1. Start with the total mechanical energy equation: $E = \frac{1}{2}mv^2 - \frac{GMm}{r}$. 2. Set the total energy to zero for escape: $0 = \frac{1}{2}mv_{esc}^2 - \frac{GMm}{r}$. 3. Solve for $v_{esc}$: $v_{esc} = \sqrt{\frac{2GM}{r}}$.

The minimum speed needed for an object to break free from a central object's gravity.

The energy an object possesses due to its position in a gravitational field, defined as zero when infinitely far away and negative as it gets closer.

The point in an orbit closest to the central object; where kinetic energy is highest and potential energy is most negative.

The point in an orbit farthest from the central object; where kinetic energy is lowest and potential energy is least negative.

The sum of the kinetic and gravitational potential energies of a satellite in orbit. It is constant in both circular and elliptical orbits.

Circular: Constant kinetic and potential energy, constant total mechanical energy. Elliptical: Fluctuating kinetic and potential energy, constant total mechanical energy.

Circular: Constant speed. Elliptical: Speed varies; highest at periapsis, lowest at apoapsis.