We’d prefer when we drive to keep the math to a minimum. In particular the whole process would ideally be understood in terms of the zeroth spatial derivative, x(t). I want to be at work by 9:30 am; I’m meeting Barbara at the restaurant at eight; let’s try to make it to Cincinnati before nightfall. In practice though, we care about velocity as well. This is a social constraint. In an Omega Man type scenario in which I was literally the only person on the road, I’d zoom around from point to point as fast as the car could take me, but here in the real world there are other drivers, limited road space, and collisions to be avoided, so we post speed limits in the hopes of keeping traffic fatalities down to a dull roar. As a concession to our social nature, then, we’d be willing to navigate by the second derivative, dx(t)/dt. You’re going down the road when you see a sign that says “Speed Limit 45”, so you tell the car, “Set your velocity to 45 miles per hour.” Later you merge onto the highway when the posted limit is 70, so you tell the vehicle, “Go 70 now.” And so on. You could imagine a front-mounted video camera linked to an optical character recognition system which read the traffic signs for you so that you could make your morning commute fast asleep. Presumably DARPA is working on this.
But no, it can’t be that easy. The physics of the situation does not conform to our desires. And why should it? A century’s worth of marketing to the contrary, an automobile is not convenience realized in metal. Instead it is an explosion, or rather a series of explosions of the sort you can create by flicking a Zippo in the vicinity of a gas can over and over again. By some miracle these explosions do not kill you as they should, but instead propel you and your passengers, groceries, and collection of fast food wrappers on the back seat floor forward. The whole business is crudely regulated by a pedal on the floor whose pressing translates to the sole desperate imperative: Burn More Gas! That’s all you can do: explode more or explode less. The explosions are translated into force which spins the wheels, and of course F = ma. All we want is to be over there, but we have no access to the zeroth or even the first derivative. Instead all we can control is dx2(t)/dt2 by means of device that–no secrets here–is even called the accelerator pedal. It is our job to translate its presses and releases into the arrivals we desire. The vehicle offers no more assistance. Every day you do integral calculus with your foot.