## Monday, September 28, 2015

### Are there any F1 drivers in the room?

When I was working through HW 3, the last question caught my attention. What forces must a car overcome to prevent sliding out while cornering?
The clear answer is the radial acceleration of the turn. If the car lost traction and could no longer exert acceleration in the radial direction, it would continue on at the current speed in the tangential direction at the instant it lost traction.
Alas, the complete answer is quite a bit more complicated. As you can see below, a wheel utilizes its static friction, even while in motion. At a given moment, the point on the wheel in contact with the ground is not moving in the direction of the wheel's movement with respect to the ground.
If the car loses traction, then the kinetic friction comes into play. If the wheel is simply spinning independent of the ground, its friction is much lower - in all directions. Once the wheels are spinning, the car is only accelerating at a rate proportional to the kinetic friction, which won't be very much. Watch an example here :
A BMW M5 should be able to accelerate quite a bit more rapidly, but in order to do so, it must utilize all of the static friction of the wheels. Once the wheels start spinning, not enough force can be transferred onto the road surface to accelerate the car much more than if he had been riding a tricycle.
The question now becomes more complicated. To what extent does a loss of friction in the tangential direction of the wheels translate into decreased friction in the lateral direction of the vehicle? Once the wheels are spinning, if the car were to slide laterally, there is a certain kinetic component to the friction applied by the tires. Over a given lateral distance, the starting and ending rotation of the wheels is not the same, even independent of the forward motion of the car. It follows then, that if a driver accelerates or brakes so rapidly going into a corner that the wheels lose traction, they are even more subject to the radial acceleration because of lower friction applied by the wheels in every direction.
But at the moment in the original problem, the driver has not lost traction with his forward acceleration. When friction isn't lost, what forces act on the car, and how should a driver approach a corner?
In fact, there are multiple approaches to this question. Many feel that all braking our acceleration should be done before and after the actual turn, so that all "available" friction is applied against the lateral g's to which the car is exposed. Intuitively, this makes sense, and it makes sense with regard to the overall system and the risk of loss of traction that we described above.
Others argue that you should accelerate through a turn. Given the equation for radial acceleration's reliance of tangential velocity, the though is that by increasing velocity through a curve, the magnitude of the acceleration in the radial direction increases, so the combined vector over time starts to point in the direction of the curve, rather than tangentially. This is an interesting thought problem, but doesn't seem to affect the instantaneous forces acting on the car.
Still other make a good point that braking in a curve can be helpful as well. Rather than addressing any of the points above, this has to do with the weight distribution of the car. You can imagine, that on it's suspension, the car's weight "shifts" rearwards during acceleration, and forwards during braking. Since only the front wheels of a car actually turn, the argument is that shifting the weight of the car onto the front wheels will increase their contact with the road as well as the downwards force on them, allowing them greater friction and therefore to withstand greater lateral force without slipping. Since they are the wheels that are causing the change of direction, this is supposed to help support that change of direction. On the other hand, if the rear wheels have insufficient friction and begin to slide laterally, the car will spin out, which seems even worse than just drifting off on the car's tangent when all wheels lose friction simultaneously. This phenomenon is called oversteer, because the car will try to spin out in the direction of the turn.
Considering the weight distribution in acceleration, the weight moves onto the rear wheels, and our steering front wheels have less weight and less contact area. This can result in what is called understeer, as the front wheels slip laterally because they have insufficient friction to change the direction of the car's travel to the direction of the wheel's orientation.
If you've made it this far, it's time to ask: if the stakes were high, and you had all the time to think about it, would you accelerate, brake, or stay constant through a turn? What factors would guide your decision in different circumstances?