Newton's cannonball.
A toy for Newton's cannonball. A cannon mounted on Earth's surface fires under inverse-square gravity, and the same equation produces every shape you'd see in nature. Turn the speed knob and watch a parabolic-looking lob become a circular orbit, then an ellipse, then a hyperbolic escape. The trajectory is always a conic section; which one you get depends on whether the launch energy sits below, at, or above the escape velocity.
What you're looking at.
Newton's insight in the Principia was that the gravity that pulls an apple to the ground is the same gravity that holds the moon in orbit. A stone thrown slowly traces an arc and crashes; thrown faster — fast enough that the curve of its fall matches the curve of the Earth — it never crashes, it orbits. There is no separate "celestial mechanics." It's one equation, sliced at different speeds.
Below circular speed (vcirc) the orbit is an ellipse whose perigee dips below the surface — so the ball crashes back. Exactly at vcirc with a horizontal launch, the ellipse becomes a circle. Between vcirc and vesc = √2 · vcirc, the orbit stays closed but elongates. Above vesc the trajectory opens into a hyperbola and the ball never returns. Every shape — parabola-ish lob, circle, ellipse, hyperbola — is a slice of a cone.
The simulation uses unit-system gravity (GM = 1, Earth's radius = 1) so vcirc = 1 and vesc = √2 ≈ 1.414. Trail style encodes the type: a thin muted line for sub-orbital lobs, a solid cobalt curve for closed orbits, a dashed cobalt line for hyperbolic escape. Multiple shots accumulate so you can compare regimes side-by-side; clear trails resets the canvas.