Talk:Newton's laws of motion/Archive 3

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Better Representation?

Hi I am only a year 7 student, and I could be very wrong, but the picture on the page with the two ice skaters pushing against each other may not be the best representation of the third law. They aren't necessarily of the same strength or weight, therefore, one may be pushed back more. I think that the recoil of a firearm would be a much better representation.Nelsondog (talk) 05:18, 16 March 2008 (UTC)Nelsondog

Thanks for checking- but Newton's 3rd Law applies even if one is more massive or is stronger. The force between them will always be the same. PhySusie (talk) 12:45, 16 March 2008 (UTC)

PhySusie, you might have misread what Nelson is saying. He is saying that a better example than two skaters would be to show the effects of equal and opposite forces on two bodies of different mass, such as a rifle and a bullet, where the smaller body is accelerated by a greater amount. The example of two skaters of roughly equal size might be misinterpreted to give the false impression that the acceleration is always equal and opposite. MarcusMaximus (talk) 04:39, 25 August 2008 (UTC)

I agree with Nelson that there should be a better way to represent the third law. I feel that standing on a frictionless surface one would never be able to exert any force on another object, in the way it is depicted in the figure. The figure implies the use of traction from the surface, which would complicate the description. As Nelson suggests recoil would be a much clearer way to illustrate the 3rd law. K.sateesh (talk) 07:01, 15 February 2009 (UTC)

Abit confused

I'm confused with what d is. From what I remember from school shouldn't say a=dv/dt be a=Δv/Δt ?? and if d==Δ then y use d instead of Δ? —Preceding unsigned comment added by Yellow Onion (talk • contribs) 05:45, 19 March 2008 (UTC)

The statement a = dv/dt refers to the first derivative of the velocity function with respect to time. It gives the slope of the line on a velocity vs time graph at that instant in time. The expression you gave using the delta symbol refers to an average over time (so it is not instantaneous) - which is used as an approximation - particularly in algebra based physics courses. PhySusie (talk) 15:25, 19 March 2008 (UTC)

Question about the Motte translation

I'm a bit confused after reading the Motte translation of Newton's second law: "If a 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"

If Newtons 'motion' is equal to modern day 'momentum' (as stated elsewhere on the page), isn't his concept of 'force' a bit different from the modern concept? Since Newtons laws state that forces generate *change* in momentum, a double force would create double the *change* in momentum. Also, the last part about applying the force altogether or gradually seems weird - if the force was not applied for the same amount of time, you would not end up with the same momentum. His concept of force here seems to be closer to what we call the 'impulse' (force x time), doesn't it? Mhc (talk) 21:44, 20 March 2008 (UTC)

Descriptive name for second law

I have changed this name from "law of acceleration" to "law of resultant force". My reasons are: (i) Newton never referred to acceleration, (ii) a more historically accurate version would refer to "rate of change of momentum" (see Feynman Lectures, vol. 1) (iii) The law refers explicitly to "resultant force" and Newton focussed on the resultant force for many pages, describing how the resultant could be found by vector addition. Moreover, this law does have the purpose of introducing the term force, which pervades all uses of mechanics. See also Williams to support the notion that force is the defining objective of the second law. It seems silly to mis-translate the law to name it in a form that is is less accurate, both historically, and from the viewpoint of relativity and, in fact, from the viewpoint of this article itself. Brews ohare (talk) 15:52, 15 April 2008 (UTC)

On Google, "Law of resultant force" gives two results (including this Wikipedia article), while "Law of acceleration" gives 58.000 results. Since Wikipedia is an online encyclopedia, and verifiability is essential (see WP:V), I propose to revert to the original "Law of acceleration". Further, regarding your edits, note this is an article made for a broad audience. Specially the lead section and first section are read by many people without an academic background. Concepts like rate of change of momentum are, in my opinion, to difficult to start with in the article lead.

Further please have a look at the discussions on this talk page and it archives. For instance, there have been several discussions on the 3rd law, resulting in the abbreviation "To every action, there is an equal and opposite reaction", which you have changed into "For every action ..." (more strongly implying a causal relationship between the two forces). Crowsnest (talk) 21:34, 16 April 2008 (UTC)

Not guilty - To my knowledge I have not tinkered with the third law.

As for momentum - a link is provided to the article on that subject. Personally, I find momentum an easier topic than acceleration, because it shows up not only in common everyday English, but in many other areas of mechanics, e.g. the law of conservation of momentum.

And, as the citations show, formulation in terms of momentum has taken place beginning with Newton in 1687, and continuing to today. It is by no means a "high brow" or "new" approach, although this formulation has shown to be compatible with relativity (1905, wasn't it?), as pointed out later in the article.

I get 7,370 hits for "law of acceleration" using "Newton" as a qualifier to eliminate bogus references. Of course, my objective in naming the "law of resultant force" was to emphasize the "force" aspect of the law. Using "resultant force" + "Newton" gives 48,100 hits, "resultant force" + "second law" provides 24,200 hits. So resultant force is a biggy. Brews ohare (talk) 15:46, 17 April 2008 (UTC)

You are right about the 3rd law change, my apologies for linking you to that change.

Regarding the 2nd law, as you stated yourself, it is your personal preference to use the momentum approach, but it is uncommon to be learned about Newton's 2nd law that way. Using a constant mass, and acceleration is simpler to start with. Regarding your "Law of resultant force", while Google hits don't say anything about the correctness of a statement, it does say something about what is common. This is not about truth, i.e. whether ultimately the 2nd law is about change of momentum or acceleration, but on how to present something in an encyclopedia, without losing most of your audience already in the lead section. So starting simple, in terms of acceleration, and expand to more general case later on. The phrasing you use is original thought, hardly found elsewhere in this formulation. Crowsnest (talk) 16:04, 17 April 2008 (UTC)

You might look at the template in the article. which shows the equation explicitly in terms of momentum

The "change of momentum" approach is very common, as the citations show. More than common, it's time-honored – it's been used since 1687!  :-)

Whether acceleration or momentum is the "easier" way is open to discussion. For some problems, momentum is easier (and/or more accurate) and for other problems it is acceleration. As for what is more natural or more easily absorbed by the reader (unrelated to advantages for any particular problem), I believe the momentum approach has the edge. Being more fundamental, the reader who goes on to the relativity section, or who wants to look into other topics, like the collisions of billiard balls etc. doesn't have to change gears. The historically interested person doesn't have to ask why Newton's correct initial statement has been changed for a less fundamental version.

As a guess, you have in mind problems where acceleration works best. However, solving such problems is not the universe of all problems, and also may neither have bearing on the ease of access to an encyclopedia article, nor to its value as introduction to other articles, nor to its value as general background of a qualitative sort (e.g. philosophical or historical value).

Have you thought a bit about the possible goals of different readers, possible backgrounds, and what would serve their purposes? Brews ohare (talk) 17:33, 17 April 2008 (UTC)

What I have in mind, when starting with acceleration, is that it directly relates to ones own experience. To get acquaintance with Newton's 2nd law most easily, is by direct experience. Which almost everybody has: in an accelerating or decelerating car, a playground swing, roller coaster, etc. You feel the force, while you see the acceleration. That is what makes it easy to relate to Newton's second law. Just because in most instances our own mass is near constant. Momentum, and momentum change rate other than by acceleration, are not in general a direct physical experience of most people. It needs explanation and verification at a higher level of abstraction. Crowsnest (talk) 18:55, 17 April 2008 (UTC)

One can just start with F=ma, and state under which conditions it applies, and later on extend to the rate of change of momentum. This is not about the fundamentally most correct formulation, but about how to present things. Crowsnest (talk) 18:59, 17 April 2008 (UTC)

You have outlined one approach and argued that it the most transparent. I have argued a different approach and argued that it is (i) at least equally transparent, (ii) more fundamental, (iii) historically predominant, (iv) used in many other articles in Wikipedia including this article, (v) adopted by reputable authors ranging from Newton to Feynman, and (vi) useful to a wider variety of readers, who undoubtedly have a wide range of agendas when consulting this article, ranging from philosophy to physics. Brews ohare (talk) 13:44, 18 April 2008 (UTC)

Since nobody else supports my view in this, I will stop with this discussion. Crowsnest (talk) 14:06, 18 April 2008 (UTC)

Recent reversions

User:Jok2000 reverted my edits without response to my explanations on talk page, and without explanations of his own. With the possible exception of the change in name for the second law, which was discussed above, the reverted changes are totally noncontroversial. I expect comment before, for example, addition of a reference is reverted. I believe these reverts were simply a "knee-jerk" reaction without justification, stated or imagined. I have added citations for the statement of Newton's second law, which is an almost literal translation of the Latin and has been used for centuries. Citations I've added go back to 1911 only. This statement of the law is used in this article itself later on. I've also added reference to Moller for relativity - a definitive reference in this area, for those who may not know these simple relations. Brews ohare (talk) 03:11, 16 April 2008 (UTC)

Tell me more about my "imagined" reasons. The 2nd law is F=ma [1]. So now take out your changes to it. Jok2000 (talk) 21:03, 16 April 2008 (UTC)

The article quotes Newton as "Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.", which I'd guess trumps Wolfram as an authority on the matter of Newton's laws. Moreover, I've provided multiple references to show that this wording persists to this day. The formulation in terms of acceleration is a more limited, less accurate formulation. You might also look at the template in the article. which shows the equation explicitly in terms of momentum Brews ohare (talk) 14:24, 17 April 2008 (UTC)

Strange, I thought it was you who removed that bit. Must have been some vandal. I retract the request. Its a bit funny actually, I repaired this once before in October 2007, if you check the history, to be what's there now. The vandals hit this thing twice a day. I can't be here all of the time, I guess. Jok2000 (talk) 15:18, 17 April 2008 (UTC)

Change the 'Classical mechanics' diagram's sub-legend ?

The ‘Classical mechanics’ diagram here and this article historically misrepresent at least Newton’s second law of motion. Here I copy below a discussion of this issue from the diagram’s Template Talk page [2] for further consideration here as perhaps the most relevant article.

Change the diagram's sub-legend

I propose the diagram's sub-legend 'Newton's second law of motion' be changed to 'The second law of motion of classical mechanics'.

The diagram is mistaken because its sub-legend 'Newton's second law of motion' is historically mistaken and if anything should be rather 'The second law of motion of classical mechanics'.

This is certainly not Newton's second law stated in the Principia, which was that THE change of motion [referred to in the first law] is proportional to the motive force impressed, i.e. Dmv @ F, or F --> Dmv (where 'D' = 'the absolute change', Delta, '@' = 'is proportional to', and '->' is the logical symbol for if... then...).

The misrepresentation of Newton's second law as F = ma or similar has the logical consequence that a = F/m and thus a = 0 when F = 0, whereby Newton's first law would be logically redundant just as Mach claimed it was.

But Newton's second law only deals with changes of motion produced by impressed force such as mentioned in the first law, and does not itself assert there is no change of motion without the action of impressed force as the law F = ma does, where F denotes impressed force rather than inertial force. And in fact both Galileo's 1590 Pisan impetus dynamics and Kepler's 'inertial' dynamics, both of which claimed motion would terminate without the continuing action of what Newton called 'impressed force', denied this principle.

But the logical function and historical purpose of Newton's first law is precisely to assert this principle, that there is no change of motion unless (i.e. If not) compelled by impressed force, and thus whereby Dmv <=> F, rather than just F --> Dmv. (Here <=> is the logical equivalence symbol for 'if and only if', and '-->' the symbol for 'If...then...') Thus Mach’s logical criticism was wrong by virtue of his ahistorical misinterpretation of Newton’s second law as F = ma.

Classical mechanics, whatever that might be, needs to be differentiated from Newton's mechanics.

--Logicus (talk) 18:20, 16 April 2008 (UTC)

Hi Logicus. Well, I don't know. The article on Newton's laws of motion says that the Newton second law is "The Rate of change of momentum is proportional to the resultant force producing it and takes place in the direction of that force". Isnt it the same thing that the formula on the template? (or maybe according to you, both article and template are wrong?) I do not oppose you change the sub-lengend, but honestly i'm not sure i see a true difference. Even if the formula is not as Newton stated it, it's greatly inspired by no :)?? And history remembers it as the Newton second law (improved?). Am I wrong? But your comment is interresting. Is this information on Newton laws of motion article???

Frédérick Lacasse (talk · contribs) 13:03, 17 April 2008 (UTC)

First of all, the relevant text is at wikisource, and I'll begin by saying that I disagree with Logicus on her/his proposal. Newton's second law is not stated, as such, mathematically. (Perhaps it is later in the Pricipia, I do not know.) For clarity, its statement under Axioms, or Laws of Motion reads:

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

(Emphasis mine, in order to make clear that this is certainly an if and only if statement. Note that Logicus has failed to quote that word in presenting his/her argument.) Since Newton used Calculus in conjunction with this law to calculate planetary orbits and such, it is not too crude to use modern calculus notation in the box, even if we choose Leibniz' $\mathrm {d} /\mathrm {d} t$ notation over Newton's dot. For that matter, we use vectorial notation when Newton had none. Therefore, clearly, in modern notation, Newton's second law reads

${\vec {F}}={\frac {\mathrm {d} }{\mathrm {d} t}}m{\vec {v}}$, and I have no problem with identifying this equation (or an equivalent one) as Newton's Second Law or Newton's Second Law of Motion. Or, see Goldstein, Poole, and Safko, Classical Mechanics (3rd ed.) page 1, where $\mathbf {F} ={\frac {d\mathbf {p} }{dt}}\equiv {\dot {\mathbf {p} }}$ is identified as Newton's second law of motion.

Regarding the other stuff you've said about Mach, historical interpretations, and other irrelevant (for the purposes here) things, perhaps this can help. Newton's second law, which can be expressed as $F=ma$ cannot imply that "Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon" (Law 1), without presupposing the existence of inertial frames, which is what the first law, in effect, does. The fundamental disconnect between Newton and Mach, as I understand it, concerns the existence of a preferred inertial frame.

If, unlike I've read in many sources over the years, you have sources that claim Newton never equated force with a time-rate-of-change of momentum, you might be able to begin to find people willing to change their ways. However, the classical mechanics template talk is not the place for that discussion. — gogobera (talk) 20:10, 17 April 2008 (UTC)

Logicus's response to Frederick Lacasse, written before Gogobera's contribution: Thanks Frederick. Yes, BOTH the article on Newton’s laws of motion and the template are wrong, because the article mistranslates the Principia’s second law’s phrase 'mutationem motus' as ‘rate of change of momentum’, whereas it should be ‘The change of motion’, with no reference to any rate of change. It referred to an absolute change of motion as produced by an impulse, as in Cartesian vortical mechanics. May I refer you to Bernard Cohen’s ‘Guide to Newton’s Principia’ in the 1999 Cohen & Whitman Principia new translation for a good discussion of this issue, which was also touched on in the recent BBC Radio 4 ‘In Our Time’ programme on Newton’s Laws of Motion and drew the following comment from a listener published on the BBC website @ http://www.bbc.co.uk/radio4/history/inourtime/inourtime_haveyoursay.shtml

“Andrew, Newton's 3 laws

Simon Schaffer might have done well to see in this archive (dating from the programme on Popper), "if you study the original version of Newton's Second Law - not the modern F=ma - you realise that Newton regarded force as a function of time, equivalent to the modern notion of an impulse. It was change of momentum: mass *or* velocity; thus even if mass increases with increased velocity so does the force required, and Newton holds." The insertion of 'rate' in 'rate of change of motion (momentum)', giving F=ma, isn't a flaw of Newton's - it's a mistranslation of 'mutationem motus'. “

The true difference, as I have already pointed out, is that rather than Newton being illogically foolish in his axiomatisation is respect of stating a logically redundant axiom, namely Law 1, as Mach implied, because it was logically entailed by his Law 2, rather his first law states a logically independent axiom which, for example, ruled out Kepler’s theory of inertia according to which the inherent force of inertia resists and terminates all motion. For in Newton’s theory which revised the keplerian theory of inertia the force of inertia only resists accelerated motion and causes uniform straight motion like impetus did in late scholastic Aristotelian and Galilean impetus dynamics.

This is all important for understanding the logic and history of scientific discovery and such as how and why ‘classical mechanics’ emerged, the project started by Duhem that was the major research project of 20th century history and philosophy of science.. But there seems to be some considerable logical confusion and contradiction in Wikipedia articles about Newton’s dynamics and about classical mechanics and what it is and how it relates to Newtonian mechanics. For example the article on ‘Classical mechanics’ says on the one hand the two are equivalent, but on the other hand they are not equivalent because classical mechanics was created later and goes well beyond Newton’s mechanics, as in the following statements:

“There are two important alternative formulations of classical mechanics: Lagrangian mechanics and Hamiltonian mechanics. They are equivalent to Newtonian mechanics, but are often more useful for solving problems.”

Versus

“While the terms classical mechanics and Newtonian mechanics are usually considered equivalent (if relativity is excluded), much of the content of classical mechanics was created in the 18th and 19th centuries and extends considerably beyond (particularly in its use of analytical mathematics) the work of Newton.”

This confusion needs sorting out, but a bigger job than I have time for. I was just trying to reduce this confusion a little, as it cropped up on the Galileo article, but I now see this diagram is pretty ubiquitous in relevant articles. Sorry just to pick on your otherwise no doubt helpful diagram.

I fear the article on Newton’s laws of motion is currently virtually wall to wall ahistorical nonsense, apparently being devoted to teaching some version of 19th century mechanics or A-level Physics rather than the history of physics.

Re your following comment

“Even if the formula is not as Newton stated it, it's greatly inspired by no :)?? And history remembers it as the Newton second law (improved?). Am I wrong?”

One problem with the first claim that Newton greatly inspired the law F = ma is that the Wikipedia classical mechanics article is currently claiming

“The proportionality between force and acceleration, am important principle in classical mechanics, was first stated by Hibat Allah Abu'l-Barakat al-Baghdaadi,[7] Ibn al-Haytham,[8] and al-Khazini.[9] “

- although I have no idea whether this is true or not.

All in all I think I should implement the proposed edit if you have no further comments or objections. But it is still unsatisfactory given it is unclear from Wikipedia what exactly classical mechanics is, whereby such as Lagrangian and Hamiltonian mechanics and even Newtonian mechanics are said to be alternative formulations of it.

The outstanding pedagogical question for your view is surely that if you claim that Newton’s second law was essentially

F = ma, then why did he think he needed to state the first law as his first axiom ? The simple answer is because it gives the equivalence between change of motion and the action of impressed force that the second law does not, because the second law is only at most Dmv α F and not Dmv = F. Logicus (talk) 20:34, 17 April 2008 (UTC)

-- Further to my observation above that this article seems to be in serious conflict between the aim of teaching some form of ‘classical mechanics’ (or A-level Physics) under its various categories on the one hand of ‘Classical mechanics’, ‘Introductory physics’ and ‘Experimental physics’ etc for example, and on the other hand that of explaining what Newton’s laws of motion were under the categories of ‘History of physics’, ‘Isaac Newton’, ‘Latin texts’ etc., I further suggest that what this article and that on ‘classical mechanics’ misleadingly call ‘Newtonian mechanics’ may in fact rather be Laplacean mechanics as distinct from ‘Newtonian mechanics’ insofar as the latter term refers to the mechanics of one Isaac Newton as expounded in his Principia, and Laplacean mechanics may possibly be what is common to both Lagrangean and Hamiltonian mechanics whereby they are inexplicably both said to be forms of some ‘classical mechanics’, but which is itself never properly identified in any corresponding axiomatic form. For example the article says

“The term classical mechanics was coined in the early 20th century to describe the system of mathematical physics begun by Isaac Newton and many contemporary 17th century workers”.

But one major difficulty of many here is that Newton himself said in his Principia Axioms Scholium that its three laws of motion were already accepted by mathematicians and such contemporary figures as Huygens, Wallace and Wren, but also by his non-contemporary Galileo and also even (essentially correctly) pointed out its first law was even current in Greek antiquity, including in Aristotle’s principle of interminable locomotion in a void (Physics 4.8.215a19-22). --Logicus (talk) 16:01, 20 April 2008 (UTC)

Excuse me if I am being a mite blunt but this appears to be a little self centred. It is not up to Wiki editors to decide what Newton intended. Rather it is to publish the balanced view i.e. on the one hand some think Newton meant F=ma on the other had they don’t. I hope ego is not getting in the way here, it is not nearly as complex as the rhetoric implies. Rolo Tamasi (talk) 20:20, 20 April 2008 (UTC)

Unfortunately, I still do not agree with your position, Logicus. Perhaps you were planning on addressing my comments before you changed the template? I believe we ought to use the term Newton's second law for the template for the reasons stated above in my previous comment. I have given a source claiming the equation to be identified as Newton's second law. Here is a list of sources that identify $F={\dot {p}}$, $F=ma$, $F=m(dv/dt)$, or some other equivalent formulation as "Newton's Second Law":

• Danby, John M.A. Fundamentals of Celestial Mechanics. Richmond, VA: Willmann-Bell, 1992. page 44.
• Feynman, Richard P., Robert B. Leighton, and Matthew Sands. The Feynman Lectures on Physics: Mainly Mechanics, Radiation, and Heat. Reading, MA: Addison-Wesley, 1977. pages 9-1,2.
• Goldstein, Herbert, Charles Poole, and John Safko Classical Mechanics. 3rd ed. New York, NY: Addison Wesley, 2002. page 1.
• Hirose, Akira and Karl E. Lonngren. Introduction to Wave Phenomena. Malabar, FL: Krieger Pub. Co, 2003. page 4.
• Marsden, Jerrold E. and Anthony J. Tromba. Vector Calculus. 5th ed. New York, NY: W.H. Freeman and Co., 2003. page 264.
• Penrose, Roger. The Road to Reality. New York, NY: Alfred A. Knopf, 2005. page 389.
• Thornton, Stephen T. and Jerry B. Marrion. Classical Dynamics of Particles and Systems. 5th ed. Belmont, CA: Brooks/Cole, 2004. page 50.

Wikipedia is not about original research. The previous sources are from advanced and introductory textbooks, as well as books for the mathematical layperson. This list is only limited by my desire to stop pulling books off my shelf. Suffice it to say that the equation presented on this template is commonly known as Newton's Second Law. I don't think there is any disagreement over that fact. The idea that "some think Newton meant F=ma" and some don't is an issue for the history of science. The fact is that today, $F=ma$ is identified as "Newton's second law" universally.

Logicus, your arguments have a tendency to sway away from the issue at hand. For instance, your comment about the inconsistency in Wikipedia's treatment of the equivalence of Newtonian mechanics and classical mechanics. However, to clarify, it should be stated that there is an equivalence between the Newtonian, Lagrangian, and Hamiltonian formulations of classical mechanics. So that statement that says "[Lagrangian and Hamiltonian mechanics] are equivalent to Newtonian mechanics" is true (if sloppily worded). Also, saying that classical mechanics has "extend[ed] considerably beyond … the work of Newton." [emphasis mine] is also true, since Newton's work was limited in scope. However, in principle, mechanical results can be derived using any of the formulations. The amount of work to do so can vary tremendously.

If Logicus wants to add cited material discussing Newton's intentions and their historical interpretation, I have no qualms with that. This discussion began, and should be focused on the use of the term "Newton's second law" to describe the equation on the template. Though my understanding is that Newton did mean his second law to be understood as $F={\dot {p}}$, the point can be made with less ambiguity by discarding historical interpretation: the equation, as written, is commonly identified as Newton's second law. Therefore, I will change the template back. — gogobera (talk) 19:26, 25 April 2008 (UTC)

Logicus to gogobera: You said "Perhaps you were planning on addressing my comments before you changed the template?", but I presumed you would appreciate your criticisms were dealt with by my reply to Lacasse. Apparently not. I shall explain later. —Preceding unsigned comment added by Logicus (talk • contribs) 18:18, 29 April 2008 (UTC)

Second Law and Impulse

I believe that it is historically incorrect to state Newton's Second Law as F=dp/dt. Motte's 1729 translation (as stated in the article) reads The alteration of motion is ever proportional to the motive force impressed.. At first sight this appears to be saying that force is proportional to motion (velocity??) which is the very concept that Newton is supposed to be discarding. However, a reading of Newton's definitions in the Principia makes it clear that by motion he means momentum and by force he means impulse in modern terminology. So what Newton actually said, using modern symbols, is;

$I\propto \Delta p$

Obviously, using SI units and differentiating yields the usual form;

${\frac {dI}{dt}}=F={\frac {dp}{dt}}$

However, it is unhelpful that the article states the law in this form right after quoting Newton's original words. The impulse version of the law should be stated first together with Newton's definition of his terms.

It has been noted in comments on this talk page already by others that what is written in the article about force does not make sense but with no satisfactory explanation ever been given. Anyone want to tell me I'm wrong? SpinningSpark 08:31, 2 May 2008 (UTC)

The set-up does suggest that the mathematical equation is a direct implementation of Newton's words. That should be fixed if your history is accurate. Can you substantiate your beliefs about the historical background?

Whether historical statements of the law should come first is debatable, as history is an interest of only a subset of readers. Also the introduction of impulse is pretty opaque to the modern reader, and a bit of a digression in applications. Its use at the beginning of the article would make it all rather indigestible. An historical section may become necessary to give room to explain about the historical role of impulse and its connection to rate of change of momentum. Brews ohare (talk) 20:37, 3 May 2008 (UTC)

I did not introduce the historical diversion into this article, it was there already. I have no view on whether it should be in this article or another. However, given that it is here, it is necessary qualify Motte's translation with the modern terms and show that the modern form of the law can be derived from Newtons form (it is only one simple step). Several comments furhter up the talk page verify that this is indeed causing confusion. As for my source, it is Newton's Principia, the definition of motion (=momentum) is given at definition 2 here [3], I am hoping you can get through the paywall (just hit cancel at login) or else try this one [4]. As for force, the text given in the article makes the point, If a force generates a motion, a double force will generate double the motion . . . whether that force be impressed altogether and at once, or gradually and successively. That only makes any sense if impulse is meant rather than force. If it really meant force a step function would produce the same motion as a ramp function - clearly nonsense. SpinningSpark 01:02, 4 May 2008 (UTC)

You will find an earnest attempt to implement what I understand from your remarks at Impulse. Brews ohare (talk) 02:13, 4 May 2008 (UTC)

You obviously put some effort into that, but I have changed it because Newton from his wording cleary does not restrict the concept to an infinitesimal time (neither does our own article, or NASA). I have changed the reference too, presumably that's what your ref says. My ref is a pretty low level one - the ones on the impulse article might be better but I have not looked them up. SpinningSpark 02:43, 4 May 2008 (UTC)

Logicus: Your discussion might well benefit from studying Cohen's analysis of this issue in his 1999 Guide to Newton's Principia in the 1999 Cohen & Whitman Principia --Logicus (talk) 14:46, 4 May 2008 (UTC)

A reference to I Cohen is added to the Impulse subsection. Brews ohare (talk) 15:55, 4 May 2008 (UTC)

Equivalence. What equivalence ?

Gogobera claimed above 25 April:"...it should be stated that there is an equivalence between the Newtonian, Lagrangian, and Hamiltonian formulations of classical mechanics. So that statement that says "[Lagrangian and Hamiltonian mechanics] are equivalent to Newtonian mechanics" is true (if sloppily worded). Also, saying that classical mechanics has "extend[ed] considerably beyond … the work of Newton." [emphasis mine] is also true, since Newton's work was limited in scope. However, in principle, mechanical results can be derived using any of the formulations. The amount of work to do so can vary tremendously."

From a logical point of view for any two theoretical systems to be equivalent every axiom and theorem of one system must be an axiom or theorem of the other. But where, when and by whom was any such logical equivalence between Lagrangian, Hamiltonian and Newtonian mechanics ever demonstrated ? If Wikipedia makes such logico-historical claims, they need source documentation. Are any of the various mechanical systems of Lagrange, Hamilton, Laplace, Newton (Principia Ed. 3) and the elusive 'classical mechanics' logically equivalent ? And what were the axioms of each of these systems, especially of the latter three (e.g. is Corollary 1 of the Principia a theorem or really an axiom, as in earlier versions i.e is Newton's proof of theoremhood valid, or invalid as Bernoulli presumed in tryit to reprove it?).--Logicus (talk) 14:46, 4 May 2008 (UTC)

To support my claim, I will quote from Section 7.7 of Stephen Thornton and Jerry Marion's Classical Mechanics (p. 257) titled Essence of Lagrangian Dynamics:

We elected to deduce Lagrange's equations by postulating Hamilton's Principle because this is the most straightforward approach and is also the formal method for unifying classical dynamics.

First, we must reiterate that Lagrangian dynamics does not constitute a new theory in any sense of the word. The results of a Lagrangian analysis or a Newtonian analysis must be the same for any given mechanical system. The only difference is the method used to obtain these results.

…

The differential statement of mechanics contained in Newton's equations or the integral statement embodied in Hamilton's Principle (and the resulting Lagrangian equations) have been shown to be entirely equivalent. Hence, no distinction exists between these viewpoints, which are based on the description of physical effects. [emphasis original]

I am not sure if I can be any clearer, except by presenting, in its entirety, section 7.6 of the same book, titled Equivalence of Lagrange's and Newton's Equations which begins: "As we have emphasized from the outset, the Lagrangian and Newtonian formulations of mechanics are equivalent: The viewpoint is different, but the content is the same. We now explicitly demonstrate this equivalence by showing that the two sets of equations of motion are in fact the same." (p. 254) The text continues with a mathematical development of the claim.

Thornton and Marion do go on to clarify that a philosophical difference exists:

In the Newtonian formulation, a certain force on a body produces a definite motion—that is, we always associate a definite effect with a certain cause. According to Hamilton's Principle, however, the motion of a body results from the attempt of nature to achieve a certain purpose, namely, to minimize the time integral of the difference between kinetic and potential energies. [N.B.] The operational solving of problems in mechanics does not depend on adopting one or the other of these views. [emphasis original](p. 258)

Does this make my position clear? I would welcome anyone else's opinions as well. — gogobera (talk) 23:39, 22 May 2008 (UTC)

Inertial frames

Any particle, regardless of what forces act on it, is at rest relative to the reference frame whose origin is defined to coincide with the particle. The non-trivial point about inertial reference frames in Newtonian mechanics is that they are the same for all physical objects. I corrected the formulation.

Taneli HUUSKONEN (talk) 14:12, 5 June 2008 (UTC)

I'm not sure what change you've made, but I hasten to add that while a particle is at rest, by definition, relative to the frame whose origin coincides with the particle, that frame is frequently not an inertial frame. — gogobera (talk) 22:54, 5 June 2008 (UTC)

Second law in intro

I maintain that F=ma is the form of the law that should be listed in the brief statements of the laws in the intro. It is the one most people are familiar with from sixth grade science class, and it does not require previous knowledge of momentum. All the stuff about special relativity and changing mass can be (and is) explained later. I don't think it matters that Newton didn't write F=ma--it is still a traditional, simple statement of the second law.

Apparently a least a couple of people disagree with me. I suggest we talk about it here rather than in revert edit summaries.Rracecarr (talk) 19:57, 1 July 2008 (UTC)

Sorry, I have reverted you again. The article has been like this for a long time after lengthy discussion on this page. I feel you ought to reach consensus here before re-inserting material that has been challenged once. I do not believe that Wikipedia should present an incorrect version of Newton's laws jsut because it is commonly taught that way in high schools. Certainly not in the lede. The article can and should explain the F=ma simplification but should not start off by giving the impression that that is actually Newton's second law. The reason it is taught that way, by the way, is not because momentum is a difficult concept, but because it avoids using differentiation. The acceleration article, for instance, correctly states that a=dv/dt. But at school I was not taught that formula, I was taught v=u+½at². It would be completely wrong to put that formula in the acceleration article lede, just as it is wrong to put the school formula in the Newton's laws lede.

Also, just as a matter of established procedure, it is completely wrong to delete a citation of the eminent physicist Richard Feynmann in favour of an obscure professor who you have only mentioned in edit summaries and with no proper citation anyone can check. At the very least some consensus is needed before anything changes. SpinningSpark 20:51, 1 July 2008 (UTC)

Ummmm... I didn't delete it. I moved it to a more appropriate location. I completely disagree with you. You are mistaken that the article has been this way for a long time--it was changed on April 15, before which it said F=ma for a much longer time. F=ma is not "wrong". Most people coming to this article are not physicists. The first thing they see re the second law should be the familiar (an adjective backed up by references) F=ma. So, sorry, I have reverted you again. Rracecarr (talk) 01:18, 2 July 2008 (UTC)

It was only that way in April 2008 because you changed it before in February 2007. As it was subsequently reverted by other editors I am not the only one who thinks Newton's second law is F=dp/dt. The very first version of this article has a statement even closer to Newton's original. In any case, it is not really relevant what the article used ot say, what is important is what is right. All five of the references state the momentum version of the second law. Those that mention the acceleration version at all, do so afterwards. You really need to find references with more status than the ones given before you make this cahnge again. SpinningSpark 21:13, 2 July 2008 (UTC)

I think F=ma is supported by the existing references. If you insist, I can add more. What I think is important is that the intro be as accessible as possible without being wrong (and again, F=ma is not wrong), and that it jogs the memory of people who haven't thought about Newton's laws for 30 years. There is plenty of space in the article to elaborate on the more universal form of the law. Rracecarr (talk) 21:22, 2 July 2008 (UTC)

I have reverted again, with the addition of a new reference (the first Google hit for "Newton's second law" as it happens), since you requested it. Rracecarr (talk) 21:30, 2 July 2008 (UTC)

You cannot be serious, a high school tutorial web site is not a suitable source to establish this point. At least it does not trump the existing references which are all solid textbooks, plus a lecture by Richard Feynmann. They all say momentum, only some of them go on to mention F=ma and certainly Newton himself does not even mention acceleration, either in the original latin or Mottes translation. SpinningSpark 21:42, 2 July 2008 (UTC)

Oh, I'm perfectly serious. You must know perfectly well that there are plenty of references to support the statement that Newton's second law is F=ma, AND ALSO that Newton's second law is F=dp/dt. Anyone can hand-pick references. That is not the issue. First, you said "the article has been this way a long time". When I pointed out that it had been the other way for much longer before, suddenly "what it used to say in the article is irrelevant". You chastised me for deleting a reference which I didn't delete. You ask for references, when there are already plenty, and when I add one, you say that I "cannot be serious". You can keep pretending to invent new issues. But the real issue is which version of Law 2 is more appropriate in a brief introductory list of traditional statements of the laws. The first law in the intro just says "no force means constant velocity", which, at face value, is redundant--it's a special case of the second law. I don't see you beefing about that. Below, it's explained that formally, the first law defines inertial reference frames. This level of technicality would be inappropriate in the intro. Similarly, the third law just says "equal and opposite reaction"--no blabber about simultaneity and signal propagation between distant particles in special relativity. You keep changing the second law to a more technical, less recognizable form that is inconsistent with the rest of the list.

Here's an idea. Instead of an edit war, let's find an intro that makes both of us happy. It could be a little longer. The second law statement could be something like: A traditional statement of the second law is that the net force on an object is equal to its mass multiplied by its acceleration (F=ma). A more general form, allowing for changing mass, is that force is equal to the rate of change of momentum.

What do you think? Rracecarr (talk) 22:21, 2 July 2008 (UTC)

Here is an even simpler idea: Get rid of the list in the intro altogether. There is another list immediately below it, with more precise statements of the laws, each of which has a "this law is often stated as" sentence for forms taught at introductory levels. Rracecarr (talk) 22:29, 2 July 2008 (UTC)

I'm glad you now want to discuss rather than revert. Finding a mutually agreeable text is the best way forward. However, I cannot accept your proposal. All serious treatments of this subject give F=dp/dt as the second law. To say anything else requires solid references. I am not picking and choosing to suite my own opinions, that's what all the textbooks say. F=dp/dt is not a generalisation of F=ma to allow for changing mass. The second law applies to bodies, any situation in which mass is changing (the classic example is a rocket using up fuel) that has the second law simplistically applied to it will result in the wrong answer. The right answer is only obtained by a consideration of the momentum of all the masses involved, rocket, fuel exhaust and reaction mass. It is rather the other way round, F=dp/dt is Newton's second law; F=ma is an approximation to the second law in non-relativistic situations. It is in exactly the same category as so many other high school formulae, such as Work Done = Force x Distance. We all know that is really $W=\int _{C}\mathbf {F} \cdot \mathrm {d} \mathbf {l}$, but schools do not want to teach calculus. Hence there is a need to find non-calculus forms for the expression of velocity, acceleration, work and force. Just because such expressions are familiar from school does not make them the accurate definitions. Certainly articles should explain where the simplified forms come from but we should not allow that form to dominate the article. We are writing an encyclopedia here, not a school homework aid. SpinningSpark 23:32, 2 July 2008 (UTC)

I always wanted to discuss. To be honest, I also want to revert, but as I am sure you are aware I have reached 3RR. There are many things I disagree with you about, but for simplicity, let's take one at a time. Consistency: why do you not have similar problems with the statements of the first and third laws. As I've already pointed out, the third law isn't true relativistically either. What are "action" and "reaction" anyway? Why no mention of force, which is what the third law is really talking about? The reason is, of course, that that is the traditional statement of the law. Rracecarr (talk) 23:44, 2 July 2008 (UTC)

I have tried another option. I put a bit of thought into it, and I think it might satisfy both our concerns. Please don't knee-jerk revert. Rracecarr (talk) 01:31, 3 July 2008 (UTC)

Dispute over statement of the second law

Should the second law be stated as Force = mass x acceleration or Force = rate of change of momentum? SpinningSpark 21:48, 2 July 2008 (UTC)

Force = rate of change of momentum. The simplified version force = mass x acceleration is only true if mass is constant. The correct version and the simplified version and the linke between them can be easily explained even in an intro paragraph. Wikipedia should not perpetuate an erroneous over-simplification by only presenting the simplified version. Gandalf61 (talk) 13:48, 3 July 2008 (UTC)

I think both forms should be shown. Most people (at least in the general public) are more familiar with the F=ma form and would be confused if it was not shown. Most introductory calc-based physics texts introduce Newton's 2nd Law as F=ma first, and then a few chapters later generalize it to the time derivative of the momentum when momentum is introduced. I am in favor of giving both the more familiar form of each of the three laws along with the more general (and more correct) forms along with an explanation.PhySusie (talk) 15:52, 3 July 2008 (UTC)

I'm a third party and I agree completely with Susie. Adding both makes the page more usable to more people, such as those looking for supplemental material for their classes. AzureFury (talk) 19:51, 3 July 2008 (UTC)

Both forms need to be given, and there is nothing wrong with giving F=ma first. In classical mechanics, it is an absolutely correct and general statement because mass is always constant in classical mechanics. Even in special relativity (which should not be the primary focus of this article) the formula can be made to be correct by letting m stand for the relativistic mass, instead of the rest mass. Loom91 (talk) 11:23, 5 July 2008 (UTC)

I'm going to agree with everyone else thus far. It's always presented at F=ma even in the beginning of calculus physics texts and classes. Later on, it is explained in more detail, introducing the calculus based component. I'd say explain both. lesthaeghet (talk) 06:29, 7 July 2008 (UTC)

Use both but start with F=ma. In addition to the reasons stated above, the simpler version, F=ma, in my experience is far more commonly used and needed then the more advanced version which is necessary only for limited cases such as rockets. It (F=ma) is more easily understood by everyone even those who have no had calculus. Finally, and most important in my PoV, is fact that introductory articles should always spiral out from easiest and simplest to most complicated and most general. Only experts benefit from going the other direction.

TStein (talk) 20:44, 7 July 2008 (UTC)

I would express it as the rate of change in momentum and as a = F/m. Writing it as F=(m)(a) implies that force is caused by an acceleration and mass interaction, rather than acceleration coming about because of an interaction between a net force and mass. LonelyBeacon (talk) 05:59, 9 July 2008 (UTC)

I agree with LonlyBeacon to an extent. Implies may be too strong of a word, though; maybe F=ma hints that the force is caused by the acceleration is better. Consider V=IR, one might either argue that I=V/R is better since the current is caused by a voltage. Alternatively one might argue that it should be R=V/I since this is a definition of R. In the end, though, because one can algebraically manipulate the equation to many different equivalent forms so the question is probably moot. F=ma has the advantage of working typesetting better than a=F/m and is slightly easier to algebraically manipulate.

TStein (talk) 03:33, 10 July 2008 (UTC)

From a basic standpoint, the layperson is much more likely to recognize or "get" a description of F=ma because it's simplified and readily applicable to most run-of-the-mill calculations. Shifting into explanations of the concept regarding non-constant masses dips into an area more esoteric but certainly more correct. The best course is to give both. The clearest general form would be, in my estimation: 1)translation of law text 2)simplified concept for stable mass is F=ma 3)wider concept involves momentum without the assumption of constant mass. Keeping the simplification and using it first allows one to wade into the waters instead of diving into the deep end. — Scientizzle 15:20, 11 July 2008 (UTC)

Exactly. F=ma is not only much simpler to understand it is by far the most common use case. If we wanted to be absolutely technically correct then we would have to use the relativistic equivalent of the Ehrenfest Theorem, anyway. Why should we settle for only an approximation like dp/dt=F? TStein (talk) 18:14, 11 July 2008 (UTC)

I suggest stating it as rate of change of momentum, but then mentioning the F=ma common form. As it happens, that's pretty much what the article currently does. As an aside, there's no need for 5 references for it though. Modest Genius ^talk 17:21, 12 July 2008 (UTC)

I agree with the majority of posts regarding the topic. Both points are equally valid. Most people researching entry-level Newtonian physics will more readily recognize F=ma, especially as it is readily manipulated using algebra. Also, this is applicable as a "stepping-stone" to those learning about gravity/acceleration equivalence within relativistic physics. However, the more appropriate definition (in my opinion) details the rate of change of momentum since it does not imply constant mass. In short, both are useful and I think both explanations should be presented on the page. That's just my 2kB. Archon Magnus^{(Talk | Home)} 19:42, 16 July 2008 (UTC)

Again, both points are valid, but I agree with the majority of posts, which suggest stating first F=ma; there is also an official guideline about this, i.e. Wikipedia:Make technical articles accessible. I also agree with User:Modest Genius about the excess of references. --Blaisorblade (talk) 00:44, 17 July 2008 (UTC)

Agreed. Start with F=ma, then explain the general form later. This article must be understandable for somebody who does not know physics at all. --Apoc2400 (talk) 11:56, 18 July 2008 (UTC)

I'm not sure if this aspect was overlooked but,

• momentum (p)= m/v;
• a= d(v)/dt (rate of change of velocity),
• and so rate of change of momentum= d(p)/dt=m*d(v)/dt (since mass is constant)

So, I think both the versions are simply corollaries of the same equation. In response to Gandalf61's point, in Newtonian (classical) physics, the mass always stays constant, right? And, for any law, even though when we tend to neglect it, but a law is always to be stated with full conditions. So, the expanded form of Newton's second law would be "With mass remaining constant, force applied on a body equals a product of mass and acceleration produced", or "With mass remaining constant, a constant force equals the rate of change of momentum of the body". —KetanPanchal^taLK 07:07, 19 July 2008 (UTC)

No - mass is not always constant in classical physics - think of rockets for example. The time rate in change of momentum is the more general equation, F=ma is true for the particular (though very common) situation of constant mass. PhySusie (talk) 12:00, 19 July 2008 (UTC)

Actually PhySusie, it's not conceptually sound to treat rockets as variable mass systems. In classical physics, mass is constant without exception. This has been discussed before. Newton's laws are about particles. In order to prove them for extended bodies, the assumption of constant composition is essential, which implies constant mass. The correct way to treat rockets is using conservation of linear momentum. Loom91 (talk) 21:47, 19 July 2008 (UTC)

I'm pretty sure that classical physics includes all of classical mechanics, including the topics of variable mass systems, extended mass systems and linear momentum. Classical physics is more than Newton's Laws. I agree that F=ma is for constant mass and treats the object as a particle. However, my point was that F=dp/dt is more general, and not necessarily (though it often is) the same equation as F=ma. PhySusie (talk) 12:28, 20 July 2008 (UTC)

"In classical physics, mass is constant without exception. This has been discussed before. Newton's Laws are about particles," is a completely ludicrous statement!! Newton never said that his Laws of Motion were about particle masses, and furthermore, in practice they are not considered to be or treated as particle masses. Newton's Laws of Motion are used all the time for doing calculations about aircraft, automobiles, spacecraft, artillery shells (which for over a century) aren't even spherical. If you wish to get disagreeable about approximations, then I reply that all of Newtonian mechanics is an approximation, anyway, though a very good one in everyday use. And then, the Special Theory of Relativity is an approximation to the General Theory....98.67.166.234 (talk) 05:01, 12 November 2009 (UTC)

One of my original reasons for starting this RFC was that Newton himself states the second law in terms of momentum and does not reference acceleration at all. Admittedly he does not use the differential form of the equation, but most likely because calculus would have been treated with suspicion by his contemporaries. Does anyone believe that if a law is named after Newton it should be true to what he actually said?

SpinningSpark 16:02, 20 July 2008 (UTC)

According to that logic, we should not use "momentum" at all, as that would be putting words in Newton's mouth. Instead we should describe it as "quantity of motion". Rracecarr (talk) 21:47, 20 July 2008 (UTC)

Of course that is not what I meant, the article is on the English Wikipedia and should be written in modern english. Actually, Newton did not use the word "motion" either, he used the Latin word motus. He does however give a very clear definition of motus, there can be no doubt that "momentum" is meant. SpinningSpark 06:42, 21 July 2008 (UTC)

That clear definition is nowhere to be found in this article. I did find it in inertia though. The Latin and the translations in this article only serves to confuse the issue, in my opinion. In any case, it is a moot point. As I state below, few if any of the named laws of physics correspond to their original form. How we teach physics should depend on sound pedagogy and not on history. Whatever way that Newton understood his laws is irrelevant to how they are understood today. TStein (talk) 22:16, 25 July 2008 (UTC)

The present form of the article is fine. It puts the momentum formulation first, which is the accurate version both historically and in practice. It then immediately discusses its reduction to F=ma, so no reader can miss that expression of the law. Momentum is arguably easier to grasp than the acceleration approach, being rooted more strongly in everyday experience. (We bump into things every day, and see what happens. We play pool, hockey, croquet. Everyday experience with momentum shaped the context Newton faced in presenting the laws.) The preference for F=ma may be just an outgrowth of learning from textbooks that use F=ma. Brews ohare (talk) 15:29, 21 July 2008 (UTC)

Both forms should be shown. It may be preferable to show F = ma first because it is better known, at least at an introductory level. In my opinion, it doesn't matter exactly how Newton defined the law, except for the parts of the article that are dealing explicitly with history. It is perfectly normal that laws and equations get redefined, new notation introduced, etc., while keeping the original name. So, what matters is, what is the most common definition today? (I'm not sure.) I disagree with using a = F/m, because, although obviously equivalent, it's not the "iconic" version of the equation. --Itub (talk) 15:45, 23 July 2008 (UTC)

I agree strongly that it doesn't matter how Newton defined the Law. There are many examples of laws associated with people that are different than the author originally wrote them. Maxwell's wrote down his equations completely different then those associated with him. Faraday expressed his law completely in terms of words. (It was Maxwell who translated Faraday's ideas into mathematics and he did so in terms of the vector potential.) Looking at all the laws that are named after people, the law that is stated the same as it was first stated is the exception (if one even exists) rather than the rule. What matters is the most common notation that is in use today with the article organized from the most simple to understand to the most general and most complicated to understand. Science is not frozen in time and Newton's laws are not limited to Newton. Both must change to fit the times or die. TStein (talk) 21:20, 25 July 2008 (UTC)

You should show both, but clearly emphasize that the law is that force exerted is equal to the time rate of change of momentum. That force is also mass times acceleration is the more familiar form, but it is only a special case of the law. RayAYang (talk) 03:20, 30 July 2008 (UTC)

Someone needs to concede and to state that this article is about Newton's Laws of Motion, as upgraded and improved by others since him, with much-improved mathematical notation and technical vocabulary -- indeed, vocabulary in a completely-different language (not Latin) -- but nevertheless, still Newton's Laws, because he stated the gist and crux of them first, and first applied them, and that we honor the memory of Newton by keeping his name on them, and by writing them like this, "Newton's Second Law of Motion", with all of the capital letters, and none of the laziness and slipshop writing that so often appears in this decade.98.67.166.234 (talk) 05:01, 12 November 2009 (UTC)

Why is the mass inside the differentiation operator: F=d(mv)/dt rather than F=m*dv/dt=ma? It is unnecessary and only results in a less familiar, more convoluted form of the equation.

Can someone give me an example of a variable mass problem in classical mechanics that requires the d(mv)/dt form of Newton's law in order to work properly? The rocket equation is not a valid example, contrary to popular myth, for reasons I can show if asked. I contend that the form F=ma is not a "special case" of Newton's Law. It is valid even for variable mass problems.

On the other hand, the F=d(mv)/dt form shown in this article yields incorrect results if you account for the time-varying mass in the only way I conceive it could be accounted for in this form. F=d(mv)/dt plus the chain rule results in F=m*dv/dt+v*dm/dt. This implies that a melting ice cube under zero net force will undergo acceleration due to its changing mass. An ice cube of m = 1 kg traveling at v = 1 m/s, but melting at a rate of dm/dt = -0.1 kg/s, would be required to accelerate at dv/dt = 0.1 m/s in order to satisfy the equation under zero net force. This is clearly a false result, since a melting ice cube doesn't accelerate under zero force.

So why does it makes sense to state this form of Newton's law in an article about classical mechanics? It seems to me the correct form for classical mechanics is the vector equation F=ma. MarcusMaximus (talk) 01:53, 15 August 2008 (UTC)

MY ACHING BACK! Cannot anyone read further back in the comments and see the answers to questions, like this one, that have already been answered?98.67.166.234 (talk) 05:51, 12 November 2009 (UTC)

Your calculations above are not correct because they are not written in vector form at all. Assume that the ice cube really is cubical, and it is in a standard atmosphere like most people breathe every day. If the ice cube is floating, motionless, in a zero-gravity environment, it will melt on all six faces evenly, giving six (equal magnitude) forces perpendicular to the faces in six different perpendicular directions. You draw the picture, but I can tell you that those six forces all sum to zero, and the ice cube does not move at all.

On the other hand, if the ice cube is resting on on a flat surface (e.g. a table), at the surface of the Earth, under standard gravity, the melting ice cube will, as a whole, accelerate in the downwards direction as it melts. (Otherwise, the water would just float in thin air.)

Please do not make up spurious examples that are based on one-dimentional equations like "F = m*dv/dt + v*dm/dt" in what are actually three-dimensional (or even two-dimensional) situations. You also need to watch (again?) Star Trek II: The Wrath of Khan, where Mr. Spock tells Captain Kirk that Khan's maneuvers in the starship Reliant given evidence of his two-dimensional thinking.98.67.166.234 (talk) 05:51, 12 November 2009 (UTC)

The form F=d(mv)/dt, which is equivalent to F=m*dv/dt+v*dm/dt, implies that an object with nonzero velocity, influenced by a nonzero external force, can either accelerate with constant mass (dv/dt is nonzero and dm/dt is zero) or experience a change in mass without accelerating (dm/dt is nonzero and dv/dt is zero), to satisfy the equation. I don't think this is a physically valid result. MarcusMaximus (talk) 02:09, 15 August 2008 (UTC)

What more, out of five references cited as the source of F=d(mv)/dt (refs 3-7), all of them state that F=ma without deriving it from F=d(mv)/dt. Only one of them mentions that F=d(mv)/dt, from which it forms F=ma, and then it proceeds to use F=ma for the rest of the chapter. The Feynmann lectures (ref 3) states: "The product ma ... is the rate of change of momentum, hence the rate of change of the momentum of a body is equal to the resultant force ... acting upon it and is proportional to the resultant force ..."

I restate that I do not see any good reason to state the law in terms of d(mv)/dt in this article. MarcusMaximus (talk) 17:28, 15 August 2008 (UTC)

Stating, "The form F = d(mv)/dt, which is equivalent to F = m*dv/dt + v*dm/dt",

is only valid in one-dimensional problems.

In the case of the vector equation, F = d(mv)/dt, everything gets a lot more complicated than this, because the time derivative has to be worked out in all of the dimensions of the coordinate system that you choose to do the problem in. Just take my word for it that it is very tough, and if you want to know more, ask an aerospace engineer, a mechanical engineer, or a physicist. (I'm an electical engineer, and haven't gone in that direction.) For those who are unfamiliar with the notation, writing the symbols for quantities in bold type announces that they are vector quantities. Also, vector quantities can be written in a lot of different ways in different coordinate systems.98.67.166.234 (talk) 05:51, 12 November 2009 (UTC)