Escape Speed and Orbital Speed

Earth's atmosphere is mostly nitrogen and oxygen, even though hydrogen and helium are by far the most abundant elements in the universe. The reason is escape speed: light gas molecules like hydrogen and helium routinely reach speeds fast enough to permanently leave Earth's gravitational pull, while the heavier nitrogen and oxygen molecules almost never do.

Defining escape speed. It is the minimum speed an object needs, launched from the Earth's surface, to escape Earth's gravitational pull for good and never fall back. Consider a mass $M$ launched from the surface with initial speed $v_i$; its initial total energy (kinetic plus gravitational potential) is

For the minimum possible launch speed, the object should just barely reach an infinite distance with zero leftover kinetic energy, i.e. $E_f = 0$ (since $U\to0$ as $r\to\infty$ too). By conservation of energy, $E_i=E_f=0$, so setting $v_i=v_e$ (the escape speed) gives

Using $g=GM_e/R_e^2$, this simplifies neatly to

$v_e=\sqrt{2gR_e}. \qquad (6.56)$ …