Grand Prix Racing - The Science of Fast Pinewood Cars

What is the speed limit?

How fast can a Grand Prix car really go? I mean, when you build a car, what is the time you are trying to beat? The answer to these questions depends on your track and the design of your car. The competing principles involved are those of greatest energy and shortest trajectory.

Potential energy and speed

Potential energy is simply formulated as weight times height. Of course, the height is measured from the center of the weight - a special point called the center of mass (CM) or sometimes center of gravity (CG) - to some reference height, usually the floor or ground. When an object is released, we can find its time and speed using the equations derived in the model of free fall at any point below.

t = \[2(H-h)/g]

v = -gt

But the key to the best time is not necessarily having the highest center of mass at the starting line!

How far can a car "fall"?

In general it is true that the greater the potential energy of your car at the starting line, the greater is the amount of that energy that can be converted to kinetic energy, therefore speed. Each car can have a different level of potential energy available at the start of the race. I'll use the words available potential energy because the height that matters is not the distance to the floor! It's the distance from the center of mass at the starting line to the center of mass at the finish line that really matters. This means, the car that has the highest center of mass is not necessarily the one with the greatest available potential energy!

As discussed in the question about the best place to put the weight, the further back and lower the center of mass is, the more available potential energy your car will have. But as you raise its center of mass off the track at the starting line, it also rises off the track at the finish line. In fact, it rises off the track at the finish line more! So the higher the center of mass is, the lower the available potential energy is and the slower that car will travel down the track.

But, it might still get there in less time! So for this question, I'm going to answer "What is the fastest a car can go?". You can click the previous link to find out the fastest time, because a slower car actually can win!

What is the fastest a car can go?

If you could actually make a car with its center of mass down on the track and right under the rear axle, the center of mass would still be higher than the starting line. It is raised the same amount as the ramp 6.4 inches in front of the starting line (the finish line is "behind" the starting line). Note, the answer to this question only depends on how high your track is and how far back your car's rear axle can be (if the center of mass was behind your rear axle, the car would wheelie and rub its bumper on the lane median). Using the Official AWANA track, this amounts to -6.4sin(-0.369627387 radians) = 2.3121 inches. Since the center of mass is on the track surface at the starting line, it will be on the track surface at the finish line too.

The starting line of the Official AWANA track is 39.92 inches above the surface of the finish line. So the total fall will be 2.3121 + 39.92 = 42.2321 inches. Using the equation for time (t) and speed (v) above and g=386.088 in/s2, we get:

t = \[2(42.2321-0)/386.088] = 0.46773 seconds

Using this time in the speed equation:

v = -386.088(0.46773) = -180.585 in/s or

15 feet and one half inch per second

You might object because racing on a track is not the same as just falling! But since we want a top speed limit, we must assume no friction. If we didn't, we'd have to haggle over what the smallest possible amount of friction is for several sources of friction. Without friction, energy is conserved during conversion to kinetic energy, so no matter the shape of the track (as long as it doesn't rise back up to the starting height somewhere) the car will have this speed at the end of the track!

You would be right that the time (0.46762) would be way too fast! Time, unlike the speed is "warped" by the shape of the track, even the angle of a ramp. So this fastest car would easily need more than half a second to cross the finish line.

But if your starting line is 39.92 inches above its finish line (vertically) the number above is your car's speed limit too. If not, you can make your own measurements and follow this example to find your track's speed limit.

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Grand Prix Racing - The Science of Fast Pinewood Cars
Copyright © 1997, 2004 by Michael Lastufka, All rights reserved worldwide.