Drag Force
F = ½ρV²·C_d·A — aerodynamic drag and the power it costs.
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The engineering
All the fluid mechanics hides in C_d; the rest is dynamic pressure times area. Reference values worth memorizing: a modern sedan ~0.28 on frontal area, a sphere ~0.47, a flat plate face-on ~1.17, a streamlined fairing under 0.05 — shape buys more than polish.
The power row is the killer: drag power goes with velocity *cubed*, so 10% more speed costs 33% more power. That cube is why cyclists draft, trucks wear skirts, and cruise speed is a fuel decision, not a schedule one.
Where this math comes from
Newton took the first analytical swing at fluid resistance in the Principia (1687) and got the V² right for the wrong microscopic reasons; Lord Rayleigh's dimensional reasoning (circa 1876) recast resistance in the coefficient form we still use. The coefficient then had to be *measured*, and Gustave Eiffel — retired from towers — did it definitively, dropping shapes from his tower and building a wind tunnel at Auteuil (1909–1912).
Eiffel's discovery that a sphere's drag coefficient suddenly *drops* near a critical Reynolds number startled Prandtl's Göttingen lab into replicating it — the boundary layer at work — and the C_d(Re) curve became aerodynamics' first great empirical database, from golf-ball dimples to reentry capsules.
- 1687Isaac NewtonPrincipia — resistance proportional to V².
- 1876Lord RayleighDimensionless resistance coefficient (circa).
- 1912Gustave EiffelWind-tunnel C_d measurements; the drag crisis found.
- 1904Ludwig PrandtlBoundary layer explains where drag comes from.
See the full timeline of the math behind every calculator →
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