Six steps for fast lap times

(Jan 13, 2021) Note: The statement I made around the relationship of lateral and longitudinal grip is false. The total force generated is not the sum of the two axes but the magnitude of the diagonal, or the length of the circle's radius.

In a previous post, I looked at how to become a faster driver. I argued that while System 1 is needed for the fastest drivers, it is System 2 that builds up the intuition over time through deliberate practice. Having a better understanding of the physics of driving around a track will accelerate a driver’s progress as she will be able to get better feedback from the car and make better use of it.

This post will provide a brief overview of the physics involved in track driving and provide some tools that a driver can use to optimize their lap times. If you are interested in track driving, I highly recommend that you sign up with a proper driving school to ensure a safe, productive environment.

To understand how to maximize the car’s potential, we start with the only point of contact between the car and the track -- the tires. Tire grip is the static friction between the tires and the track surface. A tire can generate grip longitudinally (acceleration and braking), laterally (turning left and right), or a combination of both. This is typically referred to as a grip circle but in reality, it is more of a grip diamond. The relationship between longitudinal and lateral grip is approximately linear (\(a+b=c\)), not quadratic (\(a^2+b^2=c^2\)).

A simple rule to get around a track faster is to be at the edge of the grip circle as much as possible. Given two identical racing lines, using more grip means more speed. As a corollary, time spent transitioning between opposing sides should be minimized. For example, approaching corner entry, the driver would be going from full throttle to braking. Any time spent coasting (not accelerating or braking) is time spent in the middle of the grip circle which is undesirable. This is why some racers use left foot braking to minimize the transition time from accelerating to braking, and back to accelerating. The same logic applies to lateral transitions as well.

Now that the ground rule has been established, we will unpack the cornering process step by step, in reverse order. Given that the available tire grip is maximized, a car can drive faster in bigger circles than smaller ones. This is why -- in most countries -- drivers need to slow down more for right turns than for left. When the full track width is not used, the turning radius decreases and therefore the maximum speed that the car can achieve around the corner decreases.

However, the turning radius during corner exit is not static. The car will be accelerating which means that the turning radius will necessarily increase in parallel. The ideal corner exit is one where the car has the highest possible speed at the apex¹ that maintains the ability to go full throttle for the remainder of the corner.

With differing grip and power levels, the ideal corner exit line changes for each car. High-powered cars will want to maximize the available forward thrust by straightening out the corner (at the expense of more braking) while low-powered cars will be closer to the geometric line2 with less acceleration throughout the cornering phase.

With the knowledge of what angle and speed the car needs to have at the apex, one must now get to that point as fast as possible. Going back to the grip circle, braking is at the top while apex is on the left or right. Unlike the opposing case, staying at the limit of grip while transitioning between maximal longitudinal grip and maximal lateral grip is possible.

To do this, brakes are released gradually from corner entry -- freeing up longitudinal grip to be used for lateral grip. Once the apex is reached, the brakes are fully released and the car is able to accelerate out of the corner. This is called trail braking. Trail braking creates a driving line with reducing turning radius due to the continuous braking and shift towards lateral grip.

In theory, a car can go from full braking to full turning with a quick steering input change without the gradual shift through trail braking. In reality, however, this will upset the balance of the car and cause the tires to slip and either understeer or oversteer. The reason for this is weight transfer.

Weight transfer³ (or more accurately, load transfer) is a phenomenon where the load on each wheel shifts through longitudinal or lateral acceleration. This matters because of tire load sensitivity⁴ where the maximum acceleration (Gs) that a tire can support will decrease with increasing load. Given this, it is ideal to have equal load against all 4 tires to maximize total available grip.

Applying this knowledge, a rear-wheel drive car will have an advantage compared to a front-wheel drive car during acceleration as load shifts to the rear and the available grip increases. While turning, total available grip will decrease as the load shifts laterally assuming the car is roughly balanced left to right. This is not always true, however, with tracks like in Nascar where a car turns more in one direction than the other. In this case, it could make sense to bias the lateral weight distribution to achieve 50-50 load distribution during cornering.

I had an epiphany after a race where I had an exciting oversteer moment in our front-heavy Honda Civic with a generous application of trail braking. With the load shifting to the front under braking, the rear had less grip available, making the car prone to oversteering. While oversteering may look exciting, it is generally not the fastest way to go around a corner. Recall from high school physics class that static friction is higher than kinetic friction. Exceeding the static friction through erratic inputs will lead to a loss of grip. This is the basis for the racing maxim "Slow is smooth. Smooth is fast". However, there is a grey zone in between called slip angle.

Slip angle⁵ is the angle between where the wheel is pointing and where the car is actually travelling. This generates the lateral force required to turn the car. The slip angle and lateral force generated is linear at first, and then decreases in slope until it falls off. So not only is the grip circle more of a diamond, it is a function of load and slip angle.

Even among track-oriented tires, the characteristics exhibited can vary significantly. Some will have a very high peak grip but fall off faster; some will have a more gradual drop-off. Others peak at higher slip angles which requires more, shall we say, spirited driving. It is important to match these tire characteristics to the driving style to eke out every ounce of grip.

With this knowledge, one might be tempted to maintain the slip angle required for peak grip at all times. There are some practical challenges with this however. For starters, only the front wheels turn in most cars⁶ so it is difficult to achieve identical slip angle in both the front and rear tires. It is also not clear to me whether varying loads will impact the ideal slip angle like tire load sensitivity. Lastly, even if one manages to get a car in this state, it is mighty difficult to keep it there through braking, accelerating, and ever-changing environmental factors. To drive the perfect lap is mathematically impossible -- but how are the best drivers so consistent?

The best drivers have an excellent mental model of what the perfect lap looks like. Their system 1 provide an accurate estimate of the available grip and their system 2 knows the exact braking and turn-in points. This allows them to closely follow the perfect line.

Perhaps more important than driving the perfect line, the best drivers know what to do when they veer slightly off course. Even off the optimal line, there is a best way to drive to minimize the lap time. The rules of grip circle, weight transfer, and slip angle can be applied in any situation to work out the fastest way to get around the corner.

The driver must judge whether he is driving above or below the maximal grip level by collecting data on both sides. Most drivers would be on the conservative side while driving on track. Just because I can afford to total my car, it does not mean I want to. Luckily, racing simulators can replicate driving physics quite accurately these days. With financial costs out of the equation, it is desirable to spend just as much time driving beyond the limit as below it.

The best drivers can consistently push the car to its limit through a good mental model and adapt to changing environments quickly. They respect the laws of physics and use them to their advantage. Much like in life, driving is about having a plan but course correcting decisively when things inevitably go wrong.

There are many lessons to be learned from racing. One concept that I would like to internalize in my life is spending as much time beyond the limit as below it. To understand where our own limits are, we need to go beyond it; and we may be surprised to find out what we are truly capable of.

¹The apex is the clipping point on the edge of the track where the car starts to accelerate
²The racing line that maintains a constant turning radius
³Weight transfer. (n.d). In Wikipedia. https://en.wikipedia.org/wiki/Weight_transfer, accessed January 4, 2020
⁴Tire load sensitivity. (n.d). In Wikipedia. https://en.wikipedia.org/wiki/Tire_load_sensitivity, accessed January 4, 2020
⁵Slip angle. (n.d). In Wikipedia. https://en.wikipedia.org/wiki/Slip_angle, accessed January 5, 2020
⁶A handful of cars have both front and rear-wheel steering but the ratio is nowhere close to 1:1. At lower speeds, they can even turn in the opposite direction to help the car rotate