The Sign of Total Energy Picks the Orbit
For a satellite or a comet, whether its next path is a circle, an ellipse, or an open road that never returns is decided by a single number: the total energy E = ½mv² − GMm/r. The kinetic energy ½mv² from speed is positive; the potential energy −GMm/r from gravity is negative. If their sum is negative you get a bound ellipse, if zero the borderline parabola, if positive an escaping hyperbola. This lesson watches how energy sets the fate of an orbit through five scenes: the energy well, the conic sections, the vis-viva equation, the effective potential, and a launch from a surface.
Gravity's potential energy is U(r) = −GMm/r. Far away it approaches zero; up close it plunges like a deep well. Draw the total energy E as a horizontal line and at any distance r the kinetic energy is KE = E − U(r), the height between the line and the curve. Drag the r slider to see how kinetic and potential energy split at that spot, shown as bars. Where the E line drops below the curve, KE would be negative, a forbidden region the body cannot reach.
Now sweep the total energy E across zero. A single sign fixes the whole class of orbit. If E < 0 the orbit is a bound ellipse and returns; if E = 0 it is the borderline parabola; if E > 0 it is an open hyperbola that never comes back. Push the slider up and the path unfurls from ellipse to parabola to hyperbola, with the class name changing above it. The sign of the energy is the fate of the orbit.
Rewrite the same total energy in the language of speed and you get the vis-viva equation. In v² = μ(2/r − 1/a), μ = GM and a is the semi-major axis of the orbit. Use the segments to pick circle (a = r), ellipse (a fixed), or escape (a → ∞), then drag the r slider and read the speed v as a point on the curve. At the same r the escape speed is √2 times the circular speed, visible at once. It matches the Kepler feel that a body moves fast when close and slow when far.
When angular momentum L is conserved, the radial motion can be read from a single effective potential Veff(r) = −GM/r + L²/(2r²). The first term is the attractive gravity well; the second is a centrifugal barrier that grows with L and keeps the body from plunging into the center. Raise the L slider and the barrier grows while the bottom of the well pushes outward. The minimum of the well is exactly the circular orbit, and where an energy line meets the curve are the two turning points, the periapsis and apoapsis of the orbit.
Now launch straight from a planet's surface. Two sliders, launch speed and angle, decide the outcome. Below circular speed the body falls back on a suborbital arc; exactly at circular speed and horizontal it makes a circular orbit; faster than that, in between, it clears the surface on an elliptical orbit. Cross the escape speed and it leaves forever on a hyperbola. Move the sliders and the trajectory is drawn with its class labeled.