Resonance Rider

The swing is a driven, damped oscillator: a = −ω₀²x − 2γv + F, integrated step by step with semi-implicit Euler — no scripted motion anywhere. Holding PUMP applies the constant push F in one direction, exactly like a parent standing on one side of a playground swing. The push does work F·v: positive while the swing moves away, negative while it returns. Pumping in rhythm with the natural period T = 2π/ω₀ is therefore the only way to accumulate energy — the resonance condition, played by hand.

Damping drains energy at a rate proportional to v², so the amplitude stops growing when your average power in equals the damping loss — at roughly A ≈ F/(πγω₀) for a perfect rhythm. The early levels keep γ small and that ceiling far above the gate. The later levels raise γ: the same drive buys far less arc, mistimed pushes are eaten immediately, and you feel the resonance peak flatten — the same physics that makes heavy damping the cure for wobbling bridges.

The dashed ghost line runs the very same integrator a few seconds into the future under your current pump state, so you can see where the rhythm is taking you before you commit. The strip at the bottom is the swing angle against time — a live version of the damped and driven graphs from Lessons 23.3 and 23.4.