=\frac{4}{3}\pi \rho R^3$

$\Rightarrow v_e=\sqrt{\frac{2G\left(\frac{4}{3}\right)\pi R^3\rho }{R}}$

$\therefore v_e=\sqrt{\frac{8G\pi R^2\rho }{3}}$

Hence, option (a) is the correct answer.