Here's an article on the history of those calculations.
Article
Calculators for 1/4-Mile ET & MPH vs. HP and Weight
by Jeff Lucius
Historical Perspective
In the 1950s, the late Roger Huntington (please see
http://www.mmshof.org/inductees/roger-huntington/), author of numerous books and magazine articles, was perhaps the first engineer and automotive enthusiast to empirically determine mathematical relationships between a vehicle's performance in a 1/4-mile drag race and a vehicle's power and weight. Mr. Huntington presented his graph for predicting 1/4-mile terminal speed in an article titled
Horsepower at the Drag Strip, pages 50-53 in the December 1958 issue of Rod & Custom magazine (thanks go to Earles McCaul for this information). Huntington also determined a formula for 1/4-mile elapsed time. His formulas (slightly rearranged) are:
MPH =
224 (
hp/weight)1/3 and
ET =
6.290 (
weight/hp)1/3,where
- "MPH" is the terminal speed (trap speed),
- "ET" is the elapsed time,
- "224" is the empirically determined coefficient that includes the necessary unit conversion factors,
- "hp" is the peak engine horsepower output at the clutch (net power), and
- "weight" is the the total weight of the vehicle (with driver) in pounds.
Earles McCaul fowarded to me a letter sent to him from Roger Huntington (August 1976), in which Huntington explains how he derived his empirical formula. That explanation is reproduced below with some minor editing.
"... The constant is 224 not 225, and MPH is calculated by dividing 224 by the cube root of the lbs.-per-HP ratio ... .
"... It was the result of 40 or 50 road and strip tests of many different cars. What I would do is test the car with an accelerometer - and several complex equations - to determine the true peak HP delivered at the clutch under accelerating conditions, usually in a gear giving the HP peak between 60 and 90 mph, to make it easier to get consistent accelerometer readings. (The gear effect was compensated by using a factor to allow for engine and drive line rotating inertia.) Anyway, the car was tested on a quarter-mile to accurately determine the terminal speed at the end of the quarter. Using the gross weight of the car and the tested true clutch HP, the lbs-per-HP ratio was derived ... and this was plotted on a graph vs. trap speed. The above formula is the average line through the plotted points. And actually there was amazing correlation.
"Note that I only tested cars with trap speeds between 70 and 115 mph. The curve is extrapolated above 115 mph.
"Incidentally, I did the same thing with a formula for quarter-mile e.t. That one is a constant of 6.29 times the cube root of the lbs.-per-HP ratio. However, there was more scatter in the plotted points here, due to wider variations in traction between cars. Minimum tested e.t. was 12 seconds.
"Anyway, I know I was the first to derive ths type of equation, based on true net HP. It represented many hours of work and tedious slide rule calculations over a period of 10 years." |
|
In 1964, University of Santa Clara, California Physics Professor Geoffrey T. Fox, who has been the president and CEO of Fox Racing USA in Morgan Hill, CA (maker of motorcross, BMX, and mountain bike apparel), determined the theoretical basis for Huntington's formula. Fox described this basis in 1973 in an article in The American Journal of Physics (see the References section at the end of this web page). Also in 1973, Earles McCaul published an article in the November issue of Hot Rod magazine titled
Drag Strip Dyno. In fact, it was a recent personal communication from Mr. McCaul that made me aware of the work of himself, Fox, Huntington, and others in the 1950s, 1960s, and 1970s, and of Fox's most recent (2001) empirical formulas, which are
MPH =
230 (
hp/weight)1/3 and
ET =
6.269 (
weight/hp)1/3.
In his journal article, Professor Fox lists the following key variables that affect ET and MPH.
- vehicle weight
- engine power, and the details of the torque or horsepower curve
- coefficient of friction of the tires on the track
- aerodynamic drag
- moment of inertia and frictional loss of moving parts
- drivetrain gearing
- shifting mechanisms
- location of the vehicle's vertical and horizontal center of gravity
- driver skill
Fox notes that the first two variables, weight and power, are the primary influence on MPH making it easy to use a
constant power approximation to determine a formulaic relationship. It is the variability of the remaining variables and their influence that makes it much harder to determine ET based soley on vehicle weight and engine power. What this means is: look at MPH as a determination of engine power rather than ET.
Earles McCaul provides a summary of the equations from his Hot Rod Magazine article in the MS Word document
misc/performance_mccaul_equations.doc and of Fox's equations in the MS Word document
misc/performance_fox_equations.doc. One simple relationship, the old racer's "rule of thumb", that can be extracted from these equations is that
MHP x ET = 1350; or
ET = 1350 / MHP. This linear relationship gives a first order approximation for comparing ET and MPH in the 1/4 mile. However, across the nonlinear relationships presented below, the number 1350 may go as high as the upper 1400s. Nevertheless, as a "rule of thumb" it gets you a ball park figure.
It was not until 1986 and the almost common availability of personal computers that Patrick Hale, a drag racer, engineer, and computer programmer, made available computer programs called
Quarter and
Quarter jr, which took into account nearly every variable separate from the driver that could affect acceleration, including those listed above. Hale also provided simple calculations, like those of Huntington and others, that give a first-order estimate of performance:
MPH =
234 (
hp/weight)1/3 and
ET =
5.825 (
weight/hp)1/3.
Patrick Hale is still very active in the racing community with his company Racing Systems Analysis. Among the company's many products and services are updated versions of QUARTERjr and QUARTER PRO, along with some server-side software (Dragstrip Dyno and Weather Station). You can learn more about this software and RSA at the web site
http://www.quarterjr.com/.
New Investigation
To see how well the formulas above work for modern street cars, I used Microsoft Office Excel 2003 and Poly Software PSI-Plot version 8 to compare the measured performance of 243 late model cars to the predicted performance. The data come from Road & Track Magazine road tests (as recent as the January 2008 issue) and are available, along with the charts below, in the spreadsheet
misc/performance_lrt2.xls. For acceleration testing, Road & Track uses drop-clutch starts and lift-throttle shifts. The results for acceleration are not corrected to SAE atmospheric standards. The significant difference for 0-60 mph time is 0.3 second and for the 1/4-mile time is 0.5 seconds.
In each of the four charts, I have superimposed over the road test data (the diamonds in the scatter plots) the curves from the historic equations. For the Predicted ET charts, the curve representing Huntington's formula is not shown because it graphs almost identically to Fox's curve. The plots show that Hale's formulas predict a much lower 1/4-mile elapsed time (ET) and a bit faster terminal speed (mph) than is typical for modern street cars. Hale's formulas may be more appropriate for strictly race-track cars that have better traction and drivetrains than most street cars. For the Predicted MPH charts, all of the formulas do a reasonable job of fitting the data, with Hale's formula representing the faster cars and Huntington's formula the slower cars.
I also performed new regression analysis (curve fitting) on the Road & Track test data using a power equation, y =
axb (Fox's "constant power approximation"). The dark gray curves represent a best-fit (least squares) trendline through the road test data with the power factor, b, set to 1/3 (which means multiply by the cube root of "x") or -1/3 (which means divide by the cube root of "x"). PSI-Plot was used to determine the coefficient
a for the trendline.
The black curves represent a best-fit power-equation trendline through the data with the coefficients not fixed; Excel was used and PSI-Plot verified the coefficients.
Using these new trendlines, I present here the
LRT formulas (
Lucius-Road&Track) to predict 1/4-mile track performance based on a street car's weight (curb weight plus 200 lbs) and maximum engine power (the manufacturer's quote). The coefficients of these equations may change as I periodically add additional data from Road & Track road tests to the Excel file.
LRT "ideal":
MPH =
231.3027 (
hp/weight)1/3 and
ET =
6.1178 (
weight/hp)1/3
LRT "best fit":
MPH =
215.39 (
hp/weight)0.3018 and
ET =
7.3571 (
weight/hp)0.2574
It is interesting that the LRT "ideal" equations, the ones that use the cube root of weight and hp ratios, are similar to the equations of Fox above.
The "goodness of fit" of the various curves through the data is described using R2, which is the square of the multiple correlation between the dependent and independent variables. The independent variables here are the hp and weight ratios; and the dependent variables are the measured terminal speed and elapsed time. R2 describes the proportion of the dependent variable that can be attributed to the independent variable. An R2 of 1.0 is a perfect fit to the data and an R2 of 0.0 means there is no relationship between the two variables, at least not one that can be determined using the specified equation. For example, an R2 of 0.90 means that "x" (the hp and weight ratio) explains 90% of the variability in "y" (the predicted MPH or ET) using the specified formula. The table below compares the R2 of the formulas presented here. Fox's formulas have the highest correlation of the historic formulas. The LRT formulas have the highest overall correlation. As Fox notes in his journal article, when using a power equation there is better correlation of engine power and car weight with terminal speed than there is with elapsed time.
| Goodness of Fit (R2) | | |
|---|
| | Predicted MPH | Predicted ET |
|---|
Huntington | 0.8553 | 0.7693 |
|---|
Fox | 0.9193 | 0.7870 |
|---|
Hale | 0.9124 | 0.6234 |
|---|
LRT "ideal" | 0.9214 | 0.8465 |
|---|
LRT "best fit" | 0.9301 | 0.9076 |
|---|
Comparison to 3000GT VR4 and Stealth TT Performance
So how well do these fomulas predict the actual track performance of Mitsubishi 3000GT VR4s and Stealth R/T Twin Turbos? The table below shows selected engine power ratings for a fixed test weight of 3900 lbs (with driver) and the predictions of the formulas above in comparison to typical track performance. I used the track information supplied at
http://www.team3s.com/FAQ-fastest.htm and
http://www.dragtimes.com/Mitsubishi--3000GT-Drag-Racing.html.
As you can see, actual performance for the most part falls between the predictions of the LRT and Hale formulas, with the Fox formulas doing an excellent job of matching performance when engine power exceeds 500 bhp. For the factory stock car, the LRT prediction is very close to reported performance. In general, the LRT formulas tend to underestimate track performance a little for the very-high horsepower engines, especially above 550 bhp; and the Hale formulas tend to overestimate track performance at a given horspower level. In other words, for very fast cars in the 1/4 mile the LRT formulas predict more hp than is actually needed for the highest level of performance, and the Hale formulas predict less hp than is needed based on ET but are close when mph are used. At 550 bhp and above, the Fox formulas do a good job of predicting both ET and mph.
| Comparison of Formula Estimates and Actual Track Performance | | | | | | | | |
|---|
| 3000GT VR4 / Stealth R/T TT weighing 3900 lbs with driver | | | | | | | | |
|---|
| hp | ET | mph | ET | mph | ET | mph | ET | mph |
|---|
| | LRT | Fox | Hale | Typical ranges | | | | |
|---|
| 320 | 14.003 | 101.27 | 14.427 | 99.94 | 13.405 | 101.68 | 14.2 - 13.4 | 99 - 102 |
|---|
| 350 | 13.684 | 104.05 | 14.002 | 102.97 | 13.011 | 104.77 | 13.5 - 13.0 | 103 - 105 |
|---|
| 400 | 13.221 | 108.33 | 13.393 | 107.66 | 12.444 | 109.53 | 13.3 - 12.5 | 107 - 109 |
|---|
| 450 | 12.827 | 112.25 | 12.877 | 111.97 | 11.965 | 113.92 | 12.9 - 12.0 | 111 - 114 |
|---|
| 500 | 12.483 | 115.88 | 12.433 | 115.97 | 11.552 | 117.99 | 12.2 - 11.7 | 115 - 118 |
|---|
| 550 | 12.181 | 119.26 | 12.044 | 119.72 | 11.191 | 121.80 | 12.1 - 11.3 | 118 - 122 |
|---|
| 600 | 11.911 | 122.43 | 11.700 | 123.24 | 10.871 | 125.38 | 11.5 - 11.2 | 122 - 125 |
|---|
| 650 | 11.668 | 125.42 | 11.392 | 126.57 | 10.585 | 128.78 | 11.2 - 11.0 | 125 - 128 |
|---|
| 700 | 11.448 | 128.26 | 11.114 | 129.74 | 10.326 | 132.00 | 11.0 - 10.8 | 128 - 132 |
|---|
| 750 | 11.246 | 130.96 | 10.861 | 132.76 | 10.092 | 135.07 | 10.787 | 133.9 |
|---|
| 800 | 11.061 | 133.53 | 10.630 | 135.64 | 9.877 | 138.00 | 10.561 | 135.5 |
|---|
REFERENCES
Geoffrey Fox, 1973,
On the Physics of Drag Racing: The American Journal of Physics, vol. 41, p 311-313. (reproduced in part without the explicit permission of
http://www.physnet.org/ at
misc/static_&_sliding_friction_drag_racer_design.pdf); Earles McCaul provides a summary of Fox's equations in the single page MS Word document
misc/performance_fox_equations.doc.
Roger Huntington, 1958,
Horsepower at the Drag Strip: Rod & Custom Magazine, December issue, p. 50-53.
John Lawlor, 1992,
Auto Math Handbook - Basic Calculations, Formulas, Equations and Theory for Automotive Enthusiasts: HPBooks, 146 p.
Earles McCaul, 1973,
Drag Strip Dyno: Hot Rod Magazine, November issue, p. 86.; Earles McCaul provides a summary of his equations in the single page MS Word document
misc/performance_mccaul_equations.doc.
My Excel spreadsheet containing information from Road & Track Magazine's road tests, along with the charts shown above:
misc/performance_lrt2.xls.
