Elliptical Orbits and Escape

Fire a rocket at the wrong speed or angle relative to a circular orbit, and gravity still pulls it into a closed path, but that path stretches into an ellipse. Too fast and it swings out further before gravity reels it back; too slow and it falls closer before climbing back out. The satellite still obeys the same law of gravity, it just no longer has the one special speed that keeps a circular orbit at constant radius.

Two conservation laws pin down the whole shape. Angular momentum L = mvr stays constant because gravity pulls straight along the radius and exerts no torque about the center, so at every point on the ellipse, speed and radius trade off: closer in means faster, farther out means slower. Total mechanical energy, kinetic plus gravitational potential, also stays constant, so wherever the satellite is, 0.5mv² - GMm/r equals the same fixed negative number. Combine the two statements at perigee (closest point) and apogee (farthest point) and you can solve for the unknown speed at either extreme without ever touching calculus.

Push the total energy up to exactly zero and the ellipse opens up into a parabola: the object still feels gravity but has just enough energy to coast to infinity with no speed left over. That borderline condition defines escape speed, and it connects this topic to everything before it in the unit, since it's the same energy and angular momentum bookkeeping used for circular orbits and Kepler's laws, now pushed to its limiting case.