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FFCobra Forum Question: How fast is my Cobra with this much horsepower?
This also works for all vehicles, shhhh!
INTRO
Once upon a time, in a land far away, I was a huge fan of the original Cobra and it's final originator, Shelby. I went to the plant is Southern California, but at the time was a starving student or just out of school at Cal Poly, SLO. I could not swing the 6 grand or so, so I quietly walked away. Then I bought a used Tiger. Jeeze, I am off track and have just started this. Well, anyway, I spent an entire career with the Boeing Company doing odd jobs. Some of them involved aerodynamics and such.
Now I know how you all feel about your cars, Cobras, whether or not original or a reproduction. I know that many of you are true performance fans and have hopped up your cars to the n th degree. But, after all that hopping up, you find that there is little in the way of knowing just how fast it is or can be. Roads with the public on them just aren't the way to go and the drag strip just isn't quite enough either. What I have done for my Tiger, I am gonna try and do for you. I am going to develop a set of tools that you can use to figure it all out: "Just how fast will my Cobra go?"
BASIC EQUATIONS
The math is generally pretty easy and has been developed many times by many people, so I wont go into the derivations of the equations or where they come from. At the end, I'll give you a reference text that you may or may not want to purchase (no, I don't sell books).
There are only three things that need to be considered in determining how fast you car can go. Now, mind me, in each of these things there is a plethora (I love that word!) of other factors that have to be found first.
Total Road Loads
The summation of all the forces is called road load. It is made up of rolling resistance, aerodynamic forces, and road grade. When you have determined these then you have found the power requirement for the interface between the tire and road. Here is what this equation looks like:
Total Load (pounds) = fr * W + � * rho * V * V * Cd * A + W * sin(theta)
where:
fr = is the rolling load coefficient (dimensionless)
W = the vehicle weight (pounds)
rho = air density (slugs)
V = speed (ft/sec)
Cd = drag coefficient (dimensionless)
A = frontal area of vehicle (sq ft)
theta = road grade (degrees)
Subordinate equations
Each of the terms in the above have some underlying equations that must be used. Some can be complex, but I will make some assumptions to simplify.
Tire Rolling Resistance
The rolling resistance is very complex and has to do with the road surface and the tire itself. Most work has been done in the speed regime where we drive mostly and for heavy trucks. So I am going to use the equation that fits you best: nice clean concrete roadway, tires well aired up and at the proper temperature. That equation is:
fr = fo + 3.24 * fs *( v / 100) 2. 5
where:
v = speed (mph) {note that this is little v not big V}
fo = basic coefficient
fs = Speed effect coefficient
I am going to make an assumption here that you all have warmed up the tires for about 20 miles or so and have the tires really aired up: 50 psig or so at least! Then the two coefficients fo and fs are approximately:
fo = 0.008
fs = 0.0018
Plug these back into the equation for rolling resistance:
fr = 0.008 + 3.24 * 0.0018 *( v / 100) 2. 5
Let's try a couple of examples, say 100 mph and 200 mph
fr = 0.008 + 3.24 * 0.0018 (100/100)2. 5 = 0.0138 for 100 mph
and
fr = 0.008 + 3.24 * 0.0018 (200/100)2. 5 = 0.041 for 200mph
If we multiply the coefficients by the gross vehicle weight, then we have the Tire Rolling Resistance!
for 100 mph, Tire Rolling Resistance = 0.0138 * 2700 lbs = 37.26 lbs
for 200 mph, Tire Rolling Resistance = 0.041 * 2700 lbs = 110.7 lbs
So now we know how to determine Tire Rolling Resistance.
Air Density
Air density, rho, can be rather hard to determine from what the weather news on the local station gives us. They typically use some corrected barometric pressure values and this hoses up the ability to correctly determine air density. So we will start from first principles and develop a way to get air density from real pressure and real temperature.
P = rho * g * R * T
where:
P = absolute pressure (lbs/sq ft or psf)
rho = air density (slugs)
g = local gravity (32.174 ft/sec2)
R = universal gas constant for air (53.3, you figure out the units)
T = temperature (degrees Rankine = 458.6 + F)
F = temperature (deg Fahrenheit)
Solving for rho
rho = P / (g * R * (458.6 + F))
Now I use an absolute pressure gage to measure absolute pressure, but it reads in psia, not psf. So we need to multiply the P by 144 to convert it to psf. Then rho will be in slugs:
rho = 144 * P / (g * R * (458.6 + F))
which is what we wanted in the first place. Now this is an interesting equation because it can be used to tell how much your horsepower is reduced at any altitude and any temperature and ditto for aerodynamic losses. You need only multiply the hp or drag number by the ratio of the new density divided by the old density to effect the change. Say you had your motor dynoed at (or corrected to) standard seal level conditions where the density is 0.002377 slugs and the temp is 60 degrees F. Now you are at Denver (mile high) and the temperature is about 41 degrees out. Here is how to find the ratio:
rho/rho0 = (144* 12.27psia/ (32.174 * 53.3 * (458.6 + 60)) / 0.002377
= 0.001989 / 002377 = 0.8368 or a loss of 16.32%
See how that works? If your gee whiz wham bam motor produces 550 hp at std conditions, then it will make on 460 hp at Denver on a standard day there. The above can be used for any pressure and temperature conditions.
Drag Coefficient
Boys and girls, this can be beastly to figure out, but if you want to try then see my article, drag coefficient, for how to determine the Cd using a coast down method. Analytically it is a booger! So I am going to use a published Cd of 0.42 for the open bodied Cobra. A top might reduce it a little bit, but, not much.
Frontal Area, A
This is not much of a mystery, but people always seem to get it screwed up. If you went out in front of your car and hunkered down to look straight on at it and drew an imaginary line around the perimeter of what you saw, you would see frontal area. But, how do you get it? Well, one way is to take a photograph with a ruler for scale, overlay that with a gridded paper you can see through and count squares. Another way, not as effective but a whole lot quicker and good enough is to measure the tallest point and the widest point, convert these to feet, multiply to get square feet, then take 80% of that. This will be good enough for comparisons. With the wind screen up, this amount to about 18.5 square feet for the frontal area (A) for the Cobra 427.
Theta
This is the road grade. I am assuming that most of you are smart enough not to be racing up hill or down but are on level ground. Theta in this case is 0 degrees. But if for some reason you want to go either up or down, theta is equal to the grade in percent (close enough, anyway).
Mechanical losses
There are losses between the flywheel and where the rubber meets the road. I assume that the clutch is locked up and if you are using an AOD (yeeewww, you say, but, they handle more torque) and it is in OD and torque converter is locked up, a manual tranny is in top gear, and a Fox body 8.8 inch rear end. Some of the loss numbers are: Auxiliary equipment about 2%, Manual trans about 6%, auto trans about 8%, torque converter about 3 %, rear end about 4%. Lots of variables here like fluids, temperature, so we are going to use an average of 15% for all examples to get from flywheel hp to rear wheel hp. And vice versa..
Horsepower and Drag Relationship
As torque and horsepower are related, so to are drag and horsepower. The relationship is simple and I merely present it here.
HP = Drag * V /550
SOLUTIONS!
Ok, I think we got enough to go on now. I had planned on using horsepower in the equation and solving for the maximum speed, but this quickly gets beyond the math or spreadsheet capabilities of a lot os us in a really big hurry. So what I am going to do, is finalize the equation in a manner that you can use your own particular data. I am going to solve the equation for speeds from 10 to 250 mph (yeah, right...) so that you can simply find your flywheel horsepower go accross the chart and find your top speed. Ok?
Total Load (pounds) = fr * W + � * rho * V * V * Cd * A + W * sin(theta)
Putting in all the stuff we found above, we get:
Drag = 0.008 + 3.24 * 0.0018 *( v / 100) 2. 5* W + � * rho * V * V * Cd * A + W * sin(theta)
But remember, we are racin' on flat surfaces so the last term, the theta term goes to zero and drops out.
Drag = 0.008 + 3.24 * 0.0018 *( v / 100) 2. 5* W + � * rho * V * V * Cd * A
Also remember that
HP = Drag * V / 550
So if we multiply Drag by V / 550 on each side of the equation, we have a solution for Horsepower vs the independent variable, V.
Drag * V / 550 = HP = {0.008 + 3.24 * 0.0018 *( v / 100) 2. 5* W + � * rho * V * V * Cd * A} * V / 550
I programmed this into my Excel spread sheet to find HP vs Speed. The results are shown below.
Speed (mph) Rolling Drag (lbs) Aero Drag (lbs) Total Drag (lbs) RWHP FWHP
10.0 18.0 2.0 19.9 0.53 0.61
20.0 18.1 8.0 26.1 1.39 1.60
30.0 18.6 17.9 36.5 2.92 3.36
40.0 19.2 31.8 51.1 5.45 6.27
50.0 20.2 49.7 70.0 9.33 10.73
60.0 21.6 71.6 93.2 14.91 17.15
70.0 23.3 97.5 120.8 22.55 25.93
80.0 25.4 127.4 152.8 32.59 37.48
90.0 28.0 161.2 189.1 45.40 52.21
100.0 31.0 199.0 230.0 61.34 70.54
110.0 34.5 240.8 275.3 80.77 92.88
120.0 38.5 286.6 325.1 104.05 119.66
130.0 43.1 336.3 379.4 131.55 151.28
140.0 48.2 390.0 438.2 163.65 188.20
150.0 53.9 447.7 501.7 200.71 230.81
160.0 60.2 509.4 569.6 243.10 279.57
170.0 67.1 575.1 642.2 291.21 334.90
180.0 74.7 644.8 719.4 345.41 397.22
190.0 82.9 718.4 801.3 406.08 466.99
200.0 91.8 796.0 887.8 473.60 544.64
210.0 101.4 877.6 979.0 548.35 630.60
220.0 111.7 963.2 1074.8 630.71 725.32
230.0 122.7 1052.7 1175.4 721.08 829.24
240.0 134.5 1146.2 1280.7 819.83 942.81
250.0 147.0 1243.7 1390.7 927.36 1066.47

The data for Speed vs Flywheel horsepower is plotted below:

I hope this helps all of you settle many debates and/or starts a lot of new ones!
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