Orbits, Kepler's Third Law, Energy, and Escape Speed

We now unite circular motion with gravitation to understand how satellites and planets orbit. For a satellite of mass $m$ moving in a circular orbit of radius $r$ around a much larger body of mass $M$, gravity supplies the centripetal force. Setting the gravitational force equal to the centripetal requirement gives $G\dfrac{Mm}{r^2} = \dfrac{mv^2}{r}$. The satellite mass $m$ cancels from both sides, and one power of $r$ simplifies, leaving the orbital speed $v = \sqrt{\dfrac{GM}{r}}$. This striking result shows that the required speed depends only on the central mass and the orbital radius, not on the satellite's own mass. Closer orbits demand higher speeds, which is why low satellites zip around Earth in about ninety minutes while the distant Moon takes roughly a month.

From the orbital s