You can't express the position in a polar system consistently, because whenθ isn't constant, r is not with respect to any fixed center. It could be dθ/dt and dr/dt of the velocity, though.
(θ is the angle of both the position and the velocity, but not the acceleration.)
The motion is along the curve of a circle whose center varies with the steering wheel, so what I called d²x/dt² could be seen as the tangential velocity, but then we'd have to call the steering wheel θ/x.
I've changed it to d²v/dt² to just duck the question of which direction it is.
Wait, I'm confused. d²v/dt² is the third derivative of position. I could say dv/dt, but that reduces the point I'm trying to make. So, r. (I asked a physics teacher and he said r...)
Sadly, this is not the case - if you turn the knob to "all hot", and wait, you will never reach the temperature of the sun, and maybe not even boiling.
Indeed! I missed the philosophical subtlety of the original poster. I thought that the distinction between intended uses was clear, but now I see it is muddy.
The question becomes: what, philosophically, are you doing when you "floor it"? Are you asking for a terminal speed, or an acceleration? Is the thermal control analogous? Surely I press harder when I feel I am too slow, and let up when I feel I am too fast; that is, I'm behaving as if the gas pedal controlled velocity.
What it actually controls is approximately the power output from the engine, but let's ignore the difference between that and acceleration.
In low d²x/dt² conditions, the gas pedal controls velocity because the system is dominated by opposing forces which increase with velocity (e.g. drag), so it settles to a particular velocity. But for high d²x/dt² (and typically low dx/dt), i.e. stopping and starting, the control is obviously one of acceleration.
All these are ideal approximations; your engine and your tires have limited power and response, and traction. You won't get the results this model suggests if your dx/dt is too high, or your dθ/dt too high for your dx/dt.
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(θ is the angle of both the position and the velocity, but not the acceleration.)
The motion is along the curve of a circle whose center varies with the steering wheel, so what I called d²x/dt² could be seen as the tangential velocity, but then we'd have to call the steering wheel θ/x.
I've changed it to d²v/dt² to just duck the question of which direction it is.
(Then there's the transmission...)
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The question becomes: what, philosophically, are you doing when you "floor it"? Are you asking for a terminal speed, or an acceleration? Is the thermal control analogous? Surely I press harder when I feel I am too slow, and let up when I feel I am too fast; that is, I'm behaving as if the gas pedal controlled velocity.
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In low d²x/dt² conditions, the gas pedal controls velocity because the system is dominated by opposing forces which increase with velocity (e.g. drag), so it settles to a particular velocity. But for high d²x/dt² (and typically low dx/dt), i.e. stopping and starting, the control is obviously one of acceleration.
All these are ideal approximations; your engine and your tires have limited power and response, and traction. You won't get the results this model suggests if your dx/dt is too high, or your dθ/dt too high for your dx/dt.