Orbital Motion in a Gravitational Field (AP Physics C: Mechanics)

Central force

A force that always points along the radius toward (or away from) a single point; for gravity, it points toward the attracting mass and can produce orbital motion.

Newtonian gravitational force

The magnitude of the gravitational attraction between two masses: F_g = G M m / r^2, where r is the center-to-center distance.

Universal gravitational constant (G)

The constant that sets the strength of gravity in Newton’s law of gravitation; appears in formulas like F_g = G M m / r^2.

Center-to-center distance (r)

The distance between the centers of mass of the central body and the orbiting object; for altitude h above a planet of radius R, r = R + h.

Circular orbit

An orbit with constant radius r and constant speed v, with acceleration directed inward (centripetal).

Centripetal acceleration

The inward acceleration required for circular motion, with magnitude a_c = v^2 / r.

Centripetal force condition for circular orbit

In a circular gravitational orbit, gravity supplies the centripetal force: G M m / r^2 = m v^2 / r.

Circular orbital speed

The speed needed for a circular orbit of radius r around mass M: v = sqrt(GM/r) (independent of the satellite’s mass).

Orbital period (T)

The time for one full revolution; for a circular orbit, T = 2πr / v.

Circular-orbit period formula

For a circular orbit of radius r around mass M: T = 2π sqrt(r^3/(GM)).

Kepler’s First Law (Law of Ellipses)

Planets move in ellipses with the Sun at one focus; a circle is the special case of an ellipse with zero eccentricity.

Semi-major axis (a)

Half the long diameter of an ellipse; the key size parameter for an elliptical orbit.

Kepler’s Second Law (Law of Equal Areas)

A line from the central mass to the orbiting body sweeps out equal areas in equal times, implying faster motion when closer in.

Angular momentum (L)

For a particle in orbit, L = m r vperp, where vperp is the component of velocity perpendicular to the radius.

Zero torque for central forces

For a central force like gravity, torque about the center is zero (force is along the radius), so angular momentum is conserved.

Periapsis

The point in an elliptical orbit where the object is closest to the central body and moves fastest.

Apoapsis

The point in an elliptical orbit where the object is farthest from the central body and moves slowest.

Kepler’s Third Law (Newtonian form)

For orbits around the same central mass M, T^2 = (4π^2/(GM)) a^3, so T^2 ∝ a^3.

Gravitational parameter (μ)

A common orbit constant defined by μ = GM; used to simplify orbital formulas.

Gravitational potential energy (orbital form)

Potential energy with zero at infinity: U(r) = −GMm/r; it is negative for bound gravitational systems.

Total mechanical energy (E)

The sum of kinetic and potential energy: E = K + U; its sign classifies orbital motion.

Energy classification of trajectories

E < 0: bound (elliptical, including circular); E = 0: parabolic escape; E > 0: hyperbolic flyby.

Total energy of a circular orbit

For a circular orbit of radius r: E = −GMm/(2r), and K = −(1/2)U.

Escape speed (v_esc)

The minimum speed at radius r to reach infinity with zero final speed: v_esc = sqrt(2GM/r) (independent of the object’s mass).

Vis-viva equation

A speed relation valid for any Kepler orbit: v^2 = GM(2/r − 1/a); reduces to circular speed when a = r and to escape when a → ∞.