Show that the escape speed from the surface of a planet of radius can be written as…

PHYSICS · Subjective · A Level · Physics · Fields and their consequences · Fields and their consequences

Problem

Show that the escape speed from the surface of a planet of radius

R

can be written as

v_{\text{esc}} = \sqrt{2gR}

and

v_{\text{esc}} = \sqrt{-2V}

where

g

is the gravitational field strength and

V

the gravitational potential at the planet's surface. Show that yet another expression for the escape speed is

v_{\text{esc}} = R\sqrt{\frac{8\pi G\rho}{3}}

where

\rho

is the density of the planet.