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The drag equation

Air and fluids · level 7, lesson 1 of 6

A question to start

Cycling into a headwind is hard work, and it gets much harder the faster you go. How does air resistance grow with speed?

The idea

An object moving through a fluid has to shove the fluid aside, and the fluid pushes back. This force is drag. For most everyday speeds it follows the drag equation:

\[F_D = \tfrac12 \rho v^2 C_D A,\]

where \(\rho\) is the fluid's density, \(v\) the speed, \(A\) the front-on area, and \(C_D\) the drag coefficient, a number that depends on the shape.

Shape \(C_D\), about
Flat plate, face on 1.2
Sphere 0.5
Modern car 0.3
Streamlined teardrop 0.04

The power needed to push against drag is force times speed: \(P = F_D v = \tfrac12 \rho v^3 C_D A\). Double your speed and you need eight times the power.

v2vdrag F4 times Fdrag
A cyclist at 10 m/s, \(C_D A = 0.3\) m² Mars Earth
Air density 0.016 kg/m³ 1.217 kg/m³
Drag 0.24 N 18 N
Power against drag 2.4 W 180 W

Worked examples

1. The cyclist on Earth. \(\tfrac12 \times 1.217 \times 10^2 \times 0.3 = 18\) N; power \(18 \times 10 = 180\) W.

2. A car at 30 m/s. \(C_D = 0.3\), \(A = 2.2\) m²: \(F_D = \tfrac12 \times 1.217 \times 900 \times 0.66 = 361\) N. Power against drag: \(361 \times 30 \approx 10.8\) kW.

3. A rover on Mars, crawling at 0.04 m/s with \(C_D A = 3\) m²: \(\tfrac12 \times 0.016 \times 0.0016 \times 3 = 3.8\times10^{-5}\) N. Drag does not matter to a rover; the ground does.

Try it

  1. How does drag change if the speed triples?
  2. Find the drag on a sphere 0.5 m across moving at 20 m/s through Earth air.
  3. The car above speeds up from 30 to 36 m/s. By what factor does the power against drag rise?
  4. Why does a Mars rover not need to be streamlined?
Answers
  1. It becomes \(3^2 = 9\) times bigger.
  2. \(A = \pi \times 0.25^2 = 0.196\) m²; \(F_D = \tfrac12 \times 1.217 \times 400 \times 0.5 \times 0.196 = 24\) N.
  3. \((36/30)^3 = 1.73\).
  4. It moves very slowly, and Mars air is thin, so its drag is tiny.

Common mistake

"Drag is in proportion to speed." At everyday speeds it goes with speed squared, and the power against it with speed cubed.

Where it leads