"Never accelerate when making a turn." (Or decelerate, either.) Due to my brother's participation in amateur auto racing, I studied some of this math out of curiosity. What the driver's ed teacher meant, but almost certainly couldn't put into proper mathematical language, was: Keep the _absolute value_ of the acceleration vector constant -- or at least < some bound. This is a very crude first-order estimate of the stickiness of the tires; they can't be expected to handle arbitrary accelerations in multiple directions at the same time. To aid a newbie driver, there are now accelerometers that have a little circle and a dot within the circle showing the current acceleration vector. The idea, of course, is to keep the dot within the circle. The best racing drivers can keep the dot _on_ the circle at nearly all times (this, of course, is course-dependent and auto-dependent). This is much harder than it seems, because the racing driver must also keep the suspension "loaded", otherwise there are unfortunate interactions between the suspension and the tires. Also, the moment of inertia of the race car itself about its vertical axis is non-negligible. A second-order approximation would utilize a non-circle, but the uncertainties are large enough that a simple circle is good enough. In 3 dimensions, the "constant absolute value acceleration" model is good for a solid fuel rocket, which can only be steered, but not throttled. I wanted to utilize the position curves as "splines" to try to piece together fastest-possible trajectories around a race course. I wasn't able to find closed-form solutions except for the most trivial cases. (I wasn't even trying to handle the 3D case, where the race car must climb or descend -- e.g., Laguna Seca, outside of Carmel, CA.) I can't say that I ever really conquered the math; there are a lot of sinh's and cosh's involved. Perhaps someone here has better links & references than I was able to find. At 09:28 AM 5/19/2015, James Propp wrote:
My driver's ed teacher taught us "Never accelerate when making a turn."
I objected that, in the physics sense, it is impossible make a turn without accelerating.
The teacher, who taught gym when he wasn't teaching driver's ed, wasn't amused.