A satellite orbiting Earth, the Moon orbiting Earth, Earth orbiting the Sun — all of these are really two-body systems where both objects technically orbit their common center of mass. But when one object's mass is negligible compared to the other's, that center of mass sits essentially on top of the massive object, so we simplify: the central object is treated as stationary, and all the motion belongs to the satellite.
In a circular orbit, the satellite's distance from the central object, r, never changes. Since gravitational PE and kinetic energy both depend only on r and speed — and speed is also constant in a circular orbit — every relevant quantity stays fixed for the entire trip around:
In an elliptical orbit, r constantly changes — closer to the central object at one point (perigee), farther away at another (apogee). That means gravitational PE and kinetic energy both change continuously. But two quantities still don't budge:
This is exactly conservation of energy at work: as U_g becomes more negative near perigee, KE must increase to keep the total constant — the satellite speeds up as it gets closer, and slows down as it moves away. Gravitational potential energy itself is defined so that U_g = 0 at infinite separation, which is why every bound orbit has negative U_g and negative total mechanical energy.
Drag the eccentricity slider and watch what stays constant. The central object sits at the orbit's focus (the dot doesn't move) — this is a schematic, not real astronomical units.
Elliptical orbit — r, PE, and KE all change continuously as the satellite moves. Only Total Mechanical Energy and Angular Momentum stay constant.
A satellite orbits Earth in an ellipse. At perigee, its distance from Earth's center is r_p and its speed is v_p. At apogee, its distance is r_a. Find its speed v_a at apogee in terms of v_p, r_p, and r_a.
Every bound orbit has negative total mechanical energy — the satellite is, in a sense, trapped in a gravitational "well." Escape velocity is the borderline case: the exact speed at which total mechanical energy equals zero.
At exactly this speed, the satellite moves away from the central object forever, with its speed approaching zero only in the limit as r → ∞. Any slower, and the object is bound (negative E_total, it eventually falls back or settles into an orbit); any faster, and it escapes with speed to spare (positive E_total).
Derive an expression for the escape velocity of an object launched from distance r away from a central object of mass M, using conservation of energy.