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What's interested me in my studies thus far is that in a lot of beginner undergrad mathematics and physics courses we're often told to calculate trajectories, velocities, etc. by "ignoring air resistance". Obviously this doesn't accurately model real life.

How is calculating a problem without ignoring air resistance actually done? Is there some kind of "air resistance function" which varies based on altitude?

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    $\begingroup$ You can account for it ad-hoc (with a linear or quadratic velocity dependent resistive force), or you can do it "right" and calculate it based on hydrodynamics. The level of difficulty rises very considerably as you go form ad-hoc linear to ad-hoc quadratic to hydrodynamic calculations. $\endgroup$ – CuriousOne Dec 19 '14 at 8:38
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To first order, air resistance falls into two regimes at subsonic speeds. At very low speeds it can be modeled with a linear response to velocity, while at any higher speed you generally observe a quadratic response.

The ratio of Reynold's number approximates the relative contribution for both components and has derivable values depending on the geometric figure you are dealing with and the viscosity of the fluid. I know there exist some fluid dynamic models but they are several levels more difficult than these two approximation techniques.

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Yes, there is. Usually air resistance or other kinds of resistant forces can be considered as $bv^2$ or $bv$ where $b$ is a constant that depends on many things. For example, pressure, density, and so on. These functions are just an approximation and derived experimentally. You know that friction is an actual complicated force! They are usually neglected for we have a simple differential equation.

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