How Is Newton's F=ma Formula Derived and Understood in Physics?

[DrStupid]

No, I'm not at all going outside of simple Classical Mechanics.

That depends on the type of systems your statement ("its not at all necessary to have a linear relation between m*a and force") was related to. For open systems you are right but for closed systems you already are outside classical mechanics.

When going by Newton's second law we'll get a relationship between F and m*A of the form F=K*m*A (where k is a constant).

For closed systems this is basically correct but in addition to Newtons second law you also need the third law, definition II, Galilei transformation, isotropy and maybe even more.

 

[harrylin]

Not read all thread so this may have been said. The point is often fudged in teaching.
[..] You can measure a, but you cannot in the first place measure F and m independently. So this is not an empirical law, more of a definition. [..]

As mentioned before: The equality sign F=ma is a later simplification (k=1) that came with the introduction of standard unit systems; it is due to a free choice of units.

For deriving F~ma Newton could compare identical "impressed forces" by observing an equal amount of impression of a spring, and he could also compare identical "masses" by means of equilibrium of a balance. Neither Newton nor Hooke were completely free to define "Force" as they liked.