Terminal velocity is the maximum velocity of an object falling through a fluid, reached when the forces on the object balance. For a ball in a liquid there are three forces: weight downwards, and upthrust and drag upwards. At terminal velocity, weight equals upthrust plus drag. The resultant force is zero, so the acceleration is zero. By Newton’s first law, the ball keeps moving at a constant velocity.

Upthrust equals the weight of the liquid the ball displaces. Upthrust does not depend on speed, so upthrust stays the same throughout the fall.

Drag on a small sphere moving slowly through a liquid is given by Stokes’ law, F = 6πηrv. Drag is proportional to velocity. Stokes’ law applies only when the liquid flows smoothly around the ball (laminar flow). Stokes’ law does not apply to a skydiver, who is large and fast.

Terminal velocity equation. The specification calls the falling object a particle and the liquid a fluid. So ρ_p is the density of the particle, which here is the ball, and ρ_f is the density of the fluid, which here is the liquid.
At terminal velocity, weight = upthrust + drag.
Weight = ⁴⁄₃πr³ρ_pg, upthrust = ⁴⁄₃πr³ρ_fg, drag = 6πηrv (Stokes’ law).
So ⁴⁄₃πr³(ρ_p − ρ_f)g = 6πηrv.
Dividing both sides by 6πηr gives
v = ²⁄₉ (ρ_p − ρ_f)gr² / η
For one material in one liquid, everything except r is fixed, so v is proportional to r². Double the radius and the terminal velocity is four times bigger, so the larger ball falls faster.

Comparing liquids. With the same ball, only ρ_f and η change in v = ²⁄₉(ρ_p − ρ_f)gr²/η. A more viscous liquid gives a lower terminal velocity, because v is inversely proportional to η. A denser liquid gives more upthrust, so ρ_p − ρ_f is smaller and the terminal velocity is lower. Warming glycerol from 20 °C to 30 °C more than halves its viscosity, so a ball falls more than twice as fast.

Comparing materials. With the same radius and liquid, only ρ_p changes, and v is proportional to (ρ_p − ρ_f). What matters is how much denser the ball is than the liquid. In glycerol, steel gives 7800 − 1261 = 6539 kg/m³ and glass gives 2500 − 1261 = 1239 kg/m³, so the steel ball falls about 5.3 times as fast, even though steel is only about 3.1 times as dense as glass. If the liquid is denser than the ball, the ball does not sink.

In a viscous liquid the ball reaches terminal velocity within a few milliseconds, after falling less than a millimetre. The velocity–time graph below the main graph is zoomed in on the first few milliseconds to show the curve.

In the practical, the ball is timed between rubber bands placed well below the surface. Equal times for equal distances show that the ball has reached terminal velocity. Viscosity changes a lot with temperature: glycerol is about 1.41 Pa s at 20 °C and about 0.61 Pa s at 30 °C, so the temperature must be recorded and kept steady. A narrow tube slows the ball, so a wide tube gives results closer to Stokes’ law.

The drawing is not to scale: the balls are enlarged so they can be seen. The tube holds 60 cm of liquid. Values for the liquids are typical at the stated temperature.