|Grand Prix Racing -||The Science of Fast Pinewood Cars|
How does one find out the answer to a complicated interaction between five or more moving parts a fluid (air) and a stationary surface? The answers provided here are a combination of the disciplines of logic, science and mathematics. Experience and experiment alone is inadequate to delve into the complex nature of the many factors that can effect the outcome of a race. Yet the models are sterile without the sanity checks of experience and experiment.
Each question considered in this group of links concerns one of three types of inquiry:
- Trying to understand what factors might produce the behavior under inspection
- Exploring the effect of a specific measurable factor on the race while holding all others constant
- Explaining some procedure or aspect of the race not modeled
Behavior questions are approached by first listing the factors that affect it and all the factors that affect those and so on. These factors are examined in concert to find the "trade-offs". By the mathematical process of differentiation their model equations, each behavior and factor with dependencies is made to tell us how it changes as each factor is changed by itself. This is the power of modeling, these kinds of changes cannot be so easily isolated and examined in the "real" world. The differentials for each factor are used to reach plausible mechanisms for the various behaviors. These are also illustrated by cataloging model races that demonstrate different aspects of the behavior where possible.
Trade-offs are what makes engineering an art. You will find that there are many trade-offs in Grand Prix racing. Some are "no brainers" to figure out what balance must be struck. Others are not easy. The way you evaluate the trade-off may depend on how much of a risk-taker you are!
Because such an exhaustive approach is very time consuming and solving for many of the variables under investigation is not straight forward but requires a special iterative program, another approach can sometimes bear fruit more easily. Energy relations are used with some reasonable constraints to determine a velocity expression at the finish line. The variables of interest can often be solved in these simpler expressions and then differentiated for rigor. Sometimes the energy relations can be used in logical arguments to reach a "proof" style conclusion, that is every bit as valid.
The effects of a factor are hounded out by solving the model equations for their effect on specific race outcomes like finish time or stopping on the track when possible (some factors involving the lane median are not strictly deterministic - we do the best we can with those). This process also involves differentiation and in particular the chain rule to link up the dependencies solved for in the behavior analysis. In this way we learn if there is an optimal value, or if there is only a limit of some kind, determined by the race rules or the laws of physics. As an extra benefit, it provides a way to estimate how much error is introduced in results by not being able to measure the factor perfectly, if it can be measured at all.
To this, we add some illustrative calculations from the model to show expected race times, loss distances and other data. In this way, a catalog of "proof" races is built so that actual races may be run to see if the results are true. For all the skeptics out there, the "proof" race catalog provides a way for them to build the cars and prove to themselves the things we learn from the models. It is also possible, that the "skeptics" might find some significant differences and the models would have to be changed.
Explanations are given based on the workings of the models for both types of questions.
Questions about practice, procedure, etc. are answered when possible by referring to the results of model analysis. Otherwise, the questions are handled so as to keep personal opinion to a minimum and to make it obvious where it is present.
How do we know we have the "real" answers to pinewood car questions? Indeed, it is a deep question! How do we know any answer is "real"? Even appealing to logic, science and mathematics may not conclusively determine every answer. As with all questions that really matter, the veracity of the answers to our pinewood questions ultimately rely on FAITH.
To avoid a lengthy philosophical treatise, suffice it to say that all the intellectual disciplines are set on the shifty foundations of assumptions. Propositions relate thoughts so logic begins. Succession occurs, so we begin to do math. Points and lines exist, therefore we have geometry. Interaction affects us, so we model it as physics. The form of the basic assumptions are often stated in different ways, but they are still not independent of our own reflections on them. What we use these disciplines for can affect what we see through them and in a sense we lose reality trying to catch hold of it.
But all is not lost, it just depends on what your faith is in! Having assumptions as the object of your faith is certainly not the best way to know! It would be better to have faith in the designer of the system you want to know about. The designer knows what can and can not be done, how things are meant to work, etc.. Don't put your faith in me to believe in these mathematical models; I am not infallible. But there is a master designer who is! Jesus Christ, he designed everything and continues to hold it all together! He is the only one deserving of our faith.
|[Pit Area]||[Title Page]|
|Grand Prix Racing -||The Science of Fast Pinewood Cars|
|Copyright © 1997, 2004 by Michael Lastufka, All rights reserved worldwide.|