This month, 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 bottom surface of a tyre viewed from the top as though with "X-ray vision."

The figure shows a shaded, elliptical region, where the tyre presses against the ground. All the interaction between the tyre and the ground takes place in this contact patch: that part of the tyre that touches the ground. As the tyre rolls, one bunch of tyre molecules after another move into the contact patch. But the patch itself more-or-less keeps the same shape, size, and position relative to the axis of rotation of the tyre and the car as a whole. We can use this fact to develop a simplified view of the interaction between tyre and ground. This simplified view lets us quickly and easily do approximate calculations good within a few percent. (A full-blown, mathematical analysis requires tyre coordinates that roll with the tyre, ground coordinates fixed on the ground, car coordinates fixed to the car, and many complicated equations relating these coordinate systems; the last few percent of accuracy in a mathematical model of tyre-ground interaction involves a great deal more complexity.)

You will recall that forces on the tyre from the ground are required to make a car change either its speed of motion or its direction of motion. Thinking of the X-ray vision picture, forces pointing up are required to make the car accelerate, forces pointing down are required to make it brake, and forces pointing right and left are required to make the car turn. Consider forward acceleration, for a moment. The engine applies a torque to the axle. This torque becomes a force, pointing backwards (down, on the diagram), that the tyre applies to the ground. By Newton's third law, the ground applies an equal and opposite force, therefore pointing forward (up), on the contact patch. This force is transmitted back to the car, accelerating it forward. It is easy to get confused with all this backward and forward action and reaction. Remember to think only about the forces on the tyre and to ignore the forces on the ground, which point the opposite way.