The grip circle: pushing harder

Even if you’re already blazing fast, you may be able to squeeze a little more speed out of your tires by examining the grip circle a little bit closer. I’m going to focus on accelerating while exiting a corner, but the same concepts apply to braking while entering a corner.

The grip square When you’re cornering at 100%, you have 0% of the tire’s grip left for braking or accelerating, and vice versa. So when you’re cornering at 50%, you have 50% left for accelerating or braking, right? Right. Except that the car isn’t at its limit because this describes what I call the grip square.

This is how we usually characterize the grip circle, when in reality there is a lot more grip available.

The exact amount of grip you have at any given moment is dependent on the tire, the vertical load on the tire, and its contact patch. This changes constantly as the car moves, due to weight transfer, body roll, surface imperfections, elevation changes, etc., so I’ll be referring to a unit grip circle, where the outer edge of the circle is 100% of the available traction at any given moment.

You have more grip than you think In a square, the horizontal and vertical component add to 1 at all points, but produce a vector of a magnitude that is at most 1 (along the axes). For example, at 45° on the grip square we have 0.5 + 0.5 = 1 and sqrt(0.5² + 0.5²) = 0.71. Contrast this to a circle, where the vertical and horizontal parts add to at least 1, and the magnitude of the vector is always 1. For example, sin(45°) = 0.71 and cos(45°) = 0.71, which adds to 0.71 + 0.71 = 1.4 and produces a vector of magnitude sqrt(0.71² + 0.71²) = 1, which means we have 29% more total grip available than the square leads us to believe.

Fiddle with this interactive unit circle. The cosine component is your cornering percentage, and the sine component is your braking or acceleration percentage.

This means you can apply forces in multiple directions harder than 50% each and stay within the tire’s limits. Using this example, you could corner at 71% and brake or accelerate at 71% at the same time.

Under 71% cornering, there is also 71% grip available for acceleration or braking.

Note that the values don’t change linearly. For example, sin(30°) = 0.50 and cos(30°) = 0.87. Cornering 16% harder than the previous example means your maximum braking or acceleration decreases by 21%.

The values don’t change linearly. Under 87% cornering, there is only 50% grip available for acceleration or braking.

It’s also worth noting that the horizontal and vertical components add to exactly 1 along each of the axes, which still means that under 100% cornering, braking, or acceleration, your tires can’t do anything more.

How hard can you push? I do autocross in a 2000 Toyota Celica GTS, which is a low-powered FWD car. According to Google, its 0–60 time is about 6.6 seconds. Converting units, 60 mph is 26.8 m/s, which divided by 6.6 s gives an acceleration of 4.1 m/s². Dividing by 9.81 m/s² to determine acceleration G’s, we find that my car accelerates at about 0.41 g on average for its 0–60 run. Using the formulas in Chassis Engineering by Herb Adams, my car corners at about 1.06 g (weight 2500 lbs, CG height 22 in, track width 60 in, front 64%). That means my car accelerates at approximately 39% of its peak grip on average. Using this as the vertical component of our vector, we can find the horizontal component using the Pythagorean theorem solving for b: sqrt(1² - 0.39²) = 0.92. This means I can be at (or near) full throttle at 92% cornering.

This calculation isn’t perfect, but it has an important conclusion: that you can get on the throttle soon and hard after cornering at 100%.

Where does the math fall short? Let’s count the ways. It doesn’t account for handling — full throttle too early may cause the car to run wide. It doesn’t really work for cars that burn out at 100% throttle. The exact acceleration is dependent on the engine RPMs, power curve, tires, and a whole host of other factors, so the car will rarely produce exactly 0.41 g of acceleration while exiting a corner. The assumed 1.06 g of road holding isn’t perfectly accurate since I couldn’t find a performance curve for my specific tires. I’m sure you can come up with lots of other reasons, but in the meantime let’s focus on the lesson.

The lesson is that while cornering, there is more grip available for braking and acceleration than you think — possibly a lot more grip — which takes some brain adjustment to believe and apply. For example, my brain doesn’t allow my right foot to push hard on the throttle until it thinks the car is settled enough to handle it, which is usually too late. At this point I have to specifically remember to push hard early, which will hopefully become habit over time.

Cars and corners The grip circle only tells us how much load we can apply to the tire, and doesn’t necessarily tell us how much we can apply the brakes or throttle in a specific car. A low-powered FWD car will apply a vastly different acceleration load to its tires than a high-powered RWD car. My little Celica can apply tons of throttle, but you might have to be really tender with your million-horsepower Corvette until you fully unwind the steering. The calculation above should be able to help you establish a rough guess at how hard you can push your car.

Let’s talk about corners. In a long, low-speed sweeper, you may be cornering at 100% for a long time, with only a brief moment to unwind the steering for the following straight. In this type of corner, settle in to the curve and hold a good amount of throttle to keep your speed steady, then a lot of throttle as soon as you start unwinding the steering. In the Celica I can just mash it to the floor as soon as my hands start moving, but you might have to do more of a press into the floor — still quick, but less like a button.

In a much shorter 90-degree turn, you might spend a lot less time at the 100% cornering phase and a bit more time unwinding the steering on your way to the corner exit. In this type of corner, try to be on the throttle as soon as you’ve finished braking. The 100% cornering phase will be short and hard, so you’ll want to apply a little bit of throttle (which might still be 50% or more) very briefly and roll quickly into a lot of throttle. Contrast this with the previous example, which is more like holding your cornering on a skid pad.

Practice Fire up your favorite racing sim and pick a short track with a variety of corners (not Nürburgring Nordschleife — that’s it’s own animal and won’t really be helpful for this exercise). Start by picking a low-powered car (200 hp or less, and preferably FWD so you don’t spin everywhere) and do a few normal laps to set a benchmark time. Once you’ve done that, become a lunatic on the throttle — accelerate way too early and hard at every corner. Your lap times might suffer initially, but take a few laps and try to get them close to (or better than!) your benchmark time. After that, try backing off to your normal driving technique, but try getting on the throttle harder and earlier than you think possible — as soon as you finish braking — holding your speed through the corner until you start unwinding the steering. Try this on a few different tracks with a few different cars.

Even though it feels like it goes against all the mantras about smoothness, I do in fact get my best times from mashing the throttle like a lunatic, at least with cars that aren’t especially prone to oversteering. Sometimes a little bit of traction control can be your friend!

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In the first video, I’m on-par with my normal-lap ghost best with 1:33.046. In the second, I mash the throttle a bit more and end up about a third of a second faster despite making some minor mistakes, for a 1:32.707. Obviously more speed could be found by fixing my lines and braking, but pushing harder works!

In real life This does in fact work on the autocross course. For my FWD Toyota Celica, I have to be careful that I don’t spin the tires too much by pushing hard while the steering is still at an extreme angle. Doing so causes extra wear on the insides of my tires due to my camber setup. At a Test n’ Tune event at Grissom Air Force Base I was able to get decent times using this technique, allowing me to hit my target of beating 50.000 with a 49.945 and then managed to shave almost another second off my target time by layering in smoothness, for a personal best of 49.267 (STS class). Since it wasn’t a competition event, I’m not sure how I fared in the overall standings, so I’ll be sure to add more updates later. It’s still early in the season (I’m writing this in April/May), so my brain still adjusting to this new technique. When I start to unwind the steering, my conservative driving instincts tell me I can’t accelerate yet, but then I remember that I wrote this article, so I put my foot down and the car grips and goes!