Delivered scenarios: uniform circular motion θ(t) = ωt, v = ωr for ring tracks, and the circular-orbit relations ω = √(μ/r³), T = 2π/ω for satellites at altitude h (r = R⊕ + h, μ ≈ 3.986004418×10¹⁴ m³/s², R⊕ ≈ 6,371,000 m).

r̈ = −μ·r̂ / r² [m/s²]; circular speed v_c = √(μ/r) = ωr, so a ring track with (ω, r) has μ = ω²r³.

Launch tangentially at r₀ with v = f·v_c: a = r₀/(2 − f²), e = |f² − 1| (periapsis if f > 1, apoapsis if f < 1); bound only for f < √2.

Kepler I: orbit is an ellipse. II: dA/dt = ½ r²θ̇ = h/2 = const. III: T = 2π√(a³/μ); vis-viva v² = μ(2/r − 1/a).

Reference curve: M = n·t + M₀, E − e sinE = M (Newton iteration), r = a(1 − e cosE).

Honest simplifications. Two-body point-mass motion in a plane: no drag, no J2 flattening, no third bodies, and the Earth is treated as a point (surface radius only sets r). The speed-factor control is an extension of the delivered circular scenarios. Preset values and captions are those of the original simulations.

From Ring Tracks to Elliptical Orbits: Kepler's Laws and Central-Force Motion Case-Study Lab

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