Speed and Horsepower

[AL427SBF]

Interesting article on HP requirements to achieve a certain speed with a Corvette.

More interesting to the racer is the fact that it takes 145 hp to overcome drag at 150 mph. We know that our Corvette example car has about 240 hp, so about 95 hp must be going into overcoming rolling resistance and the slight braking forces arising from internal friction in the drive train and wheel bearings. Race cars capable of going 200 mph usually have at least 650 hp, about 350 of which goes into overcoming air resistance. It is probably possible to go 200 mph with a car in the 450-500 hp range, but such a car would have very good aerodynamics; expensive, low-friction internal parts; and low rolling resistance tyres, which are designed to have the smallest possible contact patch like high performance bicycle tyres, and are therefore not good for handling.
Part 6: Speed and Horsepower

I was wondering how this stacks up with some of your real world experiences in the "not so aerodynamic" Cobra?

I'm actually trying to figure out what rear-end to use with a TKO600R .82 and 26" dia tires (335mm wide) that will get me to the theoretical top speed of my Cobra with an aluminum 427 sbf that is expected to give me 450 rwhp at redline = 6500 rpm.

[gsharapa]

more is always better!!

[Dwight]

I found this a few years back and kept a copy in my files.
Dwight



Automotive Analyses
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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:

[AL427SBF]

Gary "more is always better" doesn't answer the question lol.
Dwight, yours does, that is excellent info! - thanks.

[Zoom This]

Well.....I saw a lot of quantifying information here and I can tell you what my SPF has done. Last year my good buddy and me took an 8 day/3k mile road trip around the west. On our way back home I decided to see what the "Mighty 418" would do, on a whim. It was about 7 am on a beautiful morning. Blue sky, not a breathe of wind, 60-65degrees and very dry outside. 100 led to 135 led to 150 mph in no time. At that speed in a convertible things feel VERY busy. My favorite hat was pulled off my head at 135. The car had a passenger and about 100 lbs. of luggage, tools and spare parts in the trunk. Throttle was at about 3/4 to the floor at a 150 mph cruise. Tach read about 4400 rpm with a TKO 600 5 speed and .64 overdrive. Rear axel is 3.73:1. Motor makes 478 RWHP or around 550 at the flywheel. This was the first and last time my car will do 150 mph, with me in it anyway.

[Mac VABCH]

DAVES CAR MATH: GEARING CALCULATOR

Heres a easy calculator you can try different gear rations...

[Mac VABCH]

DAVES CAR MATH: GEARING CALCULATOR

Heres a easy calculator you can try different gear rations...

[AL427SBF]

Tire Rolling Resistance, surprised tread width is not figured in - I'm wondering how bad my 275mm fronts and 335mm rears are going to skew this data hehe.

[Dwight]

180 miles per hour in a Cobra and you need 345.41 rear wheel hp. I would think 350 tq also.

Dwight

[AL427SBF]

Thanks Mac, I use this one because it has the TKO600R I'm using.
Gear Calculator

Dwight, yes provided you are geared correctly

[Tommy]

AL427SBF - My BA degree is in aerospace engineering, but I have not used it regularly for many years. I mention that so you'll have a better idea of how much weight to give the opinion I'm about to offer. Unless you have a supercomputer and very bright people to program it (e.g., a Formula 1 team), it would be very difficult to derive an accurate horsepower required for speed curve for your specific car using just the formulae you have seen so far. If you had such a curve, you could plot your horsepower available for speed curve on the same graph. The point where the two curves crossed would be your maximum speed. ... Attaining your maximum speed would likely require a very long straight track - likely longer than any track you could actually access. Running on a curved track (e.g., Talladega Speedway) would increase drag and likely reduce your maximum speed. .... The bottom line is this. Unless you have enough money for some very expensive computer time and some very expensive track time, it is unlikely you will ever need or use the information you seek. That being the case, why not do what engineers have long done. Pick a top speed that sounds reasonable to you (e.g., 180 MPH) and select gearing that would theoretically give you that speed at 6500 RPM. No one including you will ever know if the car can actually go that fast.

[AL427SBF]

Zoom This, that info will help. Seems a roadster is not as speed friendly as a sport bike, I would have never guessed that. Any one here ride and can give a comparison between the two? I have a BMW K1200S and that thing is very stable at incredible speeds.

[digginfool]

I can tell you that my BDR has a Ford Racing 392 that makes 475 HP with a Holley 750 and the Victor, Jr. it comes with. I have 4 - 48 IDA Webers so I'm going to assume that output is pretty much the same (i.e. 475 HP). Last Thursday I had the opportunity to find out just how fast my car would go. I had three lanes of straight divided highway, walled median and wall on the outside, with nobody in sight. I ran up through the gears, hit 5th and held it until it wasn't really accelerating anymore and that was just under 165 MPH indicated. So, when somebody asks 'how fast?' I can reasonably say about 163 MPH. That's real world and the math works (RPM, gear ratios and tire circumference). Tach indicated a touch over 6,000 RPM, with .82 5th gear, 3.46 rear end, 25.7 inches as advertised by Goodyear on the tire diameter. To be fair, I did have a pretty steady breeze to my back (probably about 15 mph) so not sure how that would hold on a calm day but feel confident it would still pull 160.

[undy]

My C6 Z06 takes about 500 rwhp to bust the 200 mph mark. Aerodynamics is HUGE at that point.

[AL427SBF]

digginfool, either your assumptions on your cobra are incorrect or the math doesn't hold up.

If your "BDR has a Ford Racing 392 that makes 475 HP" then the worst case top speed for you using the Top Speed to HP table would be 190 mph (and that's using 466.99 FWHP not 475).

Speed (mph) Rolling Drag (lbs) Aero Drag (lbs) Total Drag (lbs) RWHP FWHP
190.0 82.9 718.4 801.3 406.08 466.99

[digginfool]

I'm not going to challenge the table Dwight posted because I don't have the supporting data or equations he used to derive it. What I can tell you is the real world numbers my car produced. The only assumption being made is the horsepower of my engine. Ford Racing Products advertises 475 HP with a 750 Holley and the Victor, Jr. manifold. Ford is historically conservative in their advertised power so I felt it reasonable that with the 4 Webers on my engine it's still making roughly the same. The indisputable evidence is the RPM, the gear ratios, the tire size and the math. For instance, at 6,000 RPM in 5th gear (.82:1), the driveshaft is turning 7,317 RPM. Using the 3.46 ratio in the rear end, the tires are turning 2,115 RPM. The circumference of a circle is pi x diameter, so a tire that is 25.7 inches in diameter is 6.73 feet in circumference. That means the car is travelling 14,234 feet per minute which translates into 162 MPH. The fact my engine was turning just over 6,000 RPM indicated on the tach and just under 165 MPH indicated on the speedometer, horsepower and tables be damned, my car was going approximately 163 MPH. You can't argue with facts. So, draw your conclusions from there.

[Tommy]

I believe Digginfool's results. Like I said before, you can't simulate the real world of air flow adequately with a pagefull of equations. That's why you need time on a supercomputer to calculate results that are close to real world results. ... I'll add that the last few MPH as you approach maximum speed come very slowly. That's why you need a very long track to ever reach your theoretical maximum speed.

[AL427SBF]

I believe Digginfool's results too. I think Dwight's model gives a reasonable approximation for a slippery car, something aerodynamically superior to a Cobra. His 250 mph HP requirement is in line with what a Bugatti Veyron did at Nurburgring.

[67JET]

Does it count if the Cobra is on a Dyno? My Cobra did 140 mph on a rear wheel dyno about 2 weeks ago. The guy that owns the shop was tuning it and said that it should mimic real world road and wind drag on a car. He did not have a Cobra in the computer so he used a MGB for reference.

[Tommy]

67Jet,
The effects of aerodynamic drag get very pronounced at high speeds. So relatively small things like the specific shape of the nose, the location of the mirrors and the size and shape of the tires can have a big effect. For that reason, I'd guess that simulating the drag on your Cobra with an approximation of a generic MGB would be in the ball park, but not dead on.