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Classical Mechanics

Orbits

An orbit is one of the most elegant results in mechanics: an object endlessly falling, yet never landing. Understanding it ties together every idea in this course: force, inertia, gravity, and momentum.

Newton's cannonball

Newton imagined firing a cannon horizontally from a mountaintop. Fired gently, the ball arcs down and hits the ground a short way off. Fired harder, it travels farther before landing. Fire it fast enough, he saw, and as it falls the ground curves away beneath it just as steeply, so it keeps falling forever without getting closer. That is an orbit: a projectile moving sideways so quickly that its fall matches the curve of the world.

A planet's elliptical orbit around the Sun sitting at one focus, with the velocity arrow tangent to the ellipse showing how forward motion and inward gravity combine into a closed orbit.
A planet's elliptical orbit around the Sun sitting at one focus, with the velocity arrow tangent to the ellipse showing how forward motion and inward gravity combine into a closed orbit.

The balance of fall and flight

Two things are in play. Gravity pulls the orbiting body straight toward the centre; inertia keeps it coasting forward in a straight line. Neither wins. The inward pull constantly bends the straight-line motion into a curve, and the forward motion constantly carries the body past the centre, so it circles instead of plunging in. Speed too slow and it spirals down; too fast and it flings away; just right and it settles into a stable path.

Kepler's laws

Before Newton, Johannes Kepler had already distilled the planets' motion into three rules from careful observation:

  1. Planets move in ellipses, not perfect circles, with the Sun at one focus.
  2. A planet sweeps out equal areas in equal times, so it moves faster when nearer the Sun and slower when far.
  3. The square of a planet's orbital period is proportional to the cube of its distance, tying the length of a year to the size of the orbit.

Newton's triumph

Kepler had the patterns but not the reason. Newton showed that all three laws fall straight out of universal gravitation and F=ma: given an inverse-square pull, elliptical orbits and Kepler's rules are inevitable. The same equations that describe a thrown ball, extended to a body moving fast enough, produce the motion of the planets. With that, the clockwork universe was complete: one set of laws, running the cosmos from a falling apple to the orbit of the Earth.