Today, I was watching a video, and the man on the video said that we only "accept" $F=ma$. Then said that no one can prove $F=ma$. But I know we can derive $F=ma$ by momentum with calculus. I am confused, so do we only accept $F=ma$, or do we know the proof?
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Classical mechanics is a whole theory, it's not defined by an arbitrary equation you write on a piece of paper that means nothing at all.
By theory, I mean a whole "worldview", containing primitive concepts such as the concept of space, time, mass and force, and principles/relationship about those (say, the absoluteness of space and time, galilean invariance, etc). The equation of motion F = ma is a principle too, hence why it's called a "law".
Now, in mathematics the concept of proof is something that is more or less well defined: given some "worldview" (the latter necessarily comes with an existential axiom, as in, there exists something and this something is satisfying, by decree, the following properties), and given some logic/inference rules (telling you how truth is actually preserved), you can prove if some claim about these objects are true or false. Obviously, the hard part is to chose a worldview that is consistent and useful, as you have to be very careful with this choice if you want to produce something of value.
The same is true in physics. You cannot prove a theory wrong, and even less right, because that's not what they are for: they are here to make sense of some phenomenon, like the concept of number or velocity. Is velocity "true" or "false"? It doesn't make any sense to ask this question, and the same is true for classcial mechanics, or F = ma (which can, by the way, be justified extensively).
Note that even in relativity, the "moral" of F = ma, which can be understood as "absolute motion or change is equivalent to acceleration" is entirely preserved, even if the details of the equation changes a little bit, the philosophy of it never did for the past 400 years.
sure
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