Mathematics and auto racing
Mathematics plays a crucial role in the world of auto racing, which has evolved significantly since its inception in the late 1800s. The sport encompasses various categories, including open-wheel racing exemplified by Formula 1 and the Indianapolis 500, as well as stock car racing like NASCAR. Each type of racing demands precise mathematical applications in car design, track layout, and race strategy. For instance, understanding geometry is vital for optimizing car performance on different track surfaces, which may include asphalt, dirt, or even ice.
Race teams rely on mathematical modeling to analyze data affecting vehicle dynamics, such as downforce, tire grip, and fuel efficiency. Strategy decisions during a race, like pit stops for tire changes or refueling, are informed by probability calculations and regression analysis. Additionally, advancements in technology, including computer simulations and wind tunnel testing, are frequently utilized to enhance car design and safety. As a result, the integration of mathematical principles not only influences how cars are engineered and raced but also contributes to the overall safety of the sport.
Mathematics and auto racing
Summary: Mathematics is essential in the design of race cars and racetracks, and the formulation of race strategy.
Auto racing has taken place for as long as cars have existed. While the early days of racing were related to fairly simple vehicles, it is now a very technical sport that has multiple branches with fans worldwide. Auto racing includes not only cars that are similar to those driven by the average citizen but also cars that are very sophisticated. The different branches of auto racing differ in the specifics of the car but all share a strong relationship to mathematical principles. The design of the car, its tires, the track, and the drivetrain require very careful measurement. The optimal path for a given track and weather condition requires a deep understanding of angles and geometry. Analysis of data to create probability information enables drivers and their teams to make wise decisions for a given set of conditions during a race.
Overview
Auto racing began as soon as the automobile was invented in the late 1800s. Auto racing is a broad term that includes single-seat cars or open wheel cars, which the Indianapolis 500 has made famous. Formula 1 racing is another type of open wheel racing but involves racing around courses that are not oval shaped. The National Association for Stock Car Auto Racing (NASCAR) utilizes cars that are modified from cars that can be bought by the general public. Many successful professional race car drivers began their racing careers with Kart racing, which involves vehicles that look like sophisticated go-karts.
Race Track Design
The racing surface and the track design are significant factors that affect both car design and driving strategy. Race surfaces can include asphalt, concrete, dirt, sand, and (sometimes) ice. Some tracks consist of a very short distance (1/4 mile) and are straight. These tracks are typically used for drag racing, which involves cars trying to go as fast as possible over a short distance. Many track designs have drivers travel in an oval, or near-oval, shape with some banking to help make high-speed turns easier. An understanding of geometry is imperative when determining how best to set up the car to handle the banking and the speeds. Track designs also include road courses in which racers turn both left and right and require a completely different car design to handle banking in both directions. The radius of a turn influences how fast the car can go without losing grip and crashing into the outer wall. The speed affects the size of the down force on the car (caused by spoilers), and, as such, different tracks require car designs.
Car Design
Race car designs evolve in response to technology changes and safety concerns, often as a result of mathematical or statistical analysis. Each branch of auto racing has very strict rules on car design, which are tested before—and sometimes after—each race. The testing includes very careful measurements of various components of the car from the size of various components of the engine, to the car’s width, height, and weight. The tests focus on items that affect the car’s power (the engine), response to the environment (temperature, air resistance, and gravity), and its influence on forces that are made on the car (width, height, and weight). Because they are such an important part of car performance, tires are supplied to the teams. A large amount of testing by tire companies goes into determining which type of tires will be provided for a particular track. The air and track temperatures often change drastically during a race and can affect how the tires interact with the track surface—providing more or less grip. Likewise, the gas that is put into the car is also provided to drivers. These standardizations provide a more even playing field for the teams so that the driver who wins is, presumably, the one with the greatest skill. Teams can alter the cars slightly during races to modify how the car receives forces from the track and from the air. These modifications include taking out or adding small wedges that alter the angle that the car sits on the track. The impact these small changes make on force is understood using trigonometry.
Race Strategy
Once teams have prepared their car and driver for the race, the issue of strategy plays an important role. Teams use probabilities to determine if and when to stop in the pits to change tires or to add gas. Gas mileage is estimated by using regression involving the number of laps, the speed of the car during the laps run, and the temperature. This estimation is not absolutely exact, and it is not uncommon for drivers to run out of gas near the end of some races because of an error in the team’s regression model. Some teams alter the usual pit stop, which involves replacing all of the tires and adding gas, by replacing just some of the tires or just adding gas.
Technology and Safety
Technology is playing a bigger role in auto racing in both car development and car testing. Car teams now use technology to measure a large number of factors that influence their car’s performance. For some branches of auto racing, these measurements are made during races. For other branches, the rules prohibit this during races but allow the measurements to occur during practice and research design. Because testing can be so expensive, some tests are done with a few drivers and then shared with all the teams. The use of computer simulation based on mathematical modeling is becoming more prevalent in all branches. It is not unusual for teams to use wind tunnels to test car design, and fluid dynamic modeling has been used to improve the aerodynamic properties of race cars. Off-season drivers use sophisticated driving simulators to hone their skills.
Technology has also been used to make racing safer. Race uniforms, helmets, and car interiors have become much less dangerous because of technological improvements. Additionally, track walls now include what is called a Steel and Foam Energy Reduction (SAFER) barrier, which dissipates the collision energy from a crash so that the impact force felt by the car and driver is smaller and less dangerous.
Bibliography
Beckman, Brian. “The Physics of Racing.” http://phors.locost7.info/contents.htm.
Bentley, Ross. Speed Secrets: Professional Race Driving Techniques. Osceola, WI: MBI Publishing, 1998.
Genta, Giancarlo. Motor Vehicle Dynamics: Modeling and Simulation. Singapore: World Scientific Publishing, 1997.
Gifford, Clive. Racing: The Ultimate Motorsports Encyclopedia. Boston: Kingfisher Publications, 2006.
Leslie-Pelecky, Diandra L. The Physics of NASCAR: The Science Behind the Speed. New York: Plume, 2009.