The air pushes back: drag on a football

Air is thin, but a fast ball has to shove a lot of it aside. The ball feels a drag force that points against its motion:

F_D = ½ ρ C_D A v²

• ρ is the air density, about 1.2 kg/m³ at sea level.
• A is the frontal area of the ball: for a football of 22 cm diameter, A = π r² ≈ 0.038 m².
• C_D is the drag coefficient, a pure number that describes how “draggy” the shape and surface are. For a football at free-kick speeds a value near 0.25 is typical (it is noticeably higher at low speeds, and it changes with speed, as World 2 explains).
• v is the speed relative to the air.

Put numbers in for a football (C_D ≈ 0.25):

• at 10 m/s the drag is about 0.6 N,
• at 20 m/s about 2.3 N,
• at 30 m/s about 5.1 N.

The ball weighs 0.43 kg × 9.81 m/s² ≈ 4.2 N. So at 30 m/s the air pushes back harder than gravity pulls down. Double the speed and the drag becomes four times bigger, because of the v².

Drag does not just shorten the flight. It also changes its shape:

• The ball keeps losing speed all the way, so the second half of the flight is slower than the first.
• The descent is steeper than the climb. In a simple model with C_D = 0.25, a 30 m/s kick at 30° leaves the ground at 30° but comes down at about 42°.
• The best angle for distance falls a little below 45°: in the same model it is about 40° (the top of the curve is quite flat, so angles from about 35° to 45° all land within a metre of each other).
• Range collapses at high speed. A football has a terminal speed near 27 m/s in this model (the speed at which drag equals its weight). A 30 m/s kick at 45° would travel about 92 m with no air, but only about 50 m with air.

Our 30 m/s, 20° example from the last lesson drops from 59 m to roughly 39 m.