This is a 2D companion to the 3D Buller's-drop model. Instead of only replaying one launch at a time, it numerically integrates the same equations of motion frame-by-frame with RK4 (fourth-order Runge–Kutta) — not a closed-form shortcut — and pairs the animated launch with a live chart of reach vs. spore radius that exposes the size-selection mechanism the 3D view can only hint at one radius at a time.
Surface energy released: E = γ · 4πr_d² (r_d = 0.5·r_s)
Launch kinetic energy: KE = η · E
Total post-coalescence mass: m = m_s + m_d
Launch speed: v₀ = √(2·KE / m)
Stokes drag time-constant: τ = m / (6πμr_s)
Equations of motion (2D, integrated numerically via RK4):
dv_x/dt = −v_x/τ dv_y/dt = g − v_y/τ
Closed form (used only to validate the integrator):
x(t) = v₀τ(1 − e^(−t/τ)) → x∞ = v₀τ
Scaling with radius (verified numerically, γ, ρ fixed):
v₀ ∝ r^−0.5 τ ∝ r² x∞ = v₀τ ∝ r^1.5
Note on τ: this model uses the total post-coalescence mass (spore + drop) in both the launch-speed formula and the drag time-constant, so the two stay physically consistent (about 11% larger τ than a spore-mass-only shortcut would give for a 0.5·r drop).
- The size window (chart, right): x∞ grows faster than linearly with radius (r^1.5) while the "must clear the boundary layer" threshold grows only linearly (∝3r) and the "must not hit the opposite gill" ceiling (half the gap) doesn't grow with radius at all. That squeezes viable spores into a band: below it they're too weak (stuck in the no-slip air layer), above it they overshoot into the neighboring gill — a real, physically-derived upper bound on ballistospore size.
- Spore radius — drag the slider and watch the marker slide along the reach curve; the vertical dashed line shows exactly where the current radius sits relative to both failure zones.
- Efficiency — the fraction of surface energy converted to kinetic energy; raising it stretches the reach curve upward, widening the viable window.
- Gill gap — sets the "overshoot" ceiling (gap/2) in the chart; narrower gills make large spores fail sooner.
- Gravity — governs only the vertical fall after launch (the horizontal catapult phase is gravity-independent); set it to 0 and spores drift horizontally forever without ever falling clear of the hymenium.
Real basidiospores launch at roughly 0.5–2 m/s in about 2 microseconds, then lose almost all horizontal speed within ~100 µs to air drag — matching the τ values this model computes for spores in the 1–5 µm range.