orbital energy circular elliptical hyperbolic
Whether a satellite or a comet will circle, trace an ellipse, or leave for good is settled by one number: the total energy E = ½mv² − GMm/r. The kinetic part ½mv² is always positive, and the gravitational potential part −GMm/r is always negative. If their sum is negative, the body is trapped on an ellipse; if it is exactly 0, it barely escapes along a parabola; if it is positive, it departs on a hyperbola. This lesson starts from the energy well and moves through the kinds of orbit, the vis-viva equation and the effective potential, ending with a launch straight from a planet's surface.
The gravitational potential energy is U(r) = −GMm/r. The blue curve is its shape: it approaches 0 far away and plunges like a deep well up close. The dashed gold line is the total energy E, and at any distance the kinetic energy is KE = E − U(r), the height between the line and the curve. Move the distance slider r and the orange bar shows how deep the potential is at that spot while the green bar shows the kinetic energy. Where the curve rises above the dashed line, the kinetic energy would have to be negative, so the body can never reach that distance, and the readout reports a forbidden region.
Now sweep the energy slider back and forth across 0. A single sign flips the whole class of orbit. With E < 0 the orbit is a closed ellipse that returns, with E = 0 it is the borderline parabola, and with E > 0 it is an open hyperbola that never comes back. As you raise the slider, the path around the central mass M at the focus opens from ellipse to parabola to hyperbola, and below the figure the class name appears next to the eccentricity e. The small dot on the right marks periapsis, the orbit's closest approach.
Rewrite the same energy balance in terms of speed and you get the vis-viva equation, v² = μ(2/r − 1/a), where μ = GM and a is the semi-major axis. Pick circle (a = r), ellipse (a fixed), or escape (a → ∞) with the buttons, and the gold curve plots that orbit's speed at every distance while the faint dashed curve always shows circular speed. Move the distance slider r and the gold dot reads off the speed there. The escape curve rides above the dashed one at √2 times circular speed at every distance, and every curve says that closer means faster, just as Kepler's second law does.
When the angular momentum L is conserved, the radial motion can be handled with a single effective potential, Veff(r) = −GM/r + L²/(2r²). The first term is gravity's inward-pulling well; the second is the centrifugal barrier, which grows with L and keeps the body from falling all the way in. Raise the slider L and the barrier rises while the bottom of the well moves outward and grows shallower. The gold dot at the lowest point is the circular orbit, and the two green dots where the dashed gold energy line meets the curve are periapsis and apoapsis, the orbit's nearest and farthest points.
Now launch straight from a planet's surface. Two sliders, launch speed and launch angle, decide what happens. Fire horizontally below circular speed and the path curves back into the ground as a suborbital flight; at exactly circular speed it becomes a circular orbit skimming the surface. Faster than that, it becomes an ellipse that climbs away and comes back, and beyond escape speed it leaves on a hyperbola. Tilt the launch upward and, short of escape speed, the path always meets the ground again and ends as a suborbital flight.