An elliptical orbit is the closed, bound path an object follows around a central body located at one focus of the ellipse, where total mechanical energy is negative and constant, angular momentum about the focus is conserved, and the orbiting object speeds up near the focus and slows down far from it.
An elliptical orbit is the general shape of a bound orbit under gravity. When the total mechanical energy of a satellite-planet system is negative, the satellite is trapped, and the path it traces is an ellipse with the central body sitting at one focus (not the center). That focus detail is Kepler's first law, and it matters because the orbital distance r is constantly changing, which means speed, kinetic energy, and potential energy are all changing too.
Two conservation laws run the whole show. Total mechanical energy is conserved because gravity is a conservative force, and it works out to E = -GMm/(2a), where a is the semi-major axis. Angular momentum about the central body is conserved because gravity points straight at the focus, so it exerts zero torque about that point. Put those together and you get the signature behavior of an elliptical orbit. The object moves fastest at its closest approach (perihelion) and slowest at its farthest point (aphelion), trading kinetic energy for gravitational potential energy and back again, forever.
Elliptical orbits live where gravitation meets rotation in AP Physics C: Mechanics. They pull together the universal law of gravitation from the forces material, conservation of energy from the work-energy material, and conservation of angular momentum from the rotation material. That makes them a favorite synthesis problem. The exam loves asking you to compare a satellite's speed at two points on its orbit, and the trap is that F = ma circular-motion shortcuts don't work here because r isn't constant. You have to fall back on the two conserved quantities. Understanding that E = -GMm/(2a) depends only on the semi-major axis, and that L = mvr sin θ is the same everywhere on the path, is what separates a clean FRQ solution from a dead end.
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Circular orbit (Gravitation & Orbits)
A circular orbit is just the special case of an ellipse where both foci collapse to the center and r never changes. Only in that special case can you set gravitational force equal to mv²/r, which is why that equation fails on elliptical orbits.
Kepler's laws of planetary motion (Gravitation & Orbits)
Kepler's first law says orbits are ellipses with the Sun at one focus. His second law (equal areas in equal times) is really just conservation of angular momentum in disguise, and his third law (T² ∝ a³) lets you find the period of any elliptical orbit from its semi-major axis alone.
Conservation of angular momentum (Rotation)
Gravity always points toward the focus, so the torque about the focus is zero and L stays constant. Writing L = mv_p r_p = mv_a r_a at perihelion and aphelion is the standard move for relating speeds at the two extreme points.
Escape Velocity (Gravitation & Orbits)
Escape velocity is the boundary case where total energy hits zero and the path opens up into a parabola. Elliptical orbits sit on the bound side of that line, with E < 0, which is why the object can never get infinitely far away.
No released FRQ needs the word 'elliptical' in your answer, but elliptical-orbit setups show up regularly in gravitation questions on both sections. Multiple-choice stems ask you to rank speed, kinetic energy, or potential energy at different points on the orbit, or to identify what stays constant (total energy and angular momentum) versus what changes (speed, r, KE, PE). Free-response problems typically give you the speed and distance at one point and ask for the speed at another. The expected method is to write conservation of energy and conservation of angular momentum as a system of two equations. You may also need E = -GMm/(2a) to decide whether an orbit is bound, or Kepler's third law to get a period. The classic point-loser is applying GMm/r² = mv²/r to an elliptical orbit, since that relation only holds when r is constant.
A circular orbit has constant radius, constant speed, and a net force that always equals mv²/r, so one force equation solves everything. An elliptical orbit has changing radius and changing speed, so the circular shortcut breaks and you must use conservation of energy plus conservation of angular momentum instead. Remember the hierarchy here. Every circular orbit is an ellipse with zero eccentricity, but almost no elliptical-orbit problem can be solved with circular-orbit equations.
The central body sits at one focus of the ellipse, not at the center, which is Kepler's first law.
Total mechanical energy is conserved and negative for any bound orbit, with E = -GMm/(2a) depending only on the semi-major axis.
Angular momentum about the focus is conserved because gravity exerts zero torque about that point, which forces the object to move fastest at perihelion and slowest at aphelion.
You cannot use GMm/r² = mv²/r on an elliptical orbit because the radius is not constant; that equation belongs to circular orbits only.
To find the speed at a second point on an elliptical orbit, set up conservation of energy and conservation of angular momentum as two simultaneous equations.
Kepler's third law, T² ∝ a³, applies to elliptical orbits using the semi-major axis a, not the changing orbital radius.
It is the closed path a satellite or planet follows around a central body located at one focus of the ellipse, occurring whenever the system's total mechanical energy is negative. Energy and angular momentum are conserved along the orbit, but speed and distance constantly change.
No. The Sun (or any central body) sits at one focus of the ellipse, offset from the geometric center. That is exactly what Kepler's first law states, and it is why a planet's distance from the Sun changes throughout its orbit.
No. Conservation of angular momentum (L = mvr sin θ) forces the object to move fastest at perihelion, its closest point, and slowest at aphelion, its farthest point. Only total mechanical energy and angular momentum stay constant.
In a circular orbit, the radius and speed are constant, so you can solve everything with GMm/r² = mv²/r. In an elliptical orbit, r changes continuously, so that equation fails and you must use conservation of energy plus conservation of angular momentum instead.
E = -GMm/(2a), where a is the semi-major axis. The energy is negative because the orbit is bound, and it depends only on a, so two orbits with the same semi-major axis have the same total energy and, by Kepler's third law, the same period.