Gravitation — Orbits, Escape & Satellites (Medium)

Quick Recap — Orbits, Escape & Satellites

• Orbital speed: $v_o=\sqrt{\dfrac{GM}{r}}=\sqrt{gR}$ for a near-Earth orbit ($\approx7.9$ km/s). Period $T=2\pi\sqrt{\dfrac{r^3}{GM}}$.
• Escape speed: $v_e=\sqrt{\dfrac{2GM}{R}}=\sqrt{2gR}\approx11.2$ km/s $=\sqrt2\,v_o$ (independent of the body's mass and launch direction).
• Satellite energy: $KE=+\dfrac{GMm}{2r}$, $PE=-\dfrac{GMm}{r}$, total $E=-\dfrac{GMm}{2r}=-KE$; binding energy $=+\dfrac{GMm}{2r}$.
• Geostationary satellite: period 24 h, equatorial, height $\approx36{,}000$ km. In orbit, an astronaut is weightless (free fall).

Worked mini-example. $v_e=\sqrt2\,v_o$: since a near orbit needs $\approx7.9$ km/s, escape needs $\approx11.2$ km/s.