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I wanted to go to the depth of the discovery of classical mechanics, instead of just reading, accepting and learning things. Now my question is not a single question, but it can answer many of my doubts. I completely understand motion, but I got stuck with forces. First, I want to tell all the physicists out there that I have read many books in which while describing momentum, they simply write that it is $m*v$, or they give this proof that I am not able to understand:

$p \varpropto m$

$p \varpropto v$

So $p = Kmv$, where K is constant of proportionality.

By experiments, it was found that value of k is 1.

So $p = mv$;

  • Now I don't know what kind of magical experiments they talk about while saying that $k = 1$. So, first question: Can someone explain this proof to me?

After long searching, I found a book which talked about a very unique experiment and drew a table of values for it, and before telling Newton's second law, they told that a pattern exists within the table, and that pattern was that $m*v$ in every trial was the same, with which I completely agree, and they told that this $m*v$, which remains conserved, is defined to be momentum, for easier equations. Then through momentum conservation, they came out to the fact that $F = ma$, which was completely intuitive and I moved on.

  • But, (now moving to my original question) , the equipment's used in that experiment were very new, they didn't existed in Newton's period(example light gates), or did they?

  • So my second question: How did Newton came to the fact that $m*v$ remains conserved, or simply this pattern exists?

  • Did he conducted any experiments?

  • If yes, then what were they? If no, then how he led to momentum conservation?

Now my final doubt. There are many concepts in physics which include a constant of proportionality, and books just throw their values, instead of telling how these values were found.

  • So how do these constants were found?

Can someone please tell me just the method which is used to find the value of $k$ in my first question? It will be enough to satisfy me. I have asked many questions about this, but never got the answer which was intuitive.

Note: I request the readers to not give answers involving calculus, because I don't understand it right now, I will study it in the future.

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    $\begingroup$ This might be more appropriate at History of Science and Mathematics than here. $\endgroup$ – Kyle Kanos Jan 13 '16 at 12:58
  • $\begingroup$ The problem here is about definitions. I personally would define momentum as the product of mass and velocity, and therefore avoid the "k" problem altogether. Your book calls momentum something else, and THEN proves it happens to be equal to m*v. Now, what was this "something else"? Try to understand this first. $\endgroup$ – Bzazz Jan 13 '16 at 13:26
  • $\begingroup$ What a great question! Finding proportionality constants is all about doing controlled experiments where you measure all other values than the constant itself. If you know that F is proportional to a, then you know that F=ma where m is some proportionality constant. Then do an experiment where you know both F and a (use a Newton meter to measure a force and a meterstick plus a timer to measure an acceleration). Then you just plug F and a into the expression and find m. And then you do that many times to settle it. $\endgroup$ – Steeven Jan 13 '16 at 15:03
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    $\begingroup$ Historically speaking there was a notion of impetus before this, which Descarte for example took to be conserved a priori. $\endgroup$ – Mozibur Ullah Mar 3 '16 at 21:35
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K=1

The value of K was arbitrarily defined. Actually as momentum was to be defined here, Newton had a freedom of choosing any constant except 0. He could have chosen it to be 2 but that would make the calculations difficult as compared to K = 1. This is what gives you the definition of momentum as mass times velocity. In case Newton would have chosen it to be 2, your definition would have been twice of mass times velocity.

Experiment

Actually Newton's second law is (F)(∆t) ∝ (m)(v). This was what was experimentally found by Newton by changing one of the parameters while the others remained fixed.

He used

  1. Meter scale
  2. Stopwatch
  3. Glass Surfaces to reduce friction

Conservation of momentum

This quantity of (m)(v) was defined as a new thing called Momentum. Now if F remains zero then (m)(v) doesn't change and hence momentum remains conserved.

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Newton was not the first to devise or discover the concept of momentum. This goes at least as far back as Fr. Jean Buridan (ca. 1295 - ca. 1358), professor of natural philosophy and rector of the University of Paris, who was the first to introduce the impetus theory of projectile motion. See this translated excerpt from his Quaestiones super octo physicrum libros Aristotelis.

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