I've always assumed/been told that Newton's 2nd law is an empirical law — it must be discovered by experiment. If this is the case, what experiments did Newton do to discover this? Is it related to his studies of the motion of the moon and earth? Was he able to analyze this data to see that the masses were inversely related to the acceleration, if we assume that the force the moon on the earth is equal to the force the earth exerts on the moon?
According to Wikipedia, the Principia reads:
Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.
Translated as:
Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.
My question is how did Newton come to this conclusion? I get that he knew from Galileo Galilei the idea of inertia, but this doesn't instantly tell us that the change in momentum must be proportional to the net force. Did Newton just assume this, or was there some experiment he performed to tell him this?
Answer
First of all, it would be preposterous to think that there was a simple recipe that Newton followed and that anyone else can use to deduce the laws of a similar caliber. Newton was a genius, and arguably the greatest genius in the history of science.
Second of all, Newton was inspired by the falling apple - or, more generally, by the gravity observed on the Earth. Kepler understood the elliptical orbits of the planets. One of Kepler's laws, deduced by a careful testing of simple hypotheses against the accurate data accumulated by Tycho Brahe, said that the area drawn in a unit time remains constant.
Newton realized that this is equivalent to the fact that the first derivative of the velocity i.e. the second derivative of the position - something that he already understood intuitively - has to be directed radially. In modern terms, the constant-area law is known as the conservation of the angular momentum. That's how he knew the direction of the acceleration. He also calculated the dependence on the distance - by seeing that the acceleration of the apple is 3,600 times bigger than that of the Moon.
So he systematically thought about the second derivatives of the position - the acceleration - in various contexts he has encountered - both celestial and terrestrial bodies. And he was able to determine that the second derivative could have been computed from the coordinates of the objects. He surely conjectured very quickly that all Kepler's laws can be derived from the laws for the second derivatives - and because it was true, it was straightforward to prove him this conjecture.
Obviously, he had to discover the whole theory - both $F=ma$ (or, historically more accurately, $F=dp/dt$) as well as a detailed prescription for the force - e.g. $F=Gm_1m_2/r^2$ - at the same moment because a subset of these laws is useless without the rest.
The appearance of the numerical constant in $F=ma$ or $p=mv$ is a trivial issue. The nontrivial part was of course to invent the mathematical notion of a derivative - especially because the most important one was the second derivative - and to see from the observations that the second derivative has the direction it has (from Kepler's law) and the dependence on the distance it has (from comparing the acceleration of the Moon and the apple falling from the tree).
It wasn't a straightforward task that could have been solved by anyone but it was manifestly simple enough to be solved by Newton. So he had to invent the differential calculus, $F=ma$, as well as the formula for the gravitational force at the same moment to really appreciate what any component is good for in physics.
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