Which way did Newton find F = ma?

[prosteve037]

I was under the impression that, by experiment, Newton deduced

\textit{F}\propto{m} \rightarrow \textit{F = k}_{1}\textit{m}

(where \textit{k}_{1} is some constant)

and

\textit{F}\propto{a} \rightarrow \textit{F = k}_{2}\textit{a}

(where \textit{k}_{2} is some constant)



and then found that either/both

\textit{k}_{1}\propto{a} \rightarrow \textit{k}_{1}\textit{ = c}_{1}\textit{a}

(where \textit{c}_{1} is some constant)

and/or

\textit{k}_{2}\propto{m} \rightarrow \textit{k}_{2}\textit{ = c}_{2}\textit{m}

(where \textit{c}_{2} is some constant)



thus creating

\textit{F = c}_{1}\textit{ma}

and/or

\textit{F = c}_{2}\textit{ma}



where in SI Units they would be in the form

\textit{F = ma}





However, I've read in some other forums how Newton actually meant

\textit{F}\propto{ma} \rightarrow \textit{F = kma}



Which is the case? Did he use the first method or did he simply state the second?

If he did use the first method, how did he resolve that \textit{k}_{1} is dependent on acceleration and/or that \textit{k}_{2} is dependent on mass?

 

[xts]

Actually Newton defined 'force' as F\propto ma paying little attention to proportionality constant. Before Newton the concept of "force" was only intuitive and had no precise, well defined meaning.

Newton's original works (Principia...) are pretty hard to read and understand - he used the formalism based on geometrical relations rather than on algebraic ones. A formalism we are used to, itroduced in 18th century, defined it without any proportionality constant, as F=ma.

 

[prosteve037]

Actually Newton defined 'force' as F\propto ma paying little attention to proportionality constant. Before Newton the concept of "force" was only intuitive and had no precise, well defined meaning.

Newton's original works (Principia...) are pretty hard to read and understand - he used the formalism based on geometrical relations rather than on algebraic ones. A formalism we are used to, itroduced in 18th century, defined it without any proportionality constant, as F=ma.

Hmm okay. I always wondered what those geometric methods were. I'm assuming he plotted or graphed the parameters as coordinates. And perhaps took \textit{ma} as the area and \textit{F} to be some coordinate. That's my guess at least.

Does anyone know exactly what he did?

 

[xts]

You may try to read "Principia..." just to see yourself the kind of argumentation Newton used.
There are several English translations available on-line - I recommend (as the easiest to read) - American translation from mid 19th century: http://rack1.ul.cs.cmu.edu/is/Newton/
You may want to skip all the introduction, and start from page 84 - there are corollaries to the principles - explained and illustrated geometrically.

 

[prosteve037]

Thanks xts.

I'm also curious as to what you said about Newton's view on proportionality, being that he payed little attention to the proportionality constant. Mind if I ask where you read/heard this?

This gives me a different perspective as to how proportionality was defined. Was it that proportionality was crudely defined in Newton's time? Or was it just Newton that disregarded this?

 

[xts]

I read "Principia..." - long time ago I assisted my professor with his work on history of science. I even read part of it in nasty Newton's latin...

At Newton's times proportionality was perfectly known and understood - it was used even more frequently than nowadays. Just contrary - those times the numerical values, measurement units, etc. were rather neglected. It was the effect of geometrical representation. As in Euclid "Elements" - the line segment represent the value, and twice longer section represent twice bigger value, but no one cares to say that 1 inch on the drawing represents 1 pound of mass. You may make the same Euclidean construction in different scale, and all conclusions will be the same.
Such approach is really difficult to understand for modern people, who learn on numbers, rather than on Euclodean constructions.
Newton was one of the very last scientists using such geometrical representations (but it was common till his times, Copernicus did the same). It was 18th century when numerical approach (started by Rene Descartes even a bit earlier than Newton worked) finally won popularity.

 

[xts]

You should notice that Newton worked on astronomical data. He knew precisely what were angular positions of celestial bodies. But he could only roughly estimate their absolute positions. He had no idea about mass of Earth or Moon - but he could calculate their proportion. He did not know the exact size of Earth and Mars orbits, but he could very precisely determine their proportion.

That lack of absolute scale is another reason why those times proportionality was treated as very meaningful, while nobody cared about exact value of proportionality constant. Newton had no idea what gravitational constant value may be and he was not bothered by it - it had to pass over 100 years till H.Cavendish, who lived in 'numerical era' rather than in 'geometrical times', estimated its value.

 

[prosteve037]

You should notice that Newton worked on astronomical data. He knew precisely what were angular positions of celestial bodies. But he could only roughly estimate their absolute positions. He had no idea about mass of Earth or Moon - but he could calculate their proportion. He did not know the exact size of Earth and Mars orbits, but he could very precisely determine their proportion.

I'm confused here. How was Newton able to calculate the proportions between objects/parameters if he didn't have values for either object?

And specifically what experiment did he perform to find the relation \textit{F}\propto{ma}?

EDIT:

Wait, did you mean he realized that the mass of the Earth and Moon would have to be proportional to each other and not that he could "calculate their proportion"? And similarly for the orbits?

I could understand how he would have thought the Earth's mass/orbit and Moon's mass/orbit would have to increase/decrease as the gravitational force of the Sun/Earth increased/decreased. But other than that, I can't see how he would've calculated their proportions without having values for each parameter.

 

[xts]

Try to look at it from geometrical point of view. I hope you had a course of Euclidean geometry (ruler+compass) at school. Euclidean ruler has no marks. Euclid says about proportions between legths of line segments (e.g. Thales' theorem), but he never use any absolute measure of them. It was a proportion, which made a sense of geometry. Exact measures, expressed in inches or pounds could be important for merchants and for other practical purposes, but not for reasoning.

calculate the proportions between objects/parameters if he didn't have values for either object?

If you have two segments A and B in proportion A/B=3/2 you may check it such, that using a compass you markA three times on some line, and you mark B 2 times, and finally those constructed 2A and 3B will be equal. You don't need to know how many mm or inches any of them has.

Please note that Newtonian mechanics do not depend on units of measure we chose. We may use any unit for time, distance and mass - they will be bounded by only one constant (gravitational), which was totally unknown to Newton (it got estimated by Cavendish hundred years later). All the mechanics is expressed in terms of proportions: e.g. centre of mass of two bodies is a point such that distances to both of them are in proportion reverse to proportion of their masses. That's a law true independently from units and from actual configuration of bodies.

specifically what experiment did he perform to find the relation F∝ma?

He didn't any. Once again - it is a definition of force, not an experimental knowledge.
You could however ask, what experiment he did to find the symmetry of such defined forces (3rd principle) - he utilised Galileo's observations and astronomical observations. But he also performed some laboratory experiments with elastic and half-elastic collisions. He also made a gedanken-experiment with a planet cut into two parts - and concluded, that if 3rd principle was not true, than the planet should accelerate on itself, which is absurd.

proportion of masses Earth/Moon

Honestly, I am not sure if Newton knew it - it could require a bit more accurate observations than he was able to perform. But he surely could know the proportion of masses Jupiter/Saturn, as it may be calculated from proportion of periods of their moons and the proportion between sizes of orbits of these moons. And only the periods might be measured in absolute units. Orbital sizes might be only expressed as a proportions to each other.

Please note that most of pre-Newtonian science, especially astronomy, had always been expressed in terms of proportions, never quoting the actual values of proportional constants. Take for example Kepler's laws (they inspired Newton to formulate his mechanics).

 

[harrylin]

According to Motte's translation*, it was not a definition but an axiom (a generally accepted law that is however open to disproof). I suppose that Newton did not invent the relationship but expressed what others already had experienced.

If you read his explanations:
http://gravitee.tripod.com/axioms.htm

together with his definition 2:
http://gravitee.tripod.com/definitions.htm

then it appears (or he suggests) that he first deduced the existence of a momentum (m*v) which he defined as quantity of motion (p), and about which he next claimed the force law.
That seems indeed to correspond to your second case.

More precisely (although not clearly formulated as such, I interpret "alteration" as d/dt):
F ~ dp/dt

Harald

* I find http://gravitee.tripod.com/toc.htm handy: just press "cancel".

 

[xts]

Yes and no ;)
'Principia...' are not perfectly consistent structure, even as for 17th century standards and often mix definitions with axioms (e.g. 3rd law is used in def. III)

Yes - force is introduced as a 'law of motion' (axiom), but:
No - the term 'force' is also defined quantitatively by Definitions III-VIII - III defines it as proportional to mass, and the next ones about centripetal forces are equivalent to proportionality to acceleration (compare with geometrical representation of calculus in further parts of the Book I)

I fully agree that the definition of momentum (quantity of motion) is a key point for the reasoning. I also admit that further, in the calculations, Newton equals (or rather makes proportional) the derivative of momentum with the force without deeper justification - so it may be taken as an implicit definition of both 'force' and 'alteration'.

So to answer Prosteve's question - regardless the 'force' is introduced as definition or as an axiom, it is not a subject to be tested alone. It is rather a foundation part of the structure, which must be tested against experiment (mostly against astronomical observations) as a whole.

PS. - thanks for Motte's translation better readable than the one I used!

 

[harrylin]

Yes and no ;)
'Principia...' are not perfectly consistent structure, even as for 17th century standards and often mix definitions with axioms (e.g. 3rd law is used in def. III)

Yes - force is introduced as a 'law of motion' (axiom), but:
No - the term 'force' is also defined quantitatively by Definitions III-VIII - III defines it as proportional to mass, and the next ones about centripetal forces are equivalent to proportionality to acceleration (compare with geometrical representation of calculus in further parts of the Book I)

I fully agree that the definition of momentum (quantity of motion) is a key point for the reasoning. I also admit that further, in the calculations, Newton equals (or rather makes proportional) the derivative of momentum with the force without deeper justification - so it may be taken as an implicit definition of both 'force' and 'alteration'.

So to answer Prosteve's question - regardless the 'force' is introduced as definition or as an axiom, it is not a subject to be tested alone. It is rather a foundation part of the structure, which must be tested against experiment (mostly against astronomical observations) as a whole.

PS. - thanks for Motte's translation better readable than the one I used!

- First of all, I distinguish the elaborations from the definitions. Anyway, I checked those definitions (in italics) for "proportional" and found them starting from definition VI (which I could not apply to the law of motion). Definition VII seems to say that a certain acceleration corresponds to a certain force, and that F ~ v_t. Definition VIII is unclear to me; from the elaboration it appears to mean that F ~ p_t.
Thus indeed, I now also think that the difference between definitions and axioms is very blurred!

- About the translation: regretfully the site that I use is only partial (although the most important part), so I'm also happy with yours!

 

[harrylin]

[..]
And specifically what experiment did he perform to find the relation \textit{F}\propto{ma}?
[..]

As you can read in http://gravitee.tripod.com/axioms.htm ,
he knew from experience (other people's for sure!) that "If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively."

In particular, in his elaboration on the definitions, he related to the experience with weapons:
"If a leaden ball, projected from the top of a mountain by the force of gunpowder with a given velocity, and in a direction parallel to the horizon, is carried in a curve line to the distance of two miles before it falls to the ground; the same, if the resistance of the air were taken away, with a double or decuple velocity, would fly twice or ten times as far."
- http://gravitee.tripod.com/definitions.htm
He probably knew that double the amount of powder has double as much force; and also that can be tested, with depth of penetration.

Note that air resistance is important for bullets but less so for canon balls.

Moreover, the same "impressive" force gives the same impression (deformation) of deformable objects such as springs, and to a certain degree the impression is even proportional to the force - as his competitor Hooke observed.

 

[xts]

experience with weapons [...] two miles before it falls to the ground

Don't take it as a serious experience with weapons! It is rather gedanken-experiment like, exaggerated argument. Cannons exceeding in range one mile were not available even during Napoleon Wars.

 

[sankalpmittal]

I was under the impression that, by experiment, Newton deduced

\textit{F}\propto{m} \rightarrow \textit{F = k}_{1}\textit{m}

(where \textit{k}_{1} is some constant)

and

\textit{F}\propto{a} \rightarrow \textit{F = k}_{2}\textit{a}

(where \textit{k}_{2} is some constant)
and then found that either/both

\textit{k}_{1}\propto{a} \rightarrow \textit{k}_{1}\textit{ = c}_{1}\textit{a}

(where \textit{c}_{1} is some constant)

and/or

\textit{k}_{2}\propto{m} \rightarrow \textit{k}_{2}\textit{ = c}_{2}\textit{m}

(where \textit{c}_{2} is some constant)
thus creating

\textit{F = c}_{1}\textit{ma}

and/or

\textit{F = c}_{2}\textit{ma}
where in SI Units they would be in the form

\textit{F = ma}However, I've read in some other forums how Newton actually meant

\textit{F}\propto{ma} \rightarrow \textit{F = kma}
Which is the case? Did he use the first method or did he simply state the second?

If he did use the first method, how did he resolve that \textit{k}_{1} is dependent on acceleration and/or that \textit{k}_{2} is dependent on mass?

Always remember in physics "that to prove a theory first there is experimental observation and then mathematical deductions"

Newton's laws are proved experimentally and then deducted .
He first gave his 1st law of motion which states "that every body remains in its genuine state of rest or uniform motion in a straight line unless acted upon by some external force (assuming body to be displaced linearly)"

Have you ever heard of the mathematical proof of his 1st law ? No , but it is proved by his 2nd law! But his 1st law was given prior to his 2nd law ! However his 1st law is logically explanatory. Inertia. Mass resists . Each molecule each atom of it !

Proof :
By his 2nd law we know ,
F=ma
If body remains in its original state then there is no force being applied so,
0=ma
or
a=0
So it will have no displacement.

Now let's come on to his 2nd law . What does it state ? It states "that rate of change of momentum in a body is directly proportional to the impressed force acting on it and it takes place in direction of that force assuming body to be linearly displaced in straight line "

F∝m → F = k₁m

(where k₁ is some constant)

?

How did Newton get that ?
Ans: Experimentation.

I was under the impression that, by experiment, Newton deduced

F∝m → F = k₁m

(where k₁ is some constant)

Precisely yes, this k₁ is the constant acceleration in a body.
It will be
F∝k₁m

and

F∝a → F = k₂a

(where k₂ is some constant)

You are messing the constants but precisely yes , this k₂ is the constant mass of body considering same body.

It will be
F∝k₂a

and then found that either/both

k₁∝a → k₁ = c₁a

(where c₁ is some constant)

This is where you are messing it all up and you did it wrong.

No 'twill be k₁=a
if c₁ is some constant then c₁=1

and/or

k₂∝m → k₂ = c₂m

(where c2 is some constant)

Again I would rather do same , you are messing it up .
No 'twill be k₂=m
if c₂ is some constant then c₂=1

thus creating

F = c₁ma

and/or

F = c₂ma

where c₁=c₂=k=1

where in SI Units they would be in the form

F = ma

Exactly.

However, I've read in some other forums how Newton actually meant

F∝ma → F = kma

Exactly , this is pretty cent percent correct representation of 2nd law .

Which is the case? Did he use the first method or did he simply state the second?

2nd equation is correct . He simply stated the second after deriving it mathematically.

If he did use the first method, how did he resolve that k1 is dependent on acceleration and/or that k2 is dependent on mass?

I repeat , he never used the first method. It is absurd yet I have corrected it in my "this" post above.
k1 is itself the value of acceleration if it is kept constant in a body.
k2 is itself the value of mass if it is kept constant in a body.(taking same body)____________________________________________________________________

Proof of 2nd law

Experimentally Newton found
F∝dp/dt
F∝m(v-u)/t
F∝ma
F=kma

By his experiments we know that 1 kgms^-2 force on 1 kg of body produces in it 1 ms^-2 of acceleration.

So
1=1 x k
or k=1
So
F=ma

Note : k is a universal constant which is always 1 for all the conditions. However logically his 2nd law is self explanatory.

Hope this helps.:)

 

[harrylin]

Don't take it as a serious experience with weapons! It is rather gedanken-experiment like, exaggerated argument. Cannons exceeding in range one mile were not available even during Napoleon Wars.

Good one! However, "google" tells me that the Spaniards and Venetians claimed not less than up to 3 or 4 miles:
http://www.thepirateking.com/historical/cannon_garrison_and_ship_guns.htm
http://www.angelfire.com/ga4/guilmartin.com/Weapons.html

 

[xts]

Chapeau bas!

On a quick search I found only field artillery of mid 19th century exceeding 1 mile range ;(

Anyway, I would really never expect that in 1702 Spaniards were able to kill four and wound six men on 3000 yards distance.

I must read "History of Cyprus" - I hardly believe in 3 miles range of 16th century Venetian culverines... But gentlemen don't dispute the facts... I must accept it, at least till checking the relation...

 

[prosteve037]

If you have two segments A and B in proportion A/B=3/2 you may check it such, that using a compass you markA three times on some line, and you mark B 2 times, and finally those constructed 2A and 3B will be equal. You don't need to know how many mm or inches any of them has.

Okay. I think I know where you're going with this.

See before I asked this question, I assumed some things:

1 - Newton used experiments to deduce two separate proportionality statements to arrive at the conclusion \textit{F}\propto{ma}.

2 - Subsequently, Newton assigned and used units of measurement in his experiments to arrive at this relationship.

Now from what I've read, the former ("1") is partly correct. Correct in that he arrived at the conclusion \textit{F}\propto{ma}. But according to what I understand xts to have said,

He didn't any. Once again - it is a definition of force, not an experimental knowledge. You could however ask, what experiment he did to find the symmetry of such defined forces (3rd principle) - he utilised Galileo's observations and astronomical observations. But he also performed some laboratory experiments with elastic and half-elastic collisions. He also made a gedanken-experiment with a planet cut into two parts - and concluded, that if 3rd principle was not true, than the planet should accelerate on itself, which is absurd.

that assumption was incorrect in that he didn't deduce that relationship from experiments. Specifically, 2 different experiments (\textit{F}\propto{m} and \textit{F}\propto{a}). I'm confused by this.

How was Newton able to deduce, let alone define, the relationship \textit{F}\propto{ma} without having used data to notice that the product of mass and acceleration of an object is proportional to the force applied? I'm having a tough time understanding how Newton could have made such a bold definition without math leading up to its definition.

I never meant to imply that Newton was concerned about the value of proportionality constants between parameters, but what I wanted to address was how he realized that \textit{F} is proportional to the product of mass and acceleration; not \textit{F}\propto{m} and \textit{F}\propto{a} individually of each other.

 

[harrylin]

[..] How was Newton able to deduce, let alone define, the relationship \textit{F}\propto{ma} without having used data to notice that the product of mass and acceleration of an object is proportional to the force applied? I'm having a tough time understanding how Newton could have made such a bold definition without math leading up to its definition. [..]

See the discussion of the same in this parallel thread (almost the same topic, how did that happen?):
https://www.physicsforums.com/showthread.php?t=383019

 

[xts]

what I wanted to address was how he realized that \textit{F} is proportional to the product of mass and acceleration; not \textit{F}\propto{m} and \textit{F}\propto{a} individually of each other.

That is simplest of all - pure algebraic deduction!
Any value proportional to two other values, must be also proportional to their product.

 

[harrylin]

That is simplest of all - pure algebraic deduction!
Any value proportional to two other values, must be also proportional to their product.

What I suppose prosteve meant: if that's indeed the logical way of development, as we all seem to think, then - despite the way Newton summarized it - the first approach of post #1 is what perhaps really happened, well before Newton wrote his Principia.

 

[xts]

OK. Let me try this way: I agree with one of Harrylin's posts, pointing that central concept of Newton's dynamics is momentum (quantity of motion), and observed phenomenon of preservation of momentum in closed systems (or - if you prefer: uniform motion/rest of centre of mass of closed system). Newton did some experiments to test the last: colliding penduli of different masses made of different materials (to get partially inelastic collisions).
All the rest of his dynamics (esp. F~dp/dt; F~ma; F_ab=-F_ba) is a deduction from momentum preservation principle of the closed system combined with convenient definition of force (which had no strict meaning before, so Newton was free to define it as he liked).

Of course - his theory of gravity goes beyond this - it required additional justification. And here, I believe, it was pure deduction from astronomical observations. Or even not directly from observations, but rather from Kepler's laws.

We may only regret taht Principia are not so consistent axiomatic structure as Euclid's Elements - so we have mixed definitions and axioms, and the main concept of momentum is not stressed enough - it is not even present in any of axioms.

 

[harrylin]

A few little remarks:

[..] Newton was free to define it as he liked. [..]

Not completely free, he was bound by observations: see my post #21 on https://www.physicsforums.com/showthread.php?t=383019&page=2

In general, definitions in physics only make sense (are useful) if they are compatible with the theory for which they are used.

I now come to think that perhaps Newton's Principia only appears a little inconsistent because he simply grouped together his basic definitions, which in part refer to some of the laws that he presents next. If so, that's a matter of taste, not of inconsistency.

Note also that he apparently calls descriptions of identities "definitions", while he calls descriptions of physical behaviour "laws" or "axioms"; and I now also come to agree with that.Harald

 

[sankalpmittal]

Okay. I think I know where you're going with this.

See before I asked this question, I assumed some things:

1 - Newton used experiments to deduce two separate proportionality statements to arrive at the conclusion \textit{F}\propto{ma}.

2 - Subsequently, Newton assigned and used units of measurement in his experiments to arrive at this relationship.

Now from what I've read, the former ("1") is partly correct. Correct in that he arrived at the conclusion \textit{F}\propto{ma}. But according to what I understand xts to have said,




that assumption was incorrect in that he didn't deduce that relationship from experiments. Specifically, 2 different experiments (\textit{F}\propto{m} and \textit{F}\propto{a}). I'm confused by this.

How was Newton able to deduce, let alone define, the relationship \textit{F}\propto{ma} without having used data to notice that the product of mass and acceleration of an object is proportional to the force applied? I'm having a tough time understanding how Newton could have made such a bold definition without math leading up to its definition.

I never meant to imply that Newton was concerned about the value of proportionality constants between parameters, but what I wanted to address was how he realized that \textit{F} is proportional to the product of mass and acceleration; not \textit{F}\propto{m} and \textit{F}\propto{a} individually of each other.

Will you have a look at my post #15 ? I think , it carries out the answer of your question and confusion ?

See the first paragraph in the post 15 and lay emphasis on it. The first derivation is correct but partly and the second representation is cent percent correct.

F∝m and F∝a , then F∝ma is the simple concept of proportionality.

Let me try this simple explanation :
suppose,
x=ab
Now imagine that a is a constant and a=2 , say.
then
x=2b

Now
if b=1 , x=2 . if b=2 , x=4 .if b=100 , x=200.

So we notice that b increases x also increases. if its 3b the result is x=6b

So we can say that
x∝b if a is kept constant .
Now repeat that with a ,
x∝a if b is kept constant.

so , x∝ab
or if k=1, then x=ab !

I once again adjure you to see the post #15 .
He deduced the relation by several experiments of mass and acceleration by keeping one constant at a time , and then established that F∝ma.

:)

"Again I recommend that thou shalt see post #15"

 

[xts]

(def. of 'force') Not completely free, he was bound by observations

So I'll then say it as he was free to call "the alteration of amount of movement" as he liked - and he fortunately chose the common language word 'force', which was consistent enough with it to be used.
I agree - the proportionality to mass is a part of common meaning of 'force' (two horses have twice bigger force than one horse, and they may pull two carts, or one twice bigger cart). But common meaning relation of force to acceleration is limited to monotonicity (two-horse cart starts quicker than the same cart pulled by one horse).

Actually, the 'force' is the only of Newton's terms which survived to modern times: 'alteration' got replaced by Leibnizian 'derivative over time', and 'amount of movement' occurred to be so important to earn its own name of 'momentum'.

he simply grouped together his basic definitions, which in part refer to some of the laws that he presents next. If so, that's a matter of taste, not of inconsistency

I disagree - definitions are used in axioms. So if axioms are in turn used in definitions (3rd law is used explicitely in definitions of force), he comes to circularily-self-referencing statements or idem per idem explanations. You may recover from these problems, but the structure, as presented in Principia, is quite inferior to Euclid's clarity and logical order.

 

[prosteve037]

Will you have a look at my post #15 ? I think , it carries out the answer of your question and confusion ?

See the first paragraph in the post 15 and lay emphasis on it. The first derivation is correct but partly and the second representation is cent percent correct.

F∝m and F∝a , then F∝ma is the simple concept of proportionality.

Let me try this simple explanation :
suppose,
x=ab
Now imagine that a is a constant and a=2 , say.
then
x=2b

Now
if b=1 , x=2 . if b=2 , x=4 .if b=100 , x=200.

So we notice that b increases x also increases. if its 3b the result is x=6b

So we can say that
x∝b if a is kept constant .
Now repeat that with a ,
x∝a if b is kept constant.

so , x∝ab
or if k=1, then x=ab !

I once again adjure you to see the post #15 .
He deduced the relation by several experiments of mass and acceleration by keeping one constant at a time , and then established that F∝ma.

:)

"Again I recommend that thou shalt see post #15"

Okay I read this and also I re-read post 15.

Regarding this:

Proof of 2nd law

Experimentally Newton found
F∝dp/dt
F∝m(v-u)/t
F∝ma
F=kma

By his experiments we know that 1 kgms^-2 force on 1 kg of body produces in it 1 ms^-2 of acceleration.

So
1=1 x k
or k=1
So
F=ma

Now I know you didn't say that Newton just took \textit{p = mv} and derived \textit{F = ma} from that. But if he did do that, I guess the real question that I'm asking is where did \textit{p = mv} come from?

I thought that \textit{F = ma} came before \textit{p = mv}, since momentum didn't have a solid definition at the time. Certainly it would've made Newton's job a lot easier for developing his formula.

Point is, I'm really not seeing how the second method in post 1 was how Newton found \textit{F}\propto{ma}. How could he have experimentally determined that?

At least with one of the two different relationships he could've found experimentally that the proportionality constant (either \textit{k}_{1} or \textit{k}_{2}) was proportional to the other parameter (either \textit{m} or \textit{a}), while still ending with \textit{F}\propto{ma}. Of course, that's assuming that the experiment shows that \textit{k}_{1}\propto{a} and \textit{k}_{2}\propto{m}.

 

[sankalpmittal]

My post :

Always remember in physics "that to prove a theory first there is experimental observation and then mathematical deductions"

Newton's laws are proved experimentally and then deducted .
He first gave his 1st law of motion which states "that every body remains in its genuine state of rest or uniform motion in a straight line unless acted upon by some external force (assuming body to be displaced linearly)"

Have you ever heard of the mathematical proof of his 1st law ? No , but it is proved by his 2nd law! But his 1st law was given prior to his 2nd law ! However his 1st law is logically explanatory. Inertia. Mass resists . Each molecule each atom of it !

Proof :
By his 2nd law we know ,
F=ma
If body remains in its original state then there is no force being applied so,
0=ma
or
a=0
So it will have no displacement.

Now let's come on to his 2nd law . What does it state ? It states "that rate of change of momentum in a body is directly proportional to the impressed force acting on it and it takes place in direction of that force assuming body to be linearly displaced in straight line "

F∝m → F = k₁m

(where k₁ is some constant)

?

How did Newton get that ?
Ans: Experimentation.


Precisely yes, this k₁ is the constant acceleration in a body.
It will be
F∝k₁m


You are messing the constants but precisely yes , this k₂ is the constant mass of body considering same body.

It will be
F∝k₂a

This is where you are messing it all up and you did it wrong.

No 'twill be k₁=a
if c₁ is some constant then c₁=1


Again I would rather do same , you are messing it up .
No 'twill be k₂=m
if c₂ is some constant then c₂=1


where c₁=c₂=k=1

Exactly.


Exactly , this is pretty cent percent correct representation of 2nd law .



2nd equation is correct . He simply stated the second after deriving it mathematically.

I repeat , he never used the first method. It is absurd yet I have corrected it in my "this" post above.
k1 is itself the value of acceleration if it is kept constant in a body.
k2 is itself the value of mass if it is kept constant in a body.(taking same body)


____________________________________________________________________

Proof of 2nd law

Experimentally Newton found
F∝dp/dt
F∝m(v-u)/t
F∝ma
F=kma

By his experiments we know that 1 kgms^-2 force on 1 kg of body produces in it 1 ms^-2 of acceleration.

So
1=1 x k
or k=1
So
F=ma

Note : k is a universal constant which is always 1 for all the conditions. However logically his 2nd law is self explanatory.

Hope this helps.


:)

And here is your post :

Okay I read this and also I re-read post 15.

Regarding this:




Now I know you didn't say that Newton just took \textit{p = mv} and derived \textit{F = ma} from that. But if he did do that, I guess the real question that I'm asking is where did \textit{p = mv} come from?

I thought that \textit{F = ma} came before \textit{p = mv}, since momentum didn't have a solid definition at the time. Certainly it would've made Newton's job a lot easier for developing his formula.

Point is, I'm really not seeing how the second method in post 1 was how Newton found \textit{F}\propto{ma}. How could he have experimentally determined that?

At least with one of the two different relationships he could've found experimentally that the proportionality constant (either \textit{k}_{1} or \textit{k}_{2}) was proportional to the other parameter (either \textit{m} or \textit{a}), while still ending with \textit{F}\propto{ma}. Of course, that's assuming that the experiment shows that \textit{k}_{1}\propto{a} and \textit{k}_{2}\propto{m}.

Compare them .

k₁∝a and k₂∝m are not wrong but they don't make any sense ! Correct will be to write that k₁=a and k₂=m

"Point is, I'm really not seeing how the second method in post 1 was how Newton found F∝ma. How could he have experimentally determined that?
"
Ah ! Good question. Realize that momentum was a prior research than force. However we know that the real idea of force was firstly given by Galileo before p=mv. But p=mv came before F=ma. Newton infact used (mv-mu)/t to derive it. He took a ball of mass m and applied force and did several sorts of such things - hit and try using the concept of momentum as stated in his second law.

"At least with one of the two different relationships he could've found experimentally that the proportionality constant (either k1 or k2) was proportional to the other parameter (either m or a), while still ending with F∝ma. Of course, that's assuming that the experiment shows that k1∝a and k2∝m."

As explained he used hit and trial methods.

k1∝a and k2∝m is correct but the coefficient will be 1 so k1=a and k2=m


Lets say that x=rs
Now constant is 1 if k=1
so x=krs

But you can take infinite constants !
if
a=b=c=d=e=f=g=h=i=j=l=m=n=o=p=q=t=u=v=w=x=y=z=k=1

So x=abcdefghijlmnopqtuvwxyzkrs
This is also correct, right .

I can also write this :
F=bcdefghijlnopqtuvwxyzkma

That is why don't be entrapped into the snares of delusion !

Momentum see :http://en.wikipedia.org/wiki/Momentum and http://dev.physicslab.org/Document.aspx?doctype=3&filename=Momentum_Momentum.xml

 

[prosteve037]

But doesn't \textit{k}_{1} = a and \textit{k}_{2} = m only in SI Units? And weren't SI Units developed only way after Newton's time?

With this in mind, though xts has said in his posts that units and measures were "rather neglected" in Newton's times, weren't they necessary (and arguably essential) to develop the formula? Wouldn't you need values to determine whether two things are proportional?

While at my university's bookstore today, I was looking around when I picked up a book that talked a little about \textit{F = ma} and it's formation. In it, it said that Leonhard Euler was the one who turned Newton's definition into a mathematical equation.

Here's a link to Google Books that shows the page where it says it:
http://books.google.com/books?id=IU...=onepage&q="Leonhard Euler" "F = ma"&f=false"

Also check out the very last formula in "Euler's First Law" in the "Overview" section of this Wiki article:
http://en.wikipedia.org/wiki/Euler's_laws_of_motion#Overview

Now I don't know what exactly happened in history, but I find it very plausible that it was indeed Euler who had done the world a favor by formulating Newton's Second Law. Even if Euler did formalize the equation that we accredit Newton to have made, I don't doubt that Newton had already had a rough idea in his mind of what an equation would look like. However, if Euler did indeed develop \textit{F = ma} he must have used algebraic methods right? Weren't geometric methods far outdated by that time?

I hope I've given enough of my own reasoning to show why method 2 of post 1 seems wrong to me. It's not that it's wrong or that I find it wrong, but that the logical progression that method 2 requires is method 1.

 

[sankalpmittal]

But doesn't \textit{k}_{1} = a and \textit{k}_{2} = m only in SI Units? And weren't SI Units developed only way after Newton's time?

With this in mind, though xts has said in his posts that units and measures were "rather neglected" in Newton's times, weren't they necessary (and arguably essential) to develop the formula? Wouldn't you need values to determine whether two things are proportional?

While at my university's bookstore today, I was looking around when I picked up a book that talked a little about \textit{F = ma} and it's formation. In it, it said that Leonhard Euler was the one who turned Newton's definition into a mathematical equation.

Here's a link to Google Books that shows the page where it says it:
http://books.google.com/books?id=IU...=onepage&q="Leonhard Euler" "F = ma"&f=false"

Also check out the very last formula in "Euler's First Law" in the "Overview" section of this Wiki article:
http://en.wikipedia.org/wiki/Euler's_laws_of_motion#Overview

Now I don't know what exactly happened in history, but I find it very plausible that it was indeed Euler who had done the world a favor by formulating Newton's Second Law. Even if Euler did formalize the equation that we accredit Newton to have made, I don't doubt that Newton had already had a rough idea in his mind of what an equation would look like. However, if Euler did indeed develop \textit{F = ma} he must have used algebraic methods right? Weren't geometric methods far outdated by that time?

I hope I've given enough of my own reasoning to show why method 2 of post 1 seems wrong to me. It's not that it's wrong or that I find it wrong, but that the logical progression that method 2 requires is method 1.

I think , what you say is correct. SI systems were not developed when Newton gave his laws. It was Euler who mathematically formulated Isaac Newton's three laws.

Here are certain experiments to prove F=ma which Newton must have conducted :
http://van.physics.illinois.edu/qa/listing.php?id=278
http://www.coursework.info/AS_and_A_Level/Physics/Mechanics___Radioactivity/F_ma_Experiment_L821.html
http://shep.net/physics/TOOLS/DemolabWriteup_force.pdf
http://sdsu-physics.org/physics_lab/p182A_labs/indi_labs/Newtons2ndLaw.pdf
https://www.physicsforums.com/showthread.php?t=34020

I am still not sure about the probability of the first method to be correct.

:)

 

[prosteve037]

I am still not sure about the probability of the first method to be correct.

:)

Thanks for those readings.

But to be frank, I'm still hesitant to agree that the second method was the primary method that Newton used to realize \textit{F = ma}.

It just seems more reasonable to me that he'd notice that (for example) force ∝ mass, or force ∝ acceleration, instead of force ∝ product of mass and acceleration.

 

[xts]

It just seems more reasonable to me that he'd notice that (for example) force ∝ mass, or force ∝ acceleration, instead of force ∝ product of mass and acceleration.

So try to read some of the "Principia..." to get glimpse of Newton's reasoning. Just first ten pages... You can do it! You'll then see that the core idea behind his dynamics is a momentum conservation. Idea of force is secondary to momentum: F = dp/dt. And the amount of movement (momentum) is a product of mass and velocity.

 

[Jadaav]

You may try to read "Principia..." just to see yourself the kind of argumentation Newton used.
There are several English translations available on-line - I recommend (as the easiest to read) - American translation from mid 19th century: http://rack1.ul.cs.cmu.edu/is/Newton/
You may want to skip all the introduction, and start from page 84 - there are corollaries to the principles - explained and illustrated geometrically.

The site mentioned above does not work for me please. Any idea ? I really want to read the Principia...

 

[xts]

The site mentioned above does not work for me please. Any idea ? I really want to read the Principia...

You're right... Google then for full text, or use the link (it surely works) posted by Harrylin: http://gravitee.tripod.com/ (cancel when asked for login) - it contains most important parts of the text.

There is also available English translation from early 18th century (http://books.google.com/books?id=Tm0FAAAAQAAJ), but it is hard to read (mostly due to archaic fonts, we are not used to, but the translation itself is also much worse than Motte's American edition...)

 

[Jadaav]

OK, thanks. I'll be checking the site posted by Harrylin.

 

[prosteve037]

So try to read some of the "Principia..." to get glimpse of Newton's reasoning. Just first ten pages... You can do it! You'll then see that the core idea behind his dynamics is a momentum conservation. Idea of force is secondary to momentum: F = dp/dt. And the amount of movement (momentum) is a product of mass and velocity.

Thanks xts! I re-read the definitions and some of the corollaries!

So he DID use the momentum formula. And further, he defined it as well!

But now did he define his "quantity of motion" as the product of mass and velocity for a certain reason? That is, other than the special case where \textit{m = 0} or \textit{v = 0} ?

 

[xts]

For my understanding of Newton's work, concept of momentum is derived from the concept of centre of mass (momentum it is its derivative over time) and galilean relativity applied to centre of mass. Two-arm levers, scales, etc. devices, leading to concept of centre of mass, were well known even long before Galileo.

Newton made lots of experiments with colliding penduli (different masses, elastic, semi-elastic and inelastic collisions). So he had direct experimental confirmation of momentum conservation (he knew that the angle pendulum reaches is proportional to the velocity at the lowest point if the angle is small). He could easily compare masses and find their proportions and could measure the angles pendulum reaches.

 

[prosteve037]

For my understanding of Newton's work, concept of momentum is derived from the concept of centre of mass (momentum it is its derivative over time) and galilean relativity applied to centre of mass. Two-arm levers, scales, etc. devices, leading to concept of centre of mass, were well known even long before Galileo.

Newton made lots of experiments with colliding penduli (different masses, elastic, semi-elastic and inelastic collisions). So he had direct experimental confirmation of momentum conservation (he knew that the angle pendulum reaches is proportional to the velocity at the lowest point if the angle is small). He could easily compare masses and find their proportions and could measure the angles pendulum reaches.

So in short, he used proportioning techniques to deduce the formula?

 

[xts]

So in short, he used proportioning techniques to deduce the formula?

Yes, that is what I was trying to tell you long time ago already...

 

[prosteve037]

Yes, that is what I was trying to tell you long time ago already...

Gah sorry. I always did ask more questions than I did listen to answers :/

I should ask though, does the Principia show what Newton was trying to measure when performing his pendulum experiments? What parameter did he use to measure momentum/"quantity of motion"?

 

[xts]

I rather believe that most of his ideas come from astronomical observations, especially Kepler laws. Pendulum experiments and common experience thought experiments (as well as Galileo's experiments) might help him to extend F~a into F~ma.

Newton did not measure momentum. He measured only proportions between momenta. Even further - not between momenta directly, but proportion between masses (easy with balance or combining several identical weights together) and proportions between velocities - these are in turn proportional to max angle of pendulum (in an approximation of small angles - Newton knew this relation). Measuring angles was pretty simple.

 

[prosteve037]

I rather believe that most of his ideas come from astronomical observations, especially Kepler laws. Pendulum experiments and common experience thought experiments (as well as Galileo's experiments) might help him to extend F~a into F~ma.

Newton did not measure momentum. He measured only proportions between momenta. Even further - not between momenta directly, but proportion between masses (easy with balance or combining several identical weights together) and proportions between velocities - these are in turn proportional to max angle of pendulum (in an approximation of small angles - Newton knew this relation). Measuring angles was pretty simple.

Wait... I'm confused.

So if he measured proportions between masses and proportions between velocities, where does momentum come into play?

Wouldn't proportions between masses just be a comparison? And same with velocity?

 

[xts]

So if he measured proportions between masses and proportions between velocities, where does momentum come into play?

Make two Newtonian penduli (a ball is hung on two cords mounted in two points of the ceiling, to ensure it travels in a plane) - one of some mass, the second of three times bigger mass.
Now deflect the heavier one by some angle (small enough to use sin(x)=x approx) from equilibrium, and lighter one by three times bigger angle. Release both of them simultaneously.
See that after collision they bounce to a maximum angles having the proportion 1:3, although (as the collision was not perfectly ellastic) the angles are smaller than original.

Conclusion: momentum of some body is equal to momentum of another body traveling 3 times slower but weighting 3 times more

 

[harrylin]

On a little side note, coming back to canons and what was known around then: apparently Galileo and Torricelli already accurately described the trajectories, based on initial velocity and angle. See:
http://books.google.com/books?id=gq...us rotating frame "centrifugal force"&f=false

 

[prosteve037]

Make two Newtonian penduli (a ball is hung on two cords mounted in two points of the ceiling, to ensure it travels in a plane) - one of some mass, the second of three times bigger mass.
Now deflect the heavier one by some angle (small enough to use sin(x)=x approx) from equilibrium, and lighter one by three times bigger angle. Release both of them simultaneously.
See that after collision they bounce to a maximum angles having the proportion 1:3, although (as the collision was not perfectly ellastic) the angles are smaller than original.

Conclusion: momentum of some body is equal to momentum of another body traveling 3 times slower but weighting 3 times more

Are the penduli hitting each other? I assumed they were each colliding with two identical masses/objects (two separate elastic collisions).

Written down in math, would the results look something like this?

\textit{θ}_{M}\propto{θ}_{m} --> \textit{θ}_{M}\textit{ = c}_{1}\textit{θ}_{m} (where \textit{c}_{1} is some constant, in this case \textit{c}_{1} is 3)

\textit{M}\propto{m} --> \textit{M = }\textit{c}_{2}\textit{m} (where \textit{c}_{2} is some constant, in this case \textit{c}_{2} is 3)

...

\textit{Mθ}_{m}\textit{ = mθ}_{M}

So because the angles and masses shared the same proportion (3), the "amount of motion" that each pendulum contained/generated was the same. I'm going to go ahead and assume that Newton took the thetas and did some math to replace them with the appropriate final velocities (at the colliding points) of the penduli.

Now from this, it seems like Newton just chose a definition for momentum. Because the angular displacements of the proportional penduli after collision would've been equal (two separate elastic collisions), it does make sense that he would've just defined momentum based on the displacement or "amount of motion" of the collided object.

But on what grounds does Newton stand in making such a definition? Just because the "amounts of movement" (displacements) are equal and the products of masses and velocities are equal doesn't mean that the "amounts of movement" are equal to the product of mass and velocity.

 

[prosteve037]

I hope I don't get penalized for necro-posting :[

But since last post, post I've read some documents on Google that show the thinking of someone from the time around Newton's. His name is William Whewell and he wrote books on the history of "inductive sciences". You can find some of his books on Google Books for free.

In his readings though, he talks about \textit{mv} and how it's not \textit{m + v} because of the units. Still it was simply understood, even at that time (around the 1800s I believe), that the "quantity of motion" is the product \textit{mv}.

Was \textit{mv} just defined as the "quantity of motion" of an object because it was the only similar characteristic between two objects in an experiment involving collisions? (I'm referring obviously to the Newton's cradle, or the experiments brought up by xts)

 

[xts]

I don't know Whewell's books (ok, I'll try to read some...), but,

even at that time (around the 1800s I believe)

From the perspective of our discussion 1800 is just yesterday - those were pretty modern times. Newtonian mechanics got reformulated to algebraic form in early 18th century. Whewell wrote his works after analytical mechanics by Lagrange and Hamilton.

Was mv just defined as the "quantity of motion" of an object because it was the only similar characteristic between two objects in an experiment involving collisions?

I believe - yes. Actually, in Whewell's times the other value measuring amount of motion was also used: kinetic energy, having even stranger property: it is preserved in ellastic collisions and in gravitational interactions, but it gots lost in inellastic collisions.

But on what grounds does Newton stand in making such a definition?

He found that such defined 'quantity of motion' is always preserved in isolated systems, so it may be very useful to formulate laws of motion.

Newton took the thetas and did some math to replace them with the appropriate final velocities

Yes and no. He knew that maximum velocity of the pendulum is proportional to maximum deflection angle (for a given pendulum length). But he didn't care about expressing the velocity in our modern terms (m/s or inches per second). It is yet another case, where Newton, in his euclidean approach, was focused on proportions, but not on the actual numeric values. So (you did it wrong!) he could go one step further: <br /> \theta_M/\theta_m = m/M\quad\Longrightarrow\quad<br /> v_M/v_m=m/M but he never did the next step: v_M/v_m=m/M\quad\Longrightarrow\quad Mv_M=mv_m - because for him it made no sense to multiply ounces by something else than pure number. It was also a reason why Newton never mentioned any numerical values of velocity: he could define it as a 'change of position (in time)', but he had no unit to measure it. He could measure the angle, but not the velocity: it was not only technical problem, but he lacked units of measure. He could measure angles (in degrees), but he had no unit for velocity. The idea of 'metre per second' (or rather feet per second) came 50 years later.

 

[D H]

I don't know Whewell's books (ok, I'll try to read some...), but,

From the perspective of our discussion 1800 is just yesterday - those were pretty modern times. Newtonian mechanics got reformulated to algebraic form in early 18th century. Whewell wrote his works after analytical mechanics by Lagrange and Hamilton.

I agree with your sentiment here, but not your dates. Have you tried reading physics texts written in the early 1800s? Physics then was quite different from the classical physics of today. Physics was rewritten from the ground up twice during the 19th century, first by Hamilton (born 1805) and then again by Gibbs and Heaviside in the latter part of the century. The modern notation of Newton's second law, \vec F = m\vec a, dates to the late 19th century.

And that in a sense is an answer to the key question raised in this thread, "Which way did Newton find F=ma?" He never did. To say that he did do so is in a sense a case of historical revisionism. Newton's formulation was highly geometric in nature. A rudimentary form of an algebraic interpretation of Newtonian mechanics didn't appear until several decades after Newton's death. It took another 200 years to arrive at the clean formulation of Newtonian mechanics as it is taught today. In the interim, mathematics and physics were rewritten from the ground up, multiple times.

This is typical of mathematics and science. There is a dirty little secret that underlies much of math and science: The all wrapped up with a bow picture of science that is presented to students took a long time to formulate. The initial formulations, while brilliant and revolutionary, were typically crude, clumsy, and incomplete. Newtonian mechanics is a good example of this cleanup process and after-the-fact presentation.

Regarding Newton himself, John Maynard Keynes said it best: "Newton was not the first of the age of reason. He was the last of the magicians." Some of Newton's work and reasoning are lost forever, some are in diaries and notebooks (many of which Keynes collected and later donated to science historians) that are only recently being transcribed. Does it really matter how exactly Newton came up with his laws of physics? His way of thinking was quite foreign to ours. He was limited by the mathematical tools available at his time and he was hobbled a bit by his rather antiquated views of the world. His views were a bit antiquated even by the standards of his time; he truly was the last of the magicians. And he truly was one of the greatest physicists of all times.

 

[logics]

I was under the impression that Newton deduced
\textit{F}\propto{m}
\rightarrow \textit{F = k}_{1}\textit{m}
where in SI Units they would be in the form
\textit{F = ma}...
how did he resolve that \textit{k}_{1} is dependent on acceleration and/or that \textit{k}_{2} is dependent on mass?

if you read the original Latin formulation of the second law

Lex II: "mutationem motus proportionalem esse vi motrici impressae": "change of motion is proportional to applied force",
you'll see that all interpretations are not true: Newton deduced only the obvious principle that the effect is proportional to the cause,
\textit{v}\propto{F}, and then he explains it (if a force generates a motion, a double force generates double the motion, a triple ...).

Mass was not considered, as gravity was the only known force at the time, and its effect is not influenced by mass.

The key point is that to the Latin word "motus" [redundantly] correspond English "motion", and "speed", "velocity", and corresponds also [Defintion II: "quantitas motus" = "quantity of (matter and) motion"] "momentum".
Finally, to make things worse, we must remember that "quantitas motus" for Newton [ mv = E _k ] was also the value of Kinetic Energy, and that
KE [is proportional to squared speed] = mv² was introduced only subsequently by Leibniz (vis viva).
[This is true even now, if we consider that KE = m[=1]v² \frac{1}{2} J [= (m=2 * 1 m/s)]

This should be the staring point for the discussion

 

[juanrga]

I was under the impression that, by experiment, Newton deduced

\textit{F}\propto{m} \rightarrow \textit{F = k}_{1}\textit{m}

(where \textit{k}_{1} is some constant)

and

\textit{F}\propto{a} \rightarrow \textit{F = k}_{2}\textit{a}

(where \textit{k}_{2} is some constant)



and then found that either/both

\textit{k}_{1}\propto{a} \rightarrow \textit{k}_{1}\textit{ = c}_{1}\textit{a}

(where \textit{c}_{1} is some constant)

and/or

\textit{k}_{2}\propto{m} \rightarrow \textit{k}_{2}\textit{ = c}_{2}\textit{m}

(where \textit{c}_{2} is some constant)



thus creating

\textit{F = c}_{1}\textit{ma}

and/or

\textit{F = c}_{2}\textit{ma}



where in SI Units they would be in the form

\textit{F = ma}





However, I've read in some other forums how Newton actually meant

\textit{F}\propto{ma} \rightarrow \textit{F = kma}



Which is the case? Did he use the first method or did he simply state the second?

If he did use the first method, how did he resolve that \textit{k}_{1} is dependent on acceleration and/or that \textit{k}_{2} is dependent on mass?

I read in some place that he did experiments with masses and densities and that used that data to study Jupiter motion and tests his hypothesis. Sorry by not being of more help.

 

[zoobyshoe]

if you read the original Latin formulation of the second law

Lex II: "mutationem motus proportionalem esse vi motrici impressae": "change of motion is proportional to applied force",
you'll see that all interpretations are not true: Newton deduced only the obvious principle that the effect is proportional to the cause,

I think you're short changing Newton here.

He specifically defines momentum (calling it "quantity of motion) in defintion II. It's completely clear he understands it is the product of the mass and velocity.

Thereafter almost every time he uses the word "motion" it is short for "quantity of motion". He drops the "quantity of" apparently just for convenience. When he has to refer to motion as we understand it today, he usually qualifies it as "motion in a right line".

From wiki:

"History
Newton's original Latin reads:
Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.

This was translated quite closely in Motte's 1729 translation 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.

According to modern ideas of how Newton was using his terminology,[26] this is understood, in modern terms, as an equivalent of:

The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed."

http://en.wikipedia.org/wiki/Newton's_laws_of_motion

That last translation is not a retroactive "clean up" of the type DH refers to in his excellent post. It is actually what Newton was saying. Here is the Law with it's explanation/discussion:

"The alteration of motion is ever proportional to the motive force impressd; and is made in a direction of the right line in which that force is impressed.

If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to, or subtracted from the former motion, according as they directly conspire with, or are directly contrary to each other, or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both."

The clause "whether that force be impressed altogether and at once, or gradually and successively" demonstrates that Newton understood the concept of impulse, without having a dedicated term for it.

A big hindrence in deciphering the Principia is the fact it wasn't written as a physics text aimed at the uninitiated. It was directed toward whatever people there were with enough physics education to be in, or interested in, the activities of the Royal Society, as is evident from passages like this:

"By the same, together with the third Law, Sir Christ. Wren, Dr. Wallis, and Mr. Huygens, the greatest geometers of our times, did severally determine the rules of the congress and reflexion of hard bodies, and much about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis, indeed, was something more early in the publication; then followed Sir Christopher Wren, and lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth of the thing before the Royal Society by the experiment of pendulums, which Mr. Mariotte soon after thought fit to explain in a treatise entirely upon the subject."

I think this explains why he does things like drop "quantity of" from "quantity of motion". He's pretty sure his intended audience will all have come to take the word "motion" to refer to "quantity of motion" unless otherwise qualified, from their familiarity with all the papers about and experiments concerning, conservation of momentum. The Principia comes down to us out of context.

Incidentally, Newton doesn't appear to take any credit for any of the three laws, that I can see. He ascribes the first two to Galileo:

"Hitherto I have laid down such principles as have been received by mathematicians, and are confirmed by abundance of experiments. By the first two Laws and the first two Corollaries, Galileo discovered that the descent of bodies observed the duplicate ratio of the time; and that the motion of projectiles was in the curve of a parabola…"

and the third seems to have arisen (though I'm not sure) from the collective endeavor to prove conservation of momentum he described above.