Grand Prix Racing - The Science of Fast Pinewood Cars

# What is the slowest possible time?

To find out what the slowest possible time is on a Grand Prix track, we have to find out how much friction it takes to stop the car right on the finish line and design the car so its trajectory is as long as possible. Since friction is a force opposing the forward progress of the car, any type of friction can be used to accomplish our goal (that is, no one type of friction slows the car down more, but it may be easier to create say tread friction than air resistance).

Even though tread friction is the easiest to create, let's assume the only kind of friction operating is axle friction. Of course this is unrealistic since without tread friction, the car's wheels could not turn and so any amount of axle friction no matter how great would not matter! But most people think of axle friction as the greatest source of friction in a pinewood race, so let's use that.

The model we need is the car with wheels, but no air resistance. As always, we'll use the Official AWANA track. The car will have its center of mass in the front and as high as allowed by the rules. Those determinations were made by evaluating derivatives to ensure that we really get the worst time. But in a few paragraphs, you will see intuitively why the CM must be as far forward as possible.

## Applying the model

Below then, is a table of the track and car parameter values we will use. Any parameter not listed will be taken as having the value set by a rule limit or as it comes in the kit or in the Official AWANA track plans. For example, all the wheel parameters are those of the AWANA kit.

Symbol Parameter Value Units
u Axle Friction Coefficient (front and rear) To Be Found Scalar
CMh CM above rear axle 2.4 Inches
CMx CM in front of rear axle 5.8 Inches
N Nose 0.6 Inches
B Base 5.8 Inches
u Tread Friction Coef. 0.0 Scalar
a Air Drag Coef. 0.0 Scalar
Ff Front Axle Force on Flat 4.6667 Ounces
Fr Rear Axle Force on Flat 0.0 Ounces

To use these parameters in the model equation for time and solve for the axle friction coefficient, we need to find the time for which the coasting distance is equal to the trajectory length on the flat. That is the distance the car will coast from the ramp before coming to a complete stop (nose right on top of the finish line!). For a forward weighted car, there is less available potential energy to overcome friction and the trajectory on the flat is longer so friction can fritter away more of the already lower energy. The higher CM makes the available potential energy even less despite the fact that the trajectory is shortened a bit - evenly on the ramp and flat portions.

Also, I am going to change the model slightly. As stated in the race model summary, using an initial flat velocity that is the same as the end-ramp speed will produce times closer to those of smooth transition tracks. This is because a car on a smooth transition does not lose as much energy as at an abrupt join as modeled in this manual. So v0 = vr and not vrcosO, where cosO was the amount of speed retained.

When this program is carried out, the coasting distance, yc, is set to the length of the car's trajectory on the flat. Then the yc is solved for the friction coefficient which is then used to find the time.

The result is that the

The slowest Grand Prix time is 4.9643 seconds
The highest axle friction coefficient to finish the race is 1.520

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