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If I've read Newton's Laws of Motion correctly in the Principia, it seems that Newton attributed the "change in motion" (momentum) to the "impressed force". Mathematically this would be read as $\Delta p \propto F$, right? But then how did it end up as $\frac{\Delta p}{\Delta t} \propto F$?

Also, I've read in other forums that Newton's Second Law is based on Galileo's experiments on falling bodies, where he treated acceleration and mass as distinct parameters. For example, his experiment with comparing free-fall of different masses yielded something along the lines of $\frac{F_1}{F_2} \propto \frac{m_1}{m_2}$ ($\propto a$). This threw me off even further, making me question how (and why) the heck Newton ended up with $\Delta p \propto F$ as his Second Law.

I'm not questioning its validity, I just want to understand how it came to be understood as $\frac{\Delta p}{\Delta t} \propto F$.

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Related: physics.stackexchange.com/questions/2644/… –  Dimensio1n0 May 25 '13 at 8:34
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Newton didn't say "change in momentum", he said "alteration in momentum", and whichever he said, this means clearly, and with no room for doubt, rate of change of momentum, the limit of small $\Delta t$ of $\Delta P \over \Delta t$. This was understood this way by everyone who read the book, there is no way to misinterpret if you follow the mathematical things.

The experiments of Galileo showed that bodies in gravity have the same acceleration. This means that the Earth is imparting changes in velocity to particles. The notion of "force" is already present to some extent in the theory of statics developed by Archimedes, and gravity produces a steady force in a static situtation, and this force is proportional to the mass. If you know force is proportional to the mass, and the acceleration is the same for all bodies, it is no leap to conclude that a force produces a steady acceleration inversely proportional to the mass.

The second law was not the major innovation in Newton, this was known to Hooke and Halley and Huygens for sure. Newton's innovation is the third law, and the system of the world, the special problems.

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@ProSteve037: Yes, but there is no reason to put "derived" in quotes--- it is the natural mathematical statement of the fact that force produces acceleration. I don't know why you consider the mathematical statement somehow profound or confusing, it is obvious and straightforward, and was known to everybody by the mid 1600s. This is not Newton's discovery, his discoveries are more involved, like the oblateness of the Earth, the wobble of the moon, the cause of the precession of the equinoxes and so forth. –  Ron Maimon Aug 24 '12 at 14:18
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@ProSteve037: Physics books basically make up history, in a series of outrageous fabrications, and pretent that what is confusing to students is what was confusing to people back in the 16th century. Galileo didn't really discriminate in a precise way between mass and weight, at least not as clearly as Newton does, so maybe Galileo would get a question or two wrong on a modern test. But the concept of force is known from Archimedes law of the lever--- the force times distance must balance for a lever to balance. I don't see this confusion as particularly important historically. –  Ron Maimon Aug 26 '12 at 22:16
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@ProSteve037: The way Newton was the primary figure is that he made a _respectable science+ out of Cavalieri's infinitesimal calculus, Barrows fundemental theorem, and Galileo's relativity and acceleration laws--- he showed the true scope of these discoveries, that they unlock the secrets of motion and dynamics, and give a true model of the solar system from simple first principles of Newtonian gravitation. This is an astonishing achievement, but it isn't the discovery of "F=ma", that was just known to everybody in 1666, the same way everyone knows perturbative string theory today. –  Ron Maimon Aug 28 '12 at 2:04
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Regarding the F=ma, Newton takes it as a new definition of force, which subsumes and extends the old one from statics. When things aren't moving the sum of the forces is zero and the sum of the torques is zero, and this reproduces Archimedes' law of the lever and bouyancy. So it's the same thing, but now it works for dynamics. The main discovery is how to calculate using continuous quantities like acceleration and velocity, and this discovery is modern calculus. Newton didn't choose to make a new language, he snobbishly preferred to sound like the ancients, so he fairly got bitten by Leibnitz. –  Ron Maimon Aug 28 '12 at 2:09
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@ProSteve037: It's not hard--- the works of Archimedes are widely reprinted, I read the version in "Great Books of the Western World" vol 11 (one of only two volumes worth reading, the other is 18-19th century scientists, the rest of the series is crap), Euclid Appolonius Archimedes (it's in every high school library, as part of getting kids excited about great books). Galileo's "Dialogue" and "Two New Sciences" are essential, and Barbour's "Absolute or Relative Motion" fills in the other folks from the era. That, plus Newton's Principia (after learning the modern thing) is all I read. –  Ron Maimon Aug 30 '12 at 4:32
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Isaac Newton, Philosophiae Naturalis Principia Mathematica, Axiomata, sive Leges Motus [1]:

Lex. II.
Mutationem motus proportionalem esse vi motrici impressæ, 
& fieri secundum lineam rectam qua vis illa imprimitur.
[...]

Mutationem: means change, alteration, variation, mutation, ... Motus: momentum.

So Mutationem motus means exactly the variation of momentum: $$ \frac{\Delta p}{\Delta t} $$

[1] http://la.wikisource.org/wiki/Philosophiae_Naturalis_Principia_Mathematica

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