D.1.14 (HL)—Orbital speed

Set the circular-orbit model

For a small mass $m$ in a circular orbit of radius $r$ around a much larger mass $M$, gravitational force supplies the centripetal force. The satellite mass cancels.

Worked example — lunar orbit

At $100\,\mathrm{km}$ above the Moon, $r=1.737\times10^6+0.100\times10^6=1.837\times10^6\,\mathrm{m}$. With $M=7.35\times10^{22}\,\mathrm{kg}$, $v=\sqrt{(6.67\times10^{-11})(7.35\times10^{22})/(1.837\times10^6)}=1.63\times10^3\,\mathrm{m\,s^{-1}}$.

Compare orbits

At the same central mass, orbital speed decreases as $r^{-1/2}$. A satellite at a smaller circular-orbit radius moves faster. The satellite mass does not affect the required speed in this ideal model.

Common trap

Do not use escape speed for a bound circular orbit: $v_{\rm esc}=\sqrt2\,v_{\rm orbital}$ at the same radius. Also add the planet’s radius to altitude before using $r$.