Drag equation
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In physics, the drag equation gives the drag experienced by an object moving through a fluid.
- <math>D={1 \over 2} \rho v^2 A C_d</math>
where
- D is the force of drag,
- ρ is the density of the fluid*,
- 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).
| SI | fps gravitational | fps absolute | |
|---|---|---|---|
| force | newtons | pounds force | poundals |
| density | kilograms per cubic meter | slugs per cubic foot | pounds per cubic foot |
| velocity | meters per second | feet per second | feet per second |
| area | square meters | square feet | square feet |
* Note that for the Earth's atmosphere, the density can be found using the barometric formula. In the case of air as a fluid at 15 °C and standard atmospheric pressure:
- Density= 1.225 kilograms per cubic meter or 0.002378 slugs per cubic foot.
The reference area A is related to, but not exactly equal to, 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.
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. Consider a doubling in velocity, 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. Contrast this with other types of friction that generally do not vary at all with velocity.
Another interesting relation, though it is not part of the equation, is that the power needed to push an object through a fluid increases as the cube of the velocity. A car cruising on a highway at 50 mph (80 km/h) may require only 10 horsepower (7 kW) to overcome air drag, but that same car at 100 mph (160 km/h) requires 80 hp (60 kW). With a doubling of speed the drag (force) quadruples per the formula. Since power is the rate of doing work, exerting four times the force at twice the speed requires eight times the power. However, with a doubling of velocity, the time taken to cover a given distance also halves. This means that the total energy used to overcome drag over a given distance only increases with the square of velocity.