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Drag equation

From Academic Kids

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,
Cd is the drag coefficient (a dimensionless constant, e.g. 0.25 to 0.45 for a car),
ρ is the density of the fluid*,
v is the velocity of the object relative to the fluid, and
A is the reference area.
Units used
  SI fps gravitationalfps absolute
forcenewtonspounds forcepoundals
densitykilograms per cubic meterslugs per cubic footpounds per cubic foot
velocitymeters per secondfeet per secondfeet per second
areasquare meterssquare feetsquare 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 plan 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. 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). This is because the force exerted by drag quadruples (2² = 4), and the power required equals force times velocity.

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