If the force also depends on the mass, the change in the velocity ends up being independent of mass- this is why falling objects near the surface of the Earth all drop at the same rate, regardless of mass. This is because a force is just an interaction that tries to change the momentum of an object, and the momentum depends on the mass the larger the mass, the larger the momentum, and the more force you need to change it. Air resistance is insignificant for heavy objects precisely because it doesn't depend on the mass. You might think this is because the air resistance force depends on the mass, but you'd be wrong- it's exactly the opposite. So engineers tend get a little obsessive about it. Or, for that matter, off Earth, as air resistance is a big part of the plot of The Martian. And when you're building stuff that's going to move around on the surface of the Earth, you need to worry about air resistance. Physicists are interested in fundamental laws and elegant equations, engineers are interested in building stuff. Of course, that mostly reflects a fundamental difference in mindset between physicists and (proto-)engineers. When you include air resistance, though, you have basically no choice but to solve it numerically, writing some kind of computer simulation, and every time you want to change any of the parameters, you need to go back and run the whole simulation again. One of the more memorable student course evaluations I've gotten in my years at Union was from a student in the introductory physics-for-engineers course who complained "They abstracted away all the interesting stuff, like air resistance." This strikes physicists as a weird thing to say, because air resistance is annoying- if you abstract it away, you can solve a huge range of problems with a pencil and paper, getting elegant sets of equations you can write down and manipulate algebraically.
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