|Grand Prix Racing -||The Science of Fast Pinewood Cars|
Yes! As amazing as it may seem, a slower car can win with a faster time. How can a slower car get a faster time? This paradox is solved using Newtonian physics and a little "space warp" technology. Let's join the Captain on the bridge of our AWANA Voyager space cruiser.
Captain, speaking to navigator: "What is the shortest path between these two star systems?"
Navigator: "A straight line in NORMAL space of course, but surely you want to take us out of impulse and go to warp!"
Captain, musing thoughtfully: "Oh yes, warp. Just spread a distortion in space so normal lines become great arcs, then ride the one with the greatest curvature maintaining the distortion as you go and voi-la, you get there in a blink of an eye; like racing around an athletic track and having the inside lane."
Navigator: "It sure beats running out of holodeck tokens and realigning the impulse drive every million miles or so. In fact, I remember ..."
Captain, interrupting: "We did have more time to memorize our Bible verses when our warp drive was being repaired!"
Navigator: "But Captain, surely your not suggesting that it's worth a few years ..."
Captain, smiling: "We have all eternity don't we? But for the sake of our families, we'd better get on with it. Compute a course and go to warp!"
Navigator: "Consider it done, Captain."
The "inner lane". That's what makes the distance to cover shorter! Find a path from the starting line to the finish line that's shorter than the track surface, and then engineer a car to travel on it! Simple? Create a warp field? Sort of. Actually, it's already there!
First of all, can you see that a straight line from the start to the finish (through mid-air) is shorter than the path on the track surface? But unless you build your car to raise its body (that is, center of mass) at the necessary angle and height to drive that line while its wheels remain on the track surface, your car won't be able to take advantage of that path.
But notice, the ramp is straight and the flat is straight. The place (on most tracks) that is already curved is the transition. Assume for a moment that the transition is a circular arc. Then the circle of which it is a part has a radius. The smaller the radius the shorter the distance on the arc.
What path has a shorter length than the arced surface of the transition? Any arc with a smaller radius! The "space" on the transition is already warped! Just raise the center of mass off the track into the "warp field" and the path it travels becomes SHORTER. A shorter path traveled at the same speed will be covered in less time. In fact, a shorter path can be traveled at a little slower speed and still get there in less time. This is good because raising the center of mass also decreases the available potential energy, so less speed is attained.
Sure, raising your car's center of mass doesn't produce a path that is shorter than the "bee-line" through mid-air. But it IS shorter than the path down on the track!
Now, knowing about the "warp field", how do we find the fastest time that a Grand Prix car can aim at on the Official AWANA track? First, as in the question about the speed limit assume no friction. Also, since the track is a ramp followed by a flat, the weight must be as far back as possible. Then to go to "warp" on the shorter path, the weight must be as high as the rules allow. Also, assume the wheels do spin, but that none are raised. You can see if raising a wheel or two off the track helps, but this one will have "four on the floor" to compare with those.
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. Note, the center of mass height is based on a three inch rule limit. Three inches minus the outer wheel radius gives 2.4 inches as the limit (albeit unattainable) of center of mass height above the axles.
|CMh||CM above rear axle||2.4||Inches|
|CMx||CM in front of rear axle||0.0||Inches|
The time equation to be evaluated will be similar to the one for a car with axles but no air friction.
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 evaluation is carried through, we see that the lowest time is
|Lowest Time = 2.6379 seconds|
|Speed = 180.153 in/s|
Let's compare this speed with the speed limit from the question What is the speed limit?.
Fastest speed: 180.585 in/s (with a time of 2.6409 seconds)
The difference is 180.585 - 180.153 = 0.432 in/s. This winning car is a little more than four tenths of an inch per second slower than the fastest car!
Let's look at the times; 2.6409 - 2.6379 = 0.003 seconds. At 180.585 in/s, the FASTEST car was 0.54 inches behind the SLOWER car! But it was catching up.
By using the derivatives of the race time with respect to the height of the center of mass, it can be shown that a slower car CAN NOT win on a track that has a flat more than four times its ramp length. So if your track is one like that, the height of the center of mass will not be a factor.
Also, these results only hold exactly for races with no friction! All pinewood Grand Prix races I know of involve some friction. But, computer programs using friction estimates still show the higher CM cars winning by up to half an inch over the lowest CM cars. A few thousands of a seconds is more like "a blink of an eye" than the total race time! So taking advantage of the warp in your track, may be an attempt that gets "lost in space". But it sure is something great to test your intuition about the race!
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|Grand Prix Racing -||The Science of Fast Pinewood Cars|
|Copyright © 1997, 2004 by Michael Lastufka, All rights reserved worldwide.|