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I have been thinking lately, why is it that Newton's second law of motion takes the form $F = ma$? It seems to me the idea of a force is a somewhat made up concept, at least the value we assign to it seems made up. Is the a reason that $F= ma$ instead of maybe something $F = kma$ where $k$ is some constant or $F = ma^2$ or $F = m^2a$. The fact that it relates mass and acceleration makes sense to me, but it seems as if the type of relation is somewhat arbitrary.
However upon reflection I have come up with a way that I guess you could establish this law by experiment.The basic idea of my experiment would be to get some sort of "pusher" mechanism that would "push", with constant strength, a variety of movable objects that are subject to no external interaction and then measure the acceleration of the objects while they are being pushed. I guess you could make the assumption that a falling object is being "pushed" towards earth somehow and assume that this push is constant.
Does anyone one know how Newton himself conducted the experiment? I can't seem to find an answer simply by googling it.