The Physics of Racing

The Physics of Racing: Part 18 – “Slow-in, Fast-out!” or, Advanced Racing Line Continued

In the previous instalment, we did exact calculations for a dummy line down a 650-foot entry straight, a 180-degree left-hander, and a 650-foot exit chute. Cornering radii vary from 150 feet to 200 feet, and the track is 100 feet wide all the way around. This dummy line carries constant speed around the entire left-hander. We did those calculations to provide reference times to compare against this month’s more sophisticated calculations, in which we unwind the steering wheel and accelerate at the same time. The baseline times for the dummy line over the whole course, as a function of cornering radius, are in the second-to-last column of the following table:

The Physics of Racing: Part 17 – “Slow-in, Fast-out!” or, Advanced Analysis of the Racing Line

You may remember way back in part 5 that we did some simple calculations by hand to show that the classic racing line through a 90-degree right-hander is better than the either the line that hugs the inside or the line that hugs the outside of the corner. ‘Better’ means ‘has lowest time.’ The ‘classic racing line’ was, under the assumptions of that article, the widest possible inscribed line.

In this and the next instalment of The Physics of Racing, we raise the bar. Not only do we calculate the times for all lines through a corner, but we show a new kind of analysis for the exit, accounting for simultaneously accelerating and unwinding the steering wheel after the apex. This kind of analysis requires us to search for the lowest time because we cannot calculate it directly. We apply the approximation of the traction circle-subject of part 7-to stay within the capabilities of the car. We also model a more complex segment than in part 5, including an all-important exit chute where we take advantage of improved corner-exit speed. This style of analysis applies directly to computer simulation that we now have in progress in other continuing threads of The Physics of Racing.

The whole point of this analysis is to back up the old mantra: “slow-in, fast-out.” We will find that the quickest way through the whole segment does not include the fastest line around the corner. Rather, we get the lowest overall time by cornering more slowly so we can get back on the gas earlier. It’s always tempting to corner a little faster, but it frequently does not pay off in the context of the rest of the track.

This analysis is sufficiently long that it will take two instalments of this series. In this, the first instalment, we do exact calculations on a dummy line, which is the actual line we will drive up to the apex, but just a reference line after the apex. In the next instalment, we improve on the dummy line by accelerating and unwinding, predicting the times for a line we would actually drive, but entailing some small inexactitude.

Let’s first describe the track segment. Imagine an entry straight of 650 feet, connected to a 180-degree left-hander with outer radius 200 feet and inner radius 100 feet, connected to an exit chute of 650 feet. In the following sketch, we show the segment twice with different lines. The line on the left contains the widest possible inscribed cornering radius, and therefore the greatest possible cornering speed. The sketch on the right shows the line with the lowest overall time. Although its cornering speed is slower than in the line on the left, it includes a lengthy acceleration and unwinding phase on exit that more than makes up for it.

Line with Fastest Cornering Speed	Line with Lowest Overall Time

Note that both lines begin on the extreme right-hand side of the entry straight. Such will be a feature of every corner we analyse. Lines that begin elsewhere across the entry straight may be valid in scenarios like passing. However, we focus here on lines that are more obvious candidates for lowest times. Also, throughout, we ignore the width of the car, working with the ‘bicycle line’. If we were including the width, w, of the car, we would get the same final results on a track with outer radius of 200 + w / 2 feet and inner radius of 100 - w / 2 feet.

First, we compute exact times where we can on the course: the entry straight, the braking zone, and the corner up to the apex. To have a concrete baseline for comparison, we also do a ’suboptimal’ exit computation-the dummy line-that includes completing the corner without unwinding and then running down the exit chute dead straight somewhere in the middle of the track. In the next instalment of The Physics of Racing, we compare the dummy line to the more sophisticated exit that includes simultaneously accelerating and unwinding to use up the entire width of the track in the exit chute.

Let us enter the segment in the right-hand chute at 100 mph = 146.667 fps (feet per second). We want the total times for a number of different cornering radii between two extremes. The largest extreme is a radius of 200 feet, which is the same as the radius of the outer margin of the track. It should be obvious that it is not possible to drive a circle with a radius greater than 200 feet and still stay on the track. This extreme is depicted in the following sketch:

We take the opportunity, here, to define a number of parameters that will serve throughout. First, let us call the radius of the outer edge of the track r1; this is obviously 200 feet, but, by giving it a symbolic name, we retain the option of changing its numeric value some other time. Likewise, let’s call the radius of the inner circle r0, now 100 feet. Let’s use the symbol r to denote the radius of the inscribed circle we intend to drive. In the extreme case of the widest possible line, r is the same as r1, namely, 200 feet. In the other extreme case, that of the tightest inscribed circle, r is 150 feet, as shown in the following sketch:

We’re now ready to discuss the two remaining parameters you may have noticed: h and (Greek letter alpha). Consider the following figure illustrating the general case:

h indicates the point where we must be done with braking. More precisely, h is the distance of the turn-in point below the geometric start of the corner. Its value, by inspection, is (r - r0) cos . is the angle past the geometric top where the inscribed circle-the driving line-apexes the inner edge of the track. We see two values for the horizontal distance between the centre of the inscribed circle and the centre of the inner edge, and those values are (r - r0) sin and r1 - r. Their equality allows us to solve for :

The following table shows numeric values of h and for a number of inscribed radii (Note that if we varied r0 and r1 we would have a much larger ‘book’ of values to show. For now, we’ll just vary r.):

Inscribed Corner Radius (ft) (deg) h (ft) 150 90.00 0.00 151 73.90 14.14 152 67.38 20.00 153 62.47 24.49 154 58.41 28.28 155 54.90 31.62 160 41.81 44.72 165 32.58 54.77 170 25.38 63.25 175 19.47 70.71 180 14.48 77.46 185 10.16 83.67 190 6.38 89.44 195 3.02 94.87 200 0.00 100.00

There are a couple of interesting things to notice about these numbers. First, they match up with the visually obvious values of h = 0, = 90 and h = 100, = 0 when r = 150, r = 200 respectively. This is a good check that we haven’t made a mistake. Secondly, changes very rapidly with corner radius, and this fact has major ramifications on driving line. By driving a line just one foot larger than the minimum, one is able to apex more than fifteen degrees later!

With these data, we’re now equipped to compute all the times up to the apex and beyond. First, let’s compute the speed in the corner by assuming that our car can corner at 1g = 32.1 ft / s² = v² / r, giving us . We express all speeds in miles per hour, but other lengths in feet. We won’t take the time and space to write out all the conversions explicitly, but just remind ourselves once and for all that there are 22 feet per second for every 15 miles per hour.

Now that we have the maximum cornering speed, we can compute how much braking distance we need to get down to that speed from 100 mph. Let’s assume that our car can brake at 1g also. We know that braking causes us to lose a little velocity for each little increment of time. Precisely, dv / dt = g. However, we need to understand how the velocity changes with distance, not with time. Recall that dx / dt = v, dt = dx / v, so we get dx = vdv / g. Those who remember differential and integral calculus will immediately see that is the required formula for braking distance. In any event, the braking distance goes as the square of the speed, that is, like the kinetic energy, and that’s intuitive. However, there’s a factor of two in the numerator that’s easy to miss (the origin of this factor is in the calculus, where we compute limit expressions like ).

We next subtract the braking distance from the entry straight, and also subtract h, to give us the distance in which we can go at 100 mph, top speed, before the braking zone.

Now, we need the time spent braking, and that’s easy: . All the other times are easy to compute, so here are the times for a variety of cornering lines up to the apices (or apexes for those who aren’t Latin majors):

At first glance, it appears that the widest line is a huge winner, but we must realize that these times include only driving up to the apex, and that is far earlier on the widest line, where = 0. Suppose we continued driving all the way around the corner at constant speed and then accelerated out the exit chute at 0.5g? This is the dummy line. We won’t really drive this line after the apex, but discuss it nonetheless to provide a reference time. It’s very easy to compute and provides a foundational intuition for the more advanced exit computation to follow in the next instalment:

So, we see that, driving the dummy line, the widest line yields the slowest time from the entrance up through the complete semicircle, but the quickest overall time when the exit chute is included. The widest line has lower (better) times than the tightest line in

• the entry straight by about half a second, because h is large and the entry straight is shorter for wider circles
• in the braking zone by about three tenths because the cornering speed is higher and less braking is needed
• and in the exit chute by almost a second, again because is h large and the exit chute is thereby shorter

The widest line has higher (worse) times by about a second in the circle itself because the wider circle is also longer. When the balances are all added up, the widest line is about eight tenths quicker than the tightest line, but it’s all because of the effects of the corner on the straights before and after.

Recall once again that the dummy line is not a line we would actually drive after the apex. But, with that as a framework, we’re in position to introduce the next improvement. Everything we do from here on improves just the post-apex portion of the corner and the exit chute. We will actually drive the dummy line up to the apex. So, from this point on, we need only look at the last column in the table above, where we are shocked to see that there are almost three seconds’ spread from the slowest to the quickest way out. A good deal of this ought to be available for improvement by accelerating and unwinding.

By Brian Beckman

The Physics of Racing: Part 15 – Bumps In The Road

This week, we investigate how the effects of road bumps vary with speed. Everyone has experienced that bumps are more punchy as speed increases. A bump that you barely notice at 50 mph can sting at 100 mph. But what about at 200 mph? Will it just smack a little harder, or will it knock your teeth out or, worse, cause you to lose control? Could a bump be the limiting factor in cornering speed? In an aerodynamic car, could a bump cause a sudden and catastrophic loss of downforce and adhesion? To analyse such things, we need an understanding of the variation of bump violence with speed.

The Physics of Racing: Part 14 – Why Smoothness?

Multi-part series on the Physics of Racing

Ever wondered how drivers make really fast laps when they look like they are going slow? The key to it is smoothness in driving.

This week, I’d like to understand, from first principles, why it’s so important to be smooth on the controls of a racing car. To me, “smooth” means avoiding jerkiness when applying or releasing the brakes, the gas, or steering. Most of the time, you want to roll on and off the gas, squeeze on and off the brakes, slither in and out of steering. It’s just as important to avoid jerkiness at the end of a manoeuvre as at the beginning. For example, when steering, not only should you start turning the steering wheel with a gradual, smooth push, but you want to complete the wind-up with a gradual, smooth slowing of the push. Likewise, when you unwind the wheel, you want to start and stop the unwinding smoothly. Thus, a complete steering manoeuvre consists of four gradual, slithery start-and-stop mini-manoeuvres. A complete braking event has four little mini-slithers: one each for the start and stop of the application and the releasing of the pedal. Same for the throttle.

The Physics of Racing: Part 13 – Transients

Obviously, handling is extremely important in any racing car. In an autocross car, it is critical. A poorly handling car with lots of power will not do well at all on the typical autocross course. A Miata or CRX can usually beat a 60’s muscle car like a Pontiac GTO even though the Goat may have four or five times the power. Those cars, while magnificently powerful, were designed for straight-line acceleration at the expense of cornering.

This month, we examine one aspect of handling, that of handling transient or short-lived forces. Usually, in motor sports contexts, the word “transient” means short-lived cornering forces as opposed to braking and accelerating forces. In broader contexts, it means any short-lived forces.

Transients figure prominently in autocross. Perhaps the epitome of a transient-producing autocross feature is slalom, which requires a car and driver to flick quickly from left to right and back again. Many courses also feature esses, lane changes, chicanes (dual lane changes), alternating gates, and other variations on the theme. All of these require quick cornering response to transients. Some sports cars, like Elans, MR2’s and X1/9’s, are designed specifically to have such quick response. The general rule is that these kinds of cars get you into a corner more quickly than do other kinds. They achieve their response with low weight and low polar moment of inertia (PMI). A chief goal of this article is to explain PMI.

The Physics of Racing: Part 12 – CyberCar, Every Racer’s DWIM Car?

The cybernetic DWIM car is coming. DWIM stands for “Do What I Mean.”¹ It is a commonplace term in the field of Human-machine Interfaces, and refers to systems that automatically interpret the user’s intent from his or her inputs.

Cybernetics (or at least one aspect of it) is the science of unifying humans and machines. The objective of cybernetics is usually to amplify human capability with “intelligent” machines, but sometimes the objective is the reverse. Most of the work in cybernetics has been under the aegis of defence, for building advanced tanks and aircraft. There is a modest amount of cybernetics in the automotive industry, as well. Anti-lock Braking (ABS), Acceleration Slip Reduction (ASR), Electronic Engine Management, and Automatic Traction Control (ATC) are cybernetic DWIM systems—of a kind—already in production. They all make “corrections” on the driver’s input based on an assumed intention. Steer-by-wire, Continuously Variable Transmissions (CVT), and active suspensions are on the immediate horizon. All these features are part of a distinct trend to automate the driving experience. This month, we take a break from hard physics to look at the better and the worse of increased automation, and we look at one concept of the ultimate result, CyberCar.

The Physics of Racing: Part 11 – Braking

I was recently helping to crew Mark Thornton’s effort at the Silver State Grand Prix in Nevada. Mark had built a beautiful car with a theoretical top speed of over 200 miles per hour for the 92 mile time trial from Lund to Hiko. Mark had no experience driving at these speeds and asked me as a physicist if I could predict what braking at 200 mph would be like. This month I report on the back-of-the-envelope calculations on braking I did there in the field.

There are a couple of ways of looking at this problem. Brakes work by converting the energy of motion, kinetic energy, into the energy of heat in the brakes. Converting energy from useful forms (motion, electrical, chemical, etc.) to heat is generally called dissipating the energy, because there is no easy way to get it back from heat. If we assume that brakes dissipate energy at a constant rate, then we can immediately conclude that it takes four times as much time to stop from 200 mph as from 100 mph. The reason is that kinetic energy goes up as the square of the speed. Going at twice the speed means you have four times the kinetic energy because 4 = 2². The exact formula for kinetic energy is ½mv², where m is the mass of an object and v is its speed. This was useful to Mark because braking from 100 mph was within the range of familiar driving experience.

The Physics of Racing: Part 10 – Grip Angle

In many ways, tyre mechanics is an unpleasant topic. It is shrouded in uncertainty, controversy, and trade secrecy. Both theoretical and experimental studies are extremely difficult and expensive. It is probably the most uncontrollable variable in racing today. As such, it is the source of many highs and lows. An improvement in modelling or design, even if it is found by lucky accident, can lead to several years of domination by one tyre company, as with BFGoodrich in autocrossing now. An unfortunate choice of tyre by a competitor can lead to frustration and a disastrous hole in the budget.

This month, we investigate the physics of tyre adhesion a little more deeply than in the past. In Parts 2, 4, and 7, we used the simple friction model given by F µW, where F is the maximum traction force available from a tyre; µ, assumed constant, is the coefficient of friction; and W is the instantaneous vertical load, or weight, on a tyre. While this model is adequate for a rough, intuitive feel for tyre behaviour, it is grossly inadequate for quantitative use, say, for the computer program we began in Part 8 or for race car engineering and set up.

The Physics of Racing: Part 9 – Straights

We found in part 5 of this series, “Introduction to the Racing Line,” that a driver can lose a shocking amount of time by taking a bad line in a corner. With a six-foot-wide car on a ten-foot-wide course, one can lose sixteen hundredths by “blowing” a single right-angle turn. This month, we extend the analysis of the racing line by following our example car down a straight. It is often said that the most critical corner in a course is the one before the longest straight. Let’s find out how critical it is. We calculate how much time it takes to go down a straight as a function of the speed entering the straight. The results, which are given at the end, are not terribly dramatic, but we make several, key improvements in the mathematical model that is under continuing development in this series of articles. These improvements will be used as we proceed designing the computer program begun in Part 8.

The Physics of Racing: Part 8 – Simulating Car Dynamics with a Computer Program

This week, we begin writing a computer program to simulate the physics of racing. Such a program is quite an ambitious one. A simple racing video game, such as “Pole Position,” probably took an expert programmer several months to write. A big, realistic game like “Hard Drivin’” probably took three to five people more than a year to create. The point is that the topic of writing a racing simulation is one that we will have to revisit many times in these articles, assuming your patience holds out. There are many “just physics” topics still to cover too, such as springs and dampers, transients, and thermodynamics. Your author hopes you will find the computer programming topic an enjoyable sideline and is interested, as always, in your feedback.

We will use a computer programming language called Scheme. You have probably encountered BASIC, a language that is very common on personal computers. Scheme is like BASIC in that it is interactive. An interactive computer language is the right kind to use when inventing a program as you go along. Scheme is better than BASIC, however, because it is a good deal simpler and also more powerful and modern. Scheme is available for most PCs at very modest cost (MIT Press has published a book and diskette with Scheme for IBM compatibles for about $40; I have a free version for Macintoshes). I will explain everything we need to know about Scheme as we go along. Although I assume little or no knowledge about computer programming on your part, we will ultimately learn some very advanced things.

The Physics of Racing: Part 7 – The Traction Budget

This week, we introduce the traction budget. This is a way of thinking about the traction available for car control under various conditions. It can help you make decisions about driving style, the right line around a course, and diagnosing handling problems. We introduce a diagramming technique for visualizing the traction budget and combine this with a well-known visualization tool, the “circle of traction,” also known as the circle of friction. So this month’s article is about tools, conceptual and visual, for thinking about some aspects of the physics of racing.

To introduce the traction budget, we first need to visualize a tyre in contact with the ground. Figure 1 shows how the bottom surface of a tyre might look if we could see that surface by looking down from above. In other words, this figure shows an imaginary “X-ray” view of the bottom surface of a tyre. For the rest of the discussion, we will always imagine that we view the tyre this way. From this point of view, “up” on the diagram corresponds to forward forces and motion of the tyre and the car, “down” corresponds to backward forces and motion, “left” corresponds to leftward forces and motion, and “right” on the diagram corresponds to rightward forces and motion.

The Physics of Racing: Part 6 – Speed and Horsepower

The title of this week’s article consists of two words dear to every racer’s heart. This week, we do some “back of the envelope” calculations to investigate the basic physics of speed and horsepower (the “back of the envelope” style of calculating was covered in part 3 of this series).

How much horsepower does it take to go a certain speed? At first blush, a physicist might be tempted to say “none,” because he or she remembers Newton’s first law, by which an object moving at a constant speed in a straight line continues so moving forever, even to the end of the Universe, unless acted on by an external force. Everyone knows, however, that it is necessary to keep your foot on the gas to keep a car moving at a constant speed. Keeping your foot on the gas means that you are making the engine apply a backward force to the ground, which applies a reaction force forward on the car, to keep the car moving.

The Physics of Racing: Part 5 – Introduction to the Racing Line

This week, we analyse the best way to go through a corner. “Best” means in the least time, at the greatest average speed. We ask “what is the shape of the driving line through the corner that gives the best time?” and “what are the times for some other lines, say hugging the outside or the inside of the corner?” Given the answers to these questions, we go on to ask “what shape does a corner have to be before the driving line I choose doesn’t make any time difference?” The answer is a little surprising.

The analysis presented here is the simplest I could come up with, and yet is still quite complicated. My calculations went through about thirty steps before I got the answer. Don’t worry, I won’t drag you through the mathematics; I just sketch out the analysis, trying to focus on the basic principles. Anyone who would read through thirty formulas would probably just as soon derive them for him or herself.

The Physics of Racing: Part 4 – There Is No Such Thing as Centrifugal Force

Multi-part series on the Physics of Racing

One often hears of “centrifugal force.” This is the apparent force that throws you to the outside of a turn during cornering. If there is anything loose in the car, it will immediately slide to the right in a left hand turn, and vice versa. Perhaps you have experienced what happened to me once. I had omitted to remove an empty Pepsi can hidden under the passenger seat. During a particularly aggressive run (something for which I am not unknown), this can came loose, fluttered around the cockpit for a while, and eventually flew out the passenger window in the middle of a hard left hand corner.

I shall attempt to convince you, in this week’s article, that centrifugal force is a fiction, and a consequence of the fact first noticed just over three hundred years ago by Newton that objects tend to continue moving in a straight line unless acted on by an external force.

The Physics of Racing – Part 3: Basic Calculations

In the last two articles, we plunged right into some relatively complex issues, namely weight transfer and tyre adhesion. This week, 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 autocross computer game. Eventually, we might direct this series of articles to show how to build such a game in a typical microcomputer programming language such as C or BASIC, or perhaps even my personal favourite, LISP. 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.

The Physics of Racing Part 2: Keeping your tires stuck to the ground

In last week’s article, we explained the physics behind weight transfer. That is, we explained why braking shifts weight to the front of the car, accelerating shifts weight to the rear, and cornering shifts weight to the outside of a curve. Weight transfer is a side-effect of the tyres keeping the car from flipping over during manoeuvres. We found out that a one G braking manoeuvre in our 3200 pound example car causes 640 pounds to transfer from the rear tyres to the front tyres. The explanations were given directly in terms of Newton’s fundamental laws of Nature.

This week, we investigate what causes tyres to stay stuck and what causes them to break away and slide. We will find out that you can make a tyre slide either by pushing too hard on it or by causing weight to transfer off the tyre by your control inputs of throttle, brakes, and steering. Conversely, you can cause a sliding tyre to stick again by pushing less hard on it or by transferring weight to it. The rest of this article explains all this in term of (you guessed it) physics.

The Physics of Racing – Part 1: Weight Transfer

Global Racing Schools presents a series of articles aimed at improving the understand of the physics behind racing. For novice and seasoned drivers alike, these are must read articles if you wish to improve the way you race.

Part 1: Weight Transfer

Most autocrossers and race drivers learn early in their careers the importance of balancing a car. Learning to do it consistently and automatically is one essential part of becoming a truly good driver. While the skills for balancing a car are commonly taught in drivers’ schools, the rationale behind them is not usually adequately explained. That rationale comes from simple physics. Understanding the physics of driving not only helps one be a better driver, but increases one’s enjoyment of driving as well. If you know the deep reasons why you ought to do certain things you will remember the things better and move faster toward complete internalisation of the skills.