Grand Prix Racing   The Science of Fast Pinewood Cars 
Results are obtained from the race model equations by evaluating them. Indeed, one design goal of the simple race model is to derive a system of model equations that can be evaluated using a scientific calculator. That is why some "short cuts" were made in the race model and why solutions resulting in closed equations were sought.
A few new variables have been introduced in the equations that follow to make them easier to compute "by hand". Print out this page or copy it to a text editor so you can write the values obtained in the blanks provided. This will help when you have to refer to the values of variables in the next equation you evaluate.
The process of evaluating the race equations begins by specifying the parameters of at least two cars and a track. At least two cars are needed so that the differences in the car parameters can be compared to the differences in computed behavior such as finish time or maximum speed. This way you can see from the model equations how changes in the cars affect their success in the race.
Next, the cars' trajectories are computed so they can be used to calculate the desired race behaviors. Remember it is the path taken by the cars' center of mass that determine the dynamics in the mathematical models.
Once the dynamics are known, you can compute how much distance separated the winner from the loser.
In each step of the evaluation, you will find several parameters to measure directly, some to estimate or measure experimentally and some that are computed from the others. Worksheets make organization of your measurements easier if you print them out or copy them to a text file.
To make the measurements, you will need a ruler marked in inches, preferably marked down to 16ths. A protractor may come in handy. You may need a weight scale marked in ounces (Imperial ounces, not troy ounces etc.). A caliper marked in inches is helpful, but not necessary. For the measurements deduced by experiment, see the experiment writeup for materials needed.

Here is a road map of the steps needed to "run" a virtual race. You may find it helpful to read about the notation conventions and units used in the mathematics of this manual.
Step  Description  Parameters Needed in Later Equations 

1  Specify track parameters (worksheet)  O, H, Lr and Lf 
2  Specify car parameters (worksheet)  N, B, CMx, CMy, Rf, Rr, *, M, Dr, Df, k 
3  Find the trajectory lengths  L1, L2 
4  Solve the dynamics on the ramp  Lm, vt, T, tr, vr, ar 
5  Solve the dynamics on the flat  v0, vi, Tf, yc, tc, tf 
6  Combine the results from the ramp and flat  t, vf, af, vmax, amax 
7  Compute how far the loser was behind the winner  l 
To reduce the number of repeated calculations, define x, the distance along the track (ramp or flat) to the point where the car's trajectory on the ramp intersects its trajectory on the flat.
x = ________ in = (Rr + CMycos* + CMxsin*)tan(O/2)
In the same spirit, define CMd, the horizontal distance to the car's center of mass from the front of the car.
CMd = ________ in = (N + B  CMx)cos* + CMysin*
Length of trajectory on ramp
L1 = ________ in = Lr + CMd  x
Length of trajectory on flat
L2 = ________ in = Lf  CMd  x
Distance needed to move air mass equal to car's inertial mass
Lm = ________ in = M/k
Terminal velocity on ramp
vt = ________ in/s = \[(mgsinO+Dr)/k]
Time needed to move air mass equal to car's inertia at terminal velocity on the ramp
T = ________ s = Lm/vt
Time it takes the car to reach the end of the ramp from the starting gate
tr = ________ s = Tarccosh(exp(L1/Lm)) Note: arccosh(X) = ln(X+\[XX1])
Car's speed at the end of the ramp
vr = ________ in/s = vt tanh(tr/T) Note: tanh(X) = (e^X  e^X)/(e^X + e^X)
Car's acceleration at the end of the ramp
ar = ________ in/s2 = (mgsinO+Dr) ((v0/vt)2  1)/M
Speed at start of flat accounting for energy lost at the join
v0 = ________ in/s = vrcosO (to more closely model a "smooth" transition, just use vr)
Speed of displaced air with energy matched to work done on car by drag
vi = ________ in/s = \[Df/k]
Time needed to displace an air mass equal to the car's inertial mass at the speed, vi
Tf = ________ s = Lm/vi
Coasting distance on flat
tc = ________ s = Tfarctan(v0/vi)
Time to coast to the finish line
yc = ________ in = Lm ln(cos(tc/Tf))
Time needed to coast to a stop
tf = ________ s = tc  Tf arccos(exp((L2yc)/Lm))
Time at finish line
t = ________ s = tr + tf
Speed at the finish line
vf = ________ in/s = vi tan((tct)/Tf)
Car's acceleration at the finish line (deceleration)
af = ________ in/s2 = Df((vf/vi)2 + 1)/M
Maximum speed (bottom of ramp)
vmax = ________ in/s = vr
Maximum acceleration (starting line)
amax = ________ in/s2 = (mgsinO+Dr)/M
Use the indices 1 and 2 on the variables above to denote which car we're taking about. Attatch the index 1 to the "winning" car, the one with the least total time. Notice in the equations below, there are no squares! T2 is the loser's "T" value not "T squared". The distance the loser, car 2, lagged behind when the first car crossed the finish line is:
if t1 > tr2 then the loser was on the flat  l = ________ in = L22  yc2  Lm2 ln(cos((tc2t1+tr2)/Tf2)) 

else if tr2 > t1 then the loser was on the ramp  l = ________ in = L12  Lm2 ln(cosh(t1/T2)) Note: cosh(X) = (e^X + e^X)/2 
else if t2 = tr2 then the loser was at the join  l = ________ in = L12 
Another way to compute the loss when the difference is only a few hundreths of a second, is to use the speed of the loser at the end of the race. Though not as acurate as the above method, it is much easier to evaluate! We must still arrange for t1 to be the winner's time so that t2 is greater than t1. Note again, there are no squared terms here, just the index "2".
l = ________ in = (t2t1)vf2
An easy way to do this mentally is to divide the loser's end speed, vf2, by 100. Then the loser loses by 100vf2 inches for every hundreth of a second it's behind! For closer races, 1000 can be used to figure the loss in inches for every thousandths of a second.
See an example virtual race.
[Pit Area]  [Title Page] 
Grand Prix Racing   The Science of Fast Pinewood Cars 
Copyright © 1997, 2004 by Michael Lastufka, All rights reserved worldwide. 