Energy of an Orbiting Satellite

A satellite's total mechanical energy is the sum of its kinetic energy and its gravitational potential energy, and this sum always comes out negative -- which is the mathematical statement that the satellite is permanently bound to the Earth.

Derivation. The potential energy of a satellite of mass $M_s$ at height $h$ is $U=-\dfrac{GM_sM_e}{R_e+h}$ (Eq. 6.63). Its kinetic energy uses the orbital speed found earlier, $v=\sqrt{GM_e/(R_e+h)}$:

Adding the two,

The negative sign means the satellite is bound to the Earth and cannot spontaneously escape. As $h\to\infty$, $E\to0$: physically, a satellite pushed out to an infinite distance would be completely free of Earth's gravitational influence, no longer bound to it at all. …