Orbital Energy

An orbiting satellite is always exchanging two forms of energy. Gravitational potential energy is most negative close to Earth and rises (becomes less negative) as the satellite moves farther away. Kinetic energy does the opposite: it's largest when the satellite is close and moving fast, smallest when it's far and moving slow. Around an elliptical orbit, at perigee (closest approach) the satellite has maximum kinetic energy and minimum potential energy; at apogee (farthest point) it has minimum kinetic energy and maximum potential energy. Add them at any point on the orbit and you get the same number: total orbital energy is conserved. That's also why firing engines to speed up at one point can leave the satellite slower elsewhere — the extra kinetic energy gets stored as potential energy once it climbs higher, and speed has to drop to keep the total fixed.

The sign of that total energy tells you the satellite's fate. Gravitational potential energy is defined as zero at infinite separation and negative everywhere closer, so total energy equals kinetic plus a negative number. If kinetic energy is too small to overcome that negative potential, the total stays negative — the object is bound, trapped in a closed elliptical or circular path forever. If total energy is exactly zero, the object just barely escapes, arriving at infinity with zero speed left over. If total energy is positive, it escapes with speed to spare.

This is the same potential energy curve logic from earlier in the energy unit, just applied to gravity instead of springs or generic fields. A bound orbit is a particle oscillating inside a potential well, swinging between two turning points (perigee and apogee) where kinetic energy hits zero relative to the well's shape. Escape means having enough total energy to climb out of the well entirely.