# Drag equation

The drag equation is a practical formula used to calculate the force of drag experienced by an object due to a fluid that it is moving through. The equation is attributed to Lord Rayleigh, who originally used ${\displaystyle L^{2}\ }$ in place of ${\displaystyle A\ }$ (L being some linear dimension). The force on a moving object due to a fluid is:

${\displaystyle \mathbf {F} _{d}={1 \over 2}\rho \mathbf {v} ^{2}AC_{d}}$see derivation

where

Fd is the force of drag,
ρ is the density of the fluid (Note that for the Earth's atmosphere, the density can be found using the barometric formula),
v is the velocity of the object relative to the fluid,
A is the reference area, and
Cd is the drag coefficient (a dimensionless constant, e.g. 0.25 to 0.45 for a car).

The reference area A is the area of the projection of the object on a plane perpendicular to the direction of motion (ie cross-sectional area). Sometimes different reference areas are given for the same object in which case a drag coefficient corresponding to each of these different areas must be given. The reference for a wing would be the plane area rather than the frontal area.

## Discussion

The equation is based on an idealized situation where all of the fluid impinges on the reference area and comes to a complete stop, building up stagnation pressure over the whole area. No real object exactly corresponds to this behavior. Cd is the ratio of drag for any real object to that of the ideal object. In practice a rough unstreamlined body (a bluff body) will have a Cd around 1, more or less. Smoother objects can have much lower values of Cd. The equation is precise, it is the Cd (drag coefficient) that can vary and is found by experiment.

Of particular importance is the v² dependence on velocity, meaning that fluid drag increases with the square of velocity. When velocity is doubled, for example, not only does the fluid strike with twice the velocity, but twice the mass of fluid strikes per second. Therefore the change of momentum per second is multiplied by four. Force is equivalent to the change of momentum divided by time. This is in contrast with solid-on-solid friction, which generally has very little velocity dependence.

## References

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