My question is about the appearance of a non-analytic function in the formula fo
ID: 1378927 • Letter: M
Question
My question is about the appearance of a non-analytic function in the formula for the resistive force in air or other medium. Considering the 1-dimensional case as covered by Walter Lewin in his 8.01 lecture, the magnitude of the resistive force is proportional to the square of the speed of the object, which we take to be a sphere. In other words
|F|=cv2
with the force being in the opposite direction of the object's motion.
The constant c depends on the sphere's radius, drag coefficient of air and so forth, but for convenience choose values that make c=1.
I understand this is an approximation, and in reality there is a v term as well as the v2 term, and maybe other terms as well, but I don't think that invalidates my question.
With that said, here's is a graph of the resistive force F as a function of v:
enter image description here
Because the sign of the force is opposite that of velocity, the equation can be conveniently written
F=?v|v|
Although this function looks smooth and has a derivative, 2|x|, it doesn't have a second derivative. Isn't it unusual for a simple equation of a physical phenomenon in classical mechanics to lack a second derivative? It's unintuitive to me that the force doesn't have a second derivative. Physically, it doesn't feel like anything non-smooth is going on.
My question: Is it normal to encounter functions in classical mechanics with no second derivative, and if not, what's the explanation for finding one here?
Added: For anyone interested in this topic, a related question popped up over at MathOverflow and there are several good answers.
Explanation / Answer
That function actually does have a well-defined second derivative at every point except v=0, which is a perfectly reasonable thing to have happen in physics. It's often the case that different domains of a system have different behavior, and in a continuous approximation, these domains have sharp boundaries, which can lead to second (or even first) derivatives which are not well defined at isolated points.
This isn't normally an issue because we only ever deal with approximate measurements in physics. You may be familiar with the idea that it's impossible for a measurement to produce an exact rational value; this is basically the same idea. The behavior of a function at a single isolated point (or any set of points of measure zero, I think) is irrelevant because you're never going to sample the value at that exact mathematical point in practice.
If you were talking about a function whose second derivative was not defined anywhere, that would be a different matter entirely. We don't usually deal with those sorts of functions in physics.
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