By: Brian Beckman
In the last two articles, we plunged right into some relatively complex issues, namely weight transfer and tire adhesion. This month, we regroup and review some of the basic units and dimensions needed to do dynamical calculations. Eventually, we can work up to equations sufficient for a full-blown computer simulation of car dynamics. The equations can then be "doctored" so that the computer simulation will run fast enough to be the core of an auto racing computer game. All of this is in keeping with the spirit of the series, the Physics of Racing, because so much of physics today involves computing. Software design and programming are essential skills of the modern physicist, so much so that many of us become involved in computing full time.
Physics is the science of measurement. Perhaps you have heard of highly abstract branches of physics such as quantum mechanics and relativity, in which exotic mathematics is in the forefront. But when theories are taken to the laboratory (or the race course) for testing, all the mathematics must boil down to quantities that can be measured. In racing, the fundamental quantities are distance, time, and mass. This month, we will review basic equations that will enable you to do quick calculations in your head while cooling off between runs. It is very valuable to develop a skill for estimating quantities quickly, and I will show you how.
Equations that don't involve mass are called kinematic. The first kinematic equation relates speed, time, and distance. If a car is moving at a constant speed or velocity, v, then the distance d it travels in time t is d=vt or velocity times time. This equation really expresses nothing more than the definition of velocity.
If we are to do mental calculations, the first hurdle we must jump comes from the fact that we usually measure speed in miles per hour (mph), but distance in feet and time in seconds. So, we must modify our equation with a conversion factor, like this:
If you "cancel out" the units parts of this equation, you will see that you get feet on both the left and right hand sides, as is appropriate, since equality is required of any equation. The conversion factor is 5280/3600, which happens to equal 22/15. Let's do a few quick examples. How far does a car go in one second (remember, say, "one-one-thousand, two-one-thousand," etc. to yourself to count off seconds)? At fifteen mph, we can see that we go d =15 mph times 1 sec times 22/15 = 22 feet or about 1 and a half car lengths for a 14 and 2/3 foot car like a Ferrari. So, at 30 mph, a second is three car lengths and at 60 mph it is six. If you lose a race by 1 second, you're losing by somewhere between 8 and 11 car lengths! This is because the average speed at a race at Willow Springs is between 80 and 110 mph. Every time you plow a little or get a little sideways, just visualize your competition overtaking you by a car length or so.
The next kinematic equation involves acceleration. It so happens that the distance covered by a car at constant acceleration from a standing start is given by d = 1/2 at2 or 1/2 times the acceleration times the time, squared. What conversions will help us do mental calculations with this equation? Usually, we like to measure acceleration in Gs. One G happens to be 32.1 feet per second squared. Fortunately, we don't have to deal with miles and hours here, so our equation becomes, d (feet) = 16a (Gs) t (seconds)2 roughly. So, a car accelerating from a standing start at 1/2 G, which is a typical number for a good, stock sports car, will go 8 feet in 1 second. Not very far! However, this picks up rapidly. In two seconds, the car will go 32 feet, or over two car lengths.
Just to prove to you that this isn't crazy, let's answer the questi