Drop something and it does not fall forever faster. Drag climbs with speed until it exactly cancels the weight; the net force hits zero and the acceleration stops. That steady speed is terminal velocity — and it is runnable: vt = √(2 meff g / ρ Cd A) in the quadratic regime, vt = meff g / 6πμr in the Stokes regime. The terms go in, the balance is solved, the steady speed comes out. The blue team builds it; the red team breaks it.
source G. G. Stokes, "On the Effect of the Internal Friction of Fluids on the Motion of Pendulums," Trans. Camb. Phil. Soc. 9 (1851) 8–106 — archive.org/details/b22464074 (linear/viscous case; the concept of a limiting fall speed traces to Galileo). AMBER — rendered, not quoted.
A falling body obeys m · dv/dt = meff g − Fdrag(v). Two forces, opposite signs: the effective weight meff g pulls down (mass minus buoyancy), drag Fdrag pushes up and grows with v.
Terminal velocity is simply the fixed point — the v where the two are equal and the net force is zero, so acceleration ceases:
quadratic (high Re): Fdrag = ½ ρ Cd A v² ⇒ vt = √(2 meff g / ρ Cd A).
Stokes (low Re): Fdrag = 6πμr v ⇒ vt = meff g / 6πμr.
Quadratic drag makes vt scale as √mass: quadruple the mass and vt only doubles. Denser and heavier always fall faster.
This sphere is where two neighbours reach equilibrium. the-drag-equation supplies the quadratic law F = ½ρCdAv²; the-stokes-drag supplies the linear law F = 6πμrv.
Put either drag under gravity and let it run: acceleration bleeds away as drag rises to meet the weight. Terminal velocity is their fixed point — where falling stops speeding up. Each sphere is the next one's premise.
The blue team's live check: recompute both balances, the √mass scaling, and the release-from-rest approach — and confirm them against the physics. If red flips the drag sign, this badge is where it shows.
To fix a terminal velocity you feed in the body (effective mass meff = mass − buoyancy, and its size) and the fluid (density ρ or viscosity μ), plus which drag law rules — set by the Reynolds number:
| regime | Re | drag F(v) | v grows as |
|---|---|---|---|
| Stokes | Re << 1 | 6πμr v | linear |
| quadratic | Re > ~1000 | ½ρCdA v² | quadratic |
Linear vs quadratic drag is the whole game: it decides whether vt goes like m or like √m. Feed the body and the fluid into the panel below.
The curve is v(t) released from rest; the dashed line is vt. The dot approaches it and never crosses. Every number is solved from the balance on the spot, never looked up.
What the machine produces, proven: at vt the net force is exactly zero (drag balances weight to 1e-9 in both regimes); vt scales as √mass in the quadratic regime (4× mass → 2× vt); and a body released from rest rises monotonically to vt — no overshoot — matching the closed-form tanh solution to 1e-9.
The blue team's witness (left) reconfirms these live; the red team (right) tries to make the fall run away.
The transition band (Re ~ 1–1000) obeys neither pure law. Real bodies tumble, so A and Cd flicker; compressibility bites near Mach 0.3; "added mass" makes early acceleration less than g even before drag matters. And a body only reaches vt asymptotically — over a short drop it may never get there.
"Heavier objects fall faster — always." Cut. In a vacuum all bodies fall equally (Galileo). Heavier falls faster only because more meff needs more drag to balance — a fluid effect, not gravity.
"Double the mass, double the terminal speed." Cut. Quadratic drag gives vt ∝ √m — double mass is only ×1.41; you must quadruple mass to double vt.
"Terminal velocity is a fixed number for an object." Kept, corrected. It depends on the fluid (ρ, μ), orientation, and Re-dependent Cd — a skydiver's vt varies ~2× between belly-flat and head-down.
The red team's move: flip the sign of drag so it ADDS to gravity instead of opposing it — drag now accelerates the fall. No equilibrium can exist; the ODE runs away past vt. The blue witness (window 7) is watching.
With drag added to gravity, the net force at vt is 2 meff g — never zero — and a body from rest never settles; it diverges. The witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.