Many times I have been asked about how to determine the drag coefficient of a car or even a motorcycle. I only know three ways: 1) mathematically using computational fluid dynamics and a very large computer to solve the three dimensional Navier-Stokes equations; 2) a large wind tunnel, one that can fit the full size vehicle in, or a model sized one using a lot of dimensional analyses; and 3) coast down testing. Now, unless you have access to IBM's Deep Blue or have a Cray laying about and have lots of experience in computational fluid mechanics, then method 1 is probably not your first choice. This is probably true as well for method 2. Most of us just don't have a NASA sized wind tunnel in our back yards nor even a smaller one for model testing. Sophisticated instrumentation is generally needed in wind tunnel testing also. So that leaves us with method 3. Fortunately this method can be fairly simple, yet expanded to be really complex to get the real nitty gritty numbers.

What follows is an academic exercise for me. It is not intended to be a definitive answer to anything except my curiosity. To that end, if you find it useful, amusing or down right distasteful,GREAT! You may find errors, wrong assumptions, or other nastiness and I will appreciate you pointing them out to me. Complaints should be backed up by real data, through either analyses of your own or documented information. I am ameniable to changing my thinking iffn there is good reason to. Opinions, by themselves, just don't cut it.

Getting to it, finally.

What is the goal? Well, to me, it is to take empirical data, data obtained on the salt lake (Bonneville), mud lake (El Mirage or Muroc), or concrete strip (Maxton) or even a long straight stretch of hiway in your home town, and then fittiing it to a math model such that we can find C_d.
Well, jeeze, why would we want to do that? Most of us haven't done any math since high school or at least a long time ago, so this could be real agony. Accurately knowing a drag coefficient willpermit us to determine the speed performance of a particular vehicle against horsepower available or needed to meet some goal, like breaking a record.

The "model"

The forces on a vehicle immersed in air on the ground are pretty straight forward. Any number of texts may supply you with it. I started with "Analysis of Vehicle Aerodynamics", SAE SP-1036, Paper 940420, "A Detailed Drag Study Using the Coast Down Method", by M.A. Passmore, Loughborough Univ., and G.M. Le Good of the Rover Group, LTD. It must be stated that the only part of the paper's solution I used was the following model. Solution of the model is purely my effort, no plagarism.

where:
F_T = Tractive or motive force
F_D = Sum of all resistive forces
M_edv/dt = Sum of all inertial forces
Mg sin(alpha) = Gravitational force

(note: here I departed from the text in that I am going to include additional items, later, in the model and analysis)

So,

but:
F_T = 0
Mg sin(alpha) = 0

why?
well, F_T = 0 because we have slipped the tranny into nuetral and we are coasting!
and, Mg sin(alpha) = 0 because the race track is very flat, no uphill, no downhill.

So, we are left with:

This model, then is the well spring from which all solutions must arise. We will take the simplified solution set first. Because it is the one that most racers will use. It is particular to the configuration of a given vehicle and imbedded is the sum of all forces. This is likely to make the C_d a tad off, but it will be very useful for determining the vehicle performance when changes to the power plant are made.

then:

Plugging in the terms.....

Assumptions I made are that the C_d coefficient contains all drag terms that ever could be and that the M_e factor contains all of the inertia terms that could ever be! Big jump here, but, reasonable, I think. Pressing on....

Before getting to far along, let me define the terms used:

rho is the air density in slugs (yipes, what are slugs?)
V is the vehicle velocity in feet per second (ft/sec)
A_f is the frontal area, in square feet (ft²) of the vehicle. This needs to be done accurately for your vehicle.
C_d is the dimensionless drag coefficient we are looking for
M_e is the total mass (weight/gravity or W/g in slugs) of the car, driver, fuel: in short, everything that is rolling
dv is differential velocity
dt is differential time

Now for some algebra (Aaarrrrggghhhhh....!). Rearranging the terms and separating the variables......

for simplification lets let

now our equation looks like

Now we have a full blown differential equation, which has all necessary conditions for solution. We just integrate and rearrange some terms! We all remember how to do that, right?

Now, plug in the limits of the integration and you get:

The terms here are simple:
t_i  is the time, in seconds, when you entered the measured mile or trap and or "0" if using a stop watch
t_f  is the time, in seconds, when you exited the measured mile or trap and or elapsed time if using a stop watch
V_i  is the speed when you entered the measured mile or trap and corresponding to time t_i
V_f  is the speed when you exited the measured mile or trap and corresponding to elapsed time t_f

For simplification let's let (t_f - t_i) be ET for elapsed time, in seconds.

or....