So this might not be the realm of 5-year-olds but I was curious from a mathematical standpoint how Newton came to discover specifically that [F]orce was [m]ass multiplied by [a]cceleration and not some other mathematical operation like some logarithm or polynomial or something (not sure what those are either but that’s another eli5 entirely).
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Note that “force” and “mass” are not something that can be directly observed. Taken by itself, the 2nd law merely *define* what a force is, or what a mass is (depending on your philosophical point of view). We use “force” as a convenience quantity to make it easier to state physical laws. So you can’t really consider the 2nd law by itself. “mass” by itself is a blurry concept, because once again, we can’t observe mass directly, only on what it effects is. The concept of “mass” doesn’t even take shape until the 13th century, where it is suggested that there is something else that measure quantity of matter beside volume and weight.
So it’s not meaningful to talk about the 2nd law by itself. Newton himself had no ideas what force is, constantly mixing up between what we would consider different concepts today. What you can do however, is consider the whole of 3 laws.
The concept of momentum went all the way back to Persian scientists in the 1000, and so is the idea that it’s proportional to quantity of matter and speed. By the time of Descartes, he had expressed the idea that momentum of the universe is conserved, but even before him, other philosophers had expressed similar ideas. But even at this point, “mass” is a fuzzy concept that could mean a few different things, like size or weight. “velocity” itself is not quite defined either, Descartes though that velocity is just a numerical quantity, so his “conservation of momentum” is mathematically different from our modern version, but the idea is there.
However, Galileo made a significant experiment: the resistant to change in movement is proportional to the weight of an object. Or in modern term, inertia mass is directly related to gravitational mass. This idea is adopted by Newton too.
Throughout the years after Newton, there are many arguments about what is “mass” or more precisely what is quantity of matter. It’s a very nebulous topic.
What Newton’s 3 laws expressed is just the fact that the total change in momentum is 0, where momentum is proportional to velocity and inertia (resistant to change of motion). But not just that, he make it more precisely the idea that velocity is a quantity with direction. Total momentum isn’t a numerical quantity, it’s a quantity with direction. Of course, lacking in modern mathematical concept of vector, he expressed this idea by describing force as a quantity with direction, formulate the 3rd law, and describe 2nd law also with direction. However, he had no ideas what “quantity of matter” is, precisely, and didn’t even have unit for mass, so even his 2nd law are different from today’s version. Lacking units for mass, he can only talk about force as being proportional to quantity of matter.
What Newton thought as his 3 laws are actually very different from our modern version of the 3 laws, because the concepts needed for it did not exist back then: vector calculus and mass. The modern version is formulated much later, but we attributed it to Newton anyway. The fact that momentum is proportional to quantity of matter and velocity is not original to him, it had been around since the 1000.
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