Grand Prix Racing - The Science of Fast Pinewood Cars

# How To Use The Model Equations

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.

## A virtual race

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.

## What you need

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 write-up for materials needed.

If your caluclator doesn't have a radians mode, you can use degrees, but you will have to remember to convert the "arc" function results from degrees to radians by dividing by 57.296 degrees per radian. To eliminate confusion, the worksheets use radians for all angular measurements.

## Steps to use the model equations

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

## Find the trajectory lengths

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

## Solve the dynamics on the ramp

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+\[XX-1])

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

## Solve the dynamics on the flat

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((L2-yc)/Lm))

## Combine the results from the ramp and flat

Time at finish line

t = ________ s = tr + tf

Speed at the finish line

vf = ________ in/s = vi tan((tc-t)/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

## Compute how far the loser was behind the winner

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((tc2-t1+tr2)/Tf2)) l = ________ in = L12 - Lm2 ln(cosh(t1/T2)) Note: cosh(X) = (e^X + e^-X)/2 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 = (t2-t1)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.

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