Sunday, July 3, 2011

Mathematics Through Car Racing Concepts. Physics, Car Design, and Driver's Inputs...

Car racing is a captivating sport for many of us. It is very interesting sport for kids as well and that fact can be used to motivate some important concepts in mathematics. Car racing track, with its irregular shape, dictated by urban projects requirements and geography, can be used to introduce calculus, integration, rational, irrational numbers, finding the length of the curve of arbitrary shape, finding the surface area enclosed by racing track, which is also of an arbitrary shape. Kids would be more interested in math if they can be shown the applications in things they are interested in.

Examples you can use. Speed of Formula 1 cars (256.78 km/hr), time of arrival, fuel consumption (72.59 L/km), engine temperature (985.23 C), laps counts (2.5), tire rubber temperature, pit time (58.5 sec), randomness of pit times (probability distribution, average, expectation), track length (10.25km), compare tire diameter, volume with the length of the track.

When talking about racing car related concepts, care should be taken to explain the existance of logic and functional relationship between objects that will be, possibly, quantified later. This approach is important when using any real world example for mathematics. For instance, it should be signified that business, social, geographical, financial analysis was done before a race track is built. Hence, business, social, geographical, financial analysis dictated a number that you will obtain later by measuring the length of the track. In this framework, student should be shown that there are non mathematical relationships that dictates the shape, size, volume of objects that can be quantified later.

Looking at a number of cars that participate in a race, a function, as a mathematical concept, can be introduced. Moreover, functions of several variables can be introduced considering that each car travels different path, with different speed, uses different amount of fuel, engines have different temperatures, drivers change gears different number of times per minute, pit stops are of different length, and drivers complete the race at different times. Winning a race can be shown to depend on several mathematical parameters too, plus the skill of the driver. Randomness that is present in some of these measurable parameters can be used to introduce probability concept, stochastic processes, statistics.

But, when the driver of a racing car drives the car, he is generating quantifiable entities. He is dealing with objects and events that has mutual quantitative relations in addition to other relationships, like mechanical for instance, or spatial, or temporal. He is generating numbers, giving initial and boundary conditions for a number of PDEs and IDEs in mechanics, thermodynamics. He may be not aware that he is generating numbers. He does not think too much mathematically while driving a car, nor thinks about physics laws that take place every moment. The drivers is involved in the specification of physical initial and boundary conditions for those physics laws. Pressing the gas, accelerator pedal, steering the wheel, using the breaks. The driver knows what the output will be for his inputs. There is of course, a feedback from the car measurements system, providing him with certain important numbers, like fuel gauge, speed dial, temperature, tyre pressure. Given the context of race, and driving the car, these numbers will influence his decision about physical inputs to the car system, like slowing down perhaps, or accelerating.

An engineer, car designer, engine designer, must predict the range, the envelope of values that the car will be subjected to. These values will be dictated by the driver later. However, the engineer who designed the engine and knows the racing car mechanics and functioning inside out may not be a good racing driver, and usually is not allowed to race. His intrinsic knowledge of car design, mechanics, and thermodynamics usually doesn’t help when it comes to fast thinking when to turn left or right on the racing track, when to accelerate or slow down, meaning engineers role is to design a car for specified range of possible characteristics. It is not the physics laws that win the race. It is the selection and sequence of initial and boundary conditions provided to the physics laws, or more preciously, to the car components that behave in accordance to physics laws. Physics laws are the same for all racing car drivers in the race. It is this selection of initial and boundary conditions that decide who will win the race. Note, also, how here physics and mathematics touches on social values. We value who comes first in the race, not the discovery of physics laws and the mathematics used during the design process. Good car design, brilliant engineers, physicist, and mathematicians can make an excellent car. They know very well the math and physics and engineering. But, there is no physic or mathematical law that will specify the initial and boundary conditions that will win the race. It is up to driver to input them into the car while racing, given the information he is getting on the track, during the race, about his own position, positions of other drivers, their speed, track characteristics, etc.. Moreover, same type of car, with same characteristics, driven at the same time by different drivers, in an race, will show the significance of the initial and boundary conditions selection to the PDEs and IDEs underpinning the engine performance and car functioning in general.

Every set of action while driving a racing car is to the race like an invention is to the laws of physics. Maxwell’s equations are great discovery, but they are necessary, yet not sufficient condition to make an invention or a novel engineering solution. It is the set of initial, boundary conditions, and specific configuration of elements that are part of the winning invention. And there are no laws, nor formulas, in mathematics or physics, that will let you produce inventions one after another.

Note also, how financial mathematics is linked, via logical connectives” IF…THEN” to the car racing. The winner is awarded say $500,000. So, IF your driver wins, THEN he will get $500,000. See how the context defines what will be said about truth values of events at the car racing. It is us, or rather, the Sports Governing body that specifies what will happen after that “THEN…”. That’s the axioms of the Car racing Award. It is not known what can be put there after “IF your driver wins THEN….”. It can be that he will get ice cream or, his car will be painted in red. Who knows? It is us who specify these logical statements and build a system from it.

It should be shown to students that numbers generated in a car race can come from different sources, and can be generated in different ways. They can be a product of driver's decisions, physics laws, random events, and should be shown that mathematics accepts all those numbers in the same way, as numbers only. It is us who keep track where the numbers come from, when, and why, as I have written about that in my previous posts.

[ applied math, applied mathematics, car racing, cars, cars movie, film cars, Honda, Honda Indy, math concepts, motivation, movie cars, racing sports cars ]

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