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In John Taylor's Classical Mechanics, he gives a basic treatment on air resistance/drag. In order to do this, he models air resistance as $\mathbf{f}=-f(v)\mathbf{\hat{v}}$, and then Taylor expands $f(v)$, using only the linear and quadratic terms. Of the function itself, all he says is, "The function $f(v)$ that gives the magnitude of the air resistance varies with $v$ in a complicated way, especially as the object's speed approaches the speed of sound." He gives no further resources as to what this actually is.

Does anybody know this proper formula for air resistance?

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There is no such function in general: air resistance depends not only on the velocity of the body through the medium, but also on the shape of the body and its rigidity (at least). I presume the above assumes a rigid sphere as the body, but even in that case, things are complicated.

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