Force Of Air Resistance

Aerodynamic friction exists where ever air flows over your car. The drag coeficient is a complex quantity that depends on the shape of the car, whether the air passes over it smoothly in streams or tumbles over it in millions of tiny cyclones, how close to the track the car is, etc. Even the wheels contribute some aerodynamic drag.

Aerodynamic Friction: The force opposing movement due to air being pushed out of the way and around your car.

Aerodynamic friction, can be written as a force, -apAv2/2, where a is called the aerodynamic drag coeficient, p is air density, A is the effective frontal area (crosssection) and v the speed. Note that this friction opposes the direction of the motion, v.

Daniel Bernoulli (1700-1782)

Bernoulli's principle gives us a place to start in finding a mathematical expression for the force of air resistance. Air is a fluid. Your car moves through it. Daniel Bernoulli (1700-1782) looked at fluid flows as stream lines forming a tube like strands of string cheese. Using the law of mass conservation, he characterized the relationships between pressure, fluid speed, its density and whether it was flowing up or down hill.

Bournouli's equation is

pr + pv2/2 + pgh = constant.

where pr + pgh is the static pressure in the tube at a point, and pv2/2 the dynamic pressure caused by the movement of the fluid through the tube. It is this dynamic pressure that will give us a simple model of air resistance.

Pressure is a force per unit area. F/A = pv2/2 so F = pAv2/2. This expression assumes a flat surface cutting the tube at a right angle to the fluid flow and that the flow tube does not fragment. If the surface is not flat or cut completely across the tube, the force will be altered. A simple way to allow for such deviations is to introduce a "form factor" for the surface shape, a, so F = apAv2/2.

To see if the tube remains intact as air flows around the car, we compute Reynold's number for the fastest Grand Prix car. This calculation shows that air flows around the car without turbulance. The simple force equation, F = apAv2/2, is adequate.

Air resistance is expressed in the energy relation in terms of the work it does. This is fairly easy since most if not all of the aerodynamic drag directly opposes the motion of the car. There may be some lift generated by a wing shaped car tending to change the direction of this drag, but for most cars this probably is not significant.

Work

Let k = apA/2. The units of this k show us a good insight. p has units of mass, ozs2/in, over a volume, in3. Together this is ozs2/in4. A has units of area, in2. a is scalar, so it has no units just like the pure number 2. k then has units ozs2/in2. This is a mass, ozs2/in, divided by a distance, in. Knowing that k is used in the dynamic pressure term, kv2, for air resistance, we can intrepret its units. k is the mass of air moved aside (displaced) by your vehicle for every inch it travels.

With k = apA/2, the expression for the work done is

  / y        / y
 | F.dy = k | v(y)2dy = Wa  for Wa "work of aerodynamic drag"
/ 0        / 0

However, this cannot in general be evaluated since we don't know v(y) ahead of time.

But if we take the time derivative, we can get back to the force.

          / y              / y
dWa/dt = | d(F.dy)/dt = k | d(v2dy)/dt
        / 0              / 0

Factor out m since in most cases it remains constant (rockets can be exceptions)

            / y              / y
dWa/dt = m | d(a.dy)/dt = k | d(v2dy)/dt 
          / 0              / 0

Note, a = dv/dt = (dv/dy)dy/dt = vdv/dy.

            / y                    / y
dWa/dt = m | d(v(dv/dy)dy)/dt = k | d(v2dy)/dt
          / 0                    / 0

The integration variable is changed and dy/dy = 1

            / v             / y
dWa/dt = m | d(vdv)/dt = k | d(v2dy)/dt
          / 0             / 0

The "chain" rule for integrals is used to open each of the terms

            / v             / v             / y              / y
dWa/dt = m | (dv/dt)dv + m | vd(dv)/dt = k | (dv2/dt)dy + k | v2d(dy)/dt
          / 0             / 0             / 0              / 0

Some variables can be changed and terms simplified.

             / v      / a       / y                     / v
dWa/dt = ma | dv + m | vda = k | (dv2/dv)(dv/dt)dy + k | v2dv
           / 0      / 0       / 0                     / 0

For constant acceleration the second term becomes zero. dv2/dv = 2v

             / v           / v            / v
dWa/dt = ma | dv + 0 = 2k | vdvdy/dt + k | v2dv
           / 0           / 0            / 0

Evaluate the first term and note that dy/dt = v so,

                   / v        / v
dWa/dt = mav = 2k | v2dv + k | v2dv
                 / 0        / 0

Add like terms

                   / v      
dWa/dt = mav = 3k | v2dv
                 / 0  

Evaluate this last integral

dWa/dt = mav = kv3, so that

ma = kv2 is the force of air resistance.