Fluid · Experiment

Stokes Drag & Terminal Velocity

Drop a sphere into a thick fluid and it does not accelerate forever. Drag climbs with speed until it exactly cancels gravity, and the sphere coasts at a steady terminal velocity — the balance that sizes raindrops, sorts sediment, and weighs single cells.

falling spheremeasured v(t)terminal velocity vtRe > 1 : Stokes breaks down

Controls

Terminal velocity vt—
Current speed v—
Time constant τ = m/6πηr—
Reynolds number Re—
Drag / weight-in-fluid—
Drag grows with speed until it balances the net downward pull — then the sphere coasts.
About this experiment ›

What you are looking at

A small sphere released from rest in a viscous fluid. Three vertical forces act: gravity pulls it down, buoyancy pushes it up, and viscous drag always opposes motion. At low speed in a thick fluid the drag is given by Stokes' law — proportional to speed itself.
Fdrag = 6πηr·v

The terminal balance

The net downward driving force is weight minus buoyancy, set by the density difference and the sphere's volume V = (4/3)πr³:
Fnet = (ρs − ρf)·(4/3)πr³·g
As the sphere speeds up, drag rises until it exactly cancels Fnet. Acceleration stops and the speed levels off at the terminal velocity:
vt = 2r²(ρs − ρf)g / 9η
Notice the strong r² dependence: double the radius and the sphere settles four times faster. That single fact is why fine silt hangs in a river for days while gravel drops at once, and how a centrifuge separates cells by size.

How fast it gets there

Solving m·dv/dt = Fnet − 6πηr·v gives an exponential approach with a time constant
τ = m / 6πηr = 2ρsr² / 9η  ⇒  v(t) = vt(1 − e−t/τ)
In a really thick fluid τ is tiny — the sphere reaches terminal velocity almost instantly, which is why the speed trace snaps up to the gold line and then runs flat.

When Stokes' law fails

Stokes' law assumes creeping flow — no turbulence, no wake. That holds only while the Reynolds number is small:
Re = ρf·v·(2r) / η ≲ 1
Push the radius up or the viscosity down and Re climbs past 1; the readout turns orange to warn you that a real sphere would now shed a wake and feel extra inertial drag, so the true terminal velocity would be lower than the clean Stokes value shown.

Things to try

Set the fluid to honey and drop a steel sphere (ρs ≈ 7800): it oozes down at a crawl, Re stays tiny, Stokes is exact. Now switch to water and keep the steel sphere — vt shoots up and Re blows past 1, flagging that the neat formula no longer applies. Finally set ρs below ρf and watch vt go negative: the sphere is buoyant and rises instead.