# Is speed an intensive property?

I remember being taught in elementary physics that while it makes sense to add volumes, masses, or heat, it makes no sense to add temperatures.

As I wanted to use that to illustate some other issue, I checked it on wikipedia, and discovered the concept of intensive and extensive properties of materials and systems, which is apparently a century old, though I do not remember it being taught to me (it was not as old then :-).

The concept seems to be particularly useful in material and systems thermodynamics. But I was wondering how far it extends, and I was keeping in mind my original problem of determining when adding values in a given unit (not necessarily a physical one) made sense or did not.

I noticed the fact that the ratio of two extensive variables is intensive. So I started looking, a bit ramdomly I confess, at ratios, and the first that came to mind was speed. And I wondered whether adding speeds made sense.

We do that all the time, so it should. But then, my initial problem was not about adding two quantities, but a long list of them (it was a database question). While adding a few speeds (or velocities) makes sense when I move in the bus, or analyse the motion of the Moon in the solar system, I cannot imagine it would ever make sense to add a list of a hundred or a thousand speeds (though computing the average would make sense, as it would for temperature). But I makes perfect sense to sum the masses of a thousand objects.

So there is apparently something special, not quite right, about adding speeds. Of course, I know that slightly older results state that adding speeds is not done with simple addition, but I feel that my problem is elsewhere (though there may possibly be a connection).

I am aware that temperature in some materials is related to speed of motions inside it, so I am not too surprised. But speed in general seems to go beyond that (sorry for the vagueness). Also speed is not listed by Wikipedia as an intensive property.

So, my question is : why does it seem improper to add many speeds (or velocities)?

I guess this must also be true of other physical quantities, and I am wondering what is the right way to look at this, and understand it. Is there a more general notion than intensive and extensive?

• I think intensive vs. extensive only makes any sense for true scalar quantities, while speed is (the magnitude of a velocity-) vector. So it's neither intensive nor extensive. – leftaroundabout Aug 14 '14 at 15:43
• Thermodynamics is strictly only valid for systems in equilibrium and it keeps holding for systems very near equilibrium and systems where multiple parts in equilibrium are being brought into contact with each other in suitable ways. It is not valid for any kind of dynamic system in which velocity distributions are as diverse as e.g. that of two masses at different velocities. For that one needs to find either suitable ad-hoc solutions or attempt a non-equilibrium solution. The best answer to your question is that thermodynamics is not the right tool for your kind of problem. – CuriousOne Aug 14 '14 at 18:39
• @CuriousOne Very useful remark - Thanks. I was actually forced to come to that understanding, but it is not stated in so many words in the wikipedia page, which was so far my primary source of information, as I explained. I think I actually managed to give meaning to the idea of velocity as an intensive variable, by considering motion of fluid systems that get merged (see my answer). I am not trying to solve a problem with thermodynamics, but only to understand the concepts of intensive/extensive properties, possibly to reuse them in very different contexts. – babou Aug 14 '14 at 21:51
• @leftaroundabout Wikipedia does not seems to agree with you. Magnetisation is given as an example of intensive property, and it is a vector field. – babou Aug 14 '14 at 21:52
• I developed some arguments in my own answer, and this was backed, differently, by Nathaniel's answer. I never considered mixing different velocities in the same system, where the intensive character may be concerned. I rather meant that differences of velocity imply that we have different systems.. I wonder whether you still stand by your comment. – babou Sep 19 '14 at 20:41

why does it seem improper to add many speeds (or velocities)?

Adding speeds is ofttimes inappropriate even in Newtonian mechanics. Suppose Mark is moving 3 m/s eastward with respect to Bob, and John is moving 3 m/s westward with respect to Mark. The relative velocity between Bob and John is zero rather than the 6 m/s suggested by adding speeds. You can add velocities however in this case.

Velocity however is neither an intensive nor extensive property. Consider a big brick moving at $\vec v_b$ and a grain of sand moving at $\vec v_s$. What is the velocity of the brick+grain system? Typically one would use the velocity of the center of mass, and that's $\frac {m_b \vec v_b + m_g \vec v_g}{m_b + m_g}$. That expression doesn't correspond to either the concept of an intensive property or an extensive property.

Momentum rather than velocity is additive, at least in Newtonian mechanics. The momentum of the brick+grain system is the vector sum of the momentum of the brick and the momentum of the grain of sand. Thus momentum (but not velocity or speed) is an extensive property in the context of Newtonian mechanics.

A qualification is needed here: Momentum is additive in Newtonian mechanics if those momenta are all referenced with respect to a common frame of reference. The momentum of the brick with respect to some observer and the momentum of the grain of sand with respect to the brick don't add up.

I've used "Newtonian mechanics" a number of times. The Newtonian concept of momentum as the product of mass and velocity fails when things are moving very fast. In the context of special relativity, it's relativistic momentum rather than Newtonian momentum that is additive (and once again, the concept of a common frame of reference very much applies).

In the context of general relativity, even adding relativistic momenta doesn't quite make sense. Asking about the total momentum of the Andromeda galaxy and an extremely remote galaxy is a bit nonsensical in the context of general relativity.

• This was quite helpful. Thanks. But I do not see why the formula for center of mass velocity should preclude velocity being an intensive property, as I explain in my answer. I would like to see what happens with relativity, but it may not be a major issue. The merging of two gas clouds at relativistic speed must be a rare occurrence, if possible at all. – babou Aug 14 '14 at 21:19
• I take back my last sentence. Supernova expel steller material at 10% of the speed of light, which is considered relativistic speed (it give about a 1% difference with newtonian mechanics). Now can such clouds merge with other clouds on the way and reach some equilibrium, sharing momentum. I do not know. – babou Aug 14 '14 at 22:16
• @babou: Velocity is not additive for variable mass systems. Suppose a 2 kg object moving at 100 km/s with regard to the Sun plows right into the Sun. From the perspective of an observer initially at rest wrt the Sun, the Sun's velocity does not change by 100 km/s. It changes by about $10^{-28}$ km/s. Momentum is additive, but velocity is not. – David Hammen Aug 14 '14 at 22:19
• What did I write that makes you think I might not agree with that? I am defending the idea that velocity is intensive (provided equilibrium of merged systems). That precludes additivity, since additivity is a characteristic of extensive properties. Where is our misunderstanding? – babou Aug 14 '14 at 22:52

Your question, as of right now, seems confused to me. An extensive property of a system is one that scales with the system size. An intensive property is independent of the system size. For example, consider a system $A_1$ with $N$ particles in a volume $V$, with density $\rho=\frac{N}{V}$. Now, we consider two of these systems separately, $A_1$ and $A_2$, and call the two of them together $B$. Then, $B$ has a total number of $2N$ particles, within a volume $2V$, with density $\rho=\frac{2N}{2V}=\frac{N}{V}$.

As we see, 'doubling the system' gives us twice the number of particles and twice the volume, but the same density. This shows that particle number & volume are extensive quantities, and density is an intensive quantity. Now, think of a particle in a box bouncing around at velocity $v$. Consider two such systems, and you will have two boxes with a particle, each with velocity $v$. Clearly, the velocity of each particle is unaffected by our silly thought experiment, so it is clearly not an extensive property.

Can we now say that the particle velocity is an intensive property? Not really. The notions of intensive- and extensiveness arise in thermodynamics, where they are useful in grouping certain properties of systems together based on whether or not they scale with system size. In thermodynamics, one usually considers systems which are a (large) collection of individual particles, so that one can discuss things like temperature and other notions that don't really make sense on a microscopic level.

If one considers a large collection of particles, then the velocity of a single particle really isn't a property of the system. The whole point of thermodynamics is to understand the behavior of large systems without discussing the details of what's going on with each single constituent. Therefore, the velocity of a single particle is not usually considered a property of the system to begin with, so it cannot be either intensive or extensive. I hope this clears things up a little.

However, in other contexts it of course makes perfect sense to add velocities, e.g. in Newtonian mechanics or special relativity theory.

• But I never said anywhere that I want to consider the speed of individual particles. I only made the informal, possibly unwarranted, remark that if temperature, which may correspond to particle speed, is intensive, then it may not be a surprise if velocity is too. I never intended to consider individual particles, though I would consider merging gas clouds. If the system can reach a form of statistical momentum equilibrium, this should correspond to an intensive velocity value for the system. That is what I try to explain in my answer. I do agree I was confused, and that is why I was asking. – babou Aug 14 '14 at 21:11
• @babou temperature really corresponds to average kinetic energy per particle - which is an intensive quantity. Does that answer what you are asking? – Danu Aug 14 '14 at 21:12
• Thank you, I did not remember that. As I said, it was more an informal remark. What bothered me was really the non additivity of speed, in the sens that you do not add speed of parts to get the speed of the whole. With what little I understood or read, and various remarks I had here, I did convince myself that speed/velocity is an intensive property, when it make sense at all. Hopefully I may convince you or others. I know relative speeds can be added. But how often do you have many relative speed to be all added? It is not addition in the sense of adding extensive values. – babou Aug 14 '14 at 21:40
• @babou when considering a system, and then duplicating it, the velocity of each of the particles remains unchanged, but I still think this does not mean that speed is an intensive quantity, as I explained in my answer. It really boils down to a matter of opinion on this. We can all agree that doubling the system does not change the velocities, and that is really all that should be said about it. – Danu Aug 14 '14 at 21:42
• Why do you keep coming back to these particles. I thought the whole purpose of the concepts we are duscussing was precisely to be able to ignore them. At the level of particles, nothing is homogeneous, and everything falls to pieces (literally so). What is density or pressure at the particle level? (not asking how it relates to particles .. it just ceases to exist as such). If it is only a matter of opinion, then it is not longer science. If you use the concept to do science, it must have a definition. And whether a property fits that definition is a matter of science, not of opinion. – babou Aug 14 '14 at 22:05

Taking answers and comments into account, my own current conclusion is that velocity is an intensive property, provided the system considered is homogeneous, at least with respect to speed. Like other intensive properties, this may depend on scale, and cease to have meaning at molecular level.

I did not intend to write an answer to my own question but ... writing the question is usually a good way to better understand the issue and to find the answer. I could have added this to the question, as there are remaining issues at the end. But the question is already long, and this is long too. Hence it seemed better to do it this way.

I initially looked at speed as a ratio because I know it as the ratio of a distance to a time. But that was much without thinking, and is the wrong way to look at it, especially since I do not know too much what distance and time may be. I actually stayed with speed, or rather velocity, when I realized I would not see the meaning of adding many speeds, which was a variant of my original database problem.

But then, I realized that velocity is a ratio in a different way: $\overrightarrow{V}=\overrightarrow{P}/M$, i.e., velocity is the ratio of the momentum (vector) by the mass.

And then, David Hammen's answer timely reminded me that momentum is additive in Newtonian mechanics, when considering the motion of several bodies in the same frame, and also that velocity is not ... which confirms that, if anything, it is not extensive.

Indeed, if I take the answer of Nathaniel to another question, as well as the definition of the IUPAC Green Book, recalled in the wikipedia page, additivity is what characterizes an extensive property. I must then conclude that (linear) momentum is an extensive property, and velocity is not.

The same reasonning and conclusion applies to mass (this seems generally agreed upon :-).

Since velocity is the ratio of the momentum vector by the mass, it ensues that velocity must be an intensive property of a system.

This justifies that velocity (or speed) is not additive, in the sense that you cannot make a large system from smaller parts, and add the velocities of the parts to get the velocity of the whole.

But this conclusion is in contradiction with David Hammen's view that velocity is not intensive, though he does not really say why, and I do not see why his formula for the velocity of the center of mass should be a problem.

However, I think that a requirement (stated only in examples, but not in definitions in the documents I have) is that intensive properties are meaningful only in homogeneous systems in equilibrium (which was just confirmed by the comment of CuriousOne. With regard to velocity, that would mean that it can apply only when the system considered is velocity-homogeneous. or somehow merge into a single homogenous system. Maybe it could apply to the collision of gas clouds in space, or of water blobs in zero gravity (in a space ship). There are probably other examples of fluids that can merge and balance momentum to reach equilibrium, without breaking apart.

That should make velocity an intensive property.

Velocities may be added when considering relative velocities of different systems with respect to each other. But that is a completely different situation, and it has nothing to do with merging several systems into a single system, and properly "combining" their velocities. It is unlikely to lead anyone to add together a large number of velocities, as one might add many momenta when having a large number of bodies merged into a single system.

In the above discussion I have been considering blobs or clouds of fluid as exemple, as that seem necessary to be able to talk of merging two systems. However, there is no need for such restriction when just considering a single system, say a stone or a cube of iron. If velocity as an intensive property makes sense at all, it does make sense for any velocity-homogeneous, and that certainly applies to a solid.

When considering relativistic velocities, there is no reason the question should lose its meaning. A supernova can expel clouds of material at 10% of the speed of light, which may be considered a relativistic speed as it corresponds to a 1% discrepency between Newtonian and relativistic mechanics. I guess that such a cloud, or part of it, could hit and other cloud and merge with it into an ultimately homogeneous cloud (but I have zero expertise on such phenomena). The penultimate paragraph of David Hammen's answer, as well as the comment by Void, seem to indicate that things still work at relativistic speed, provided the right properties are used (such as relativistic momentum according to David Hammen, or four-momentum and four-velocity according to Void).).

Finally, a supported comment by leftaroundabout suggests that "intensive vs. extensive only makes any sense for true scalar quantities", but without further justification. I am wondering why that should be. Besides, this seems in disagreement with the Wikipedia page that lists magnetization as an intensive property. But magnetization is a vector field.

Similarly, I do understand Danu's answer that the concepts of intensive and extensive properties were developed as useful tools for studying thermodynamics (I did read a bit before asking my question). But according to the wikipedia page, they are already used in other contexts, and I do not see why that should exclude considering velocity, when statistically as homogeneous, as I explained above.

• Yes, momentum is extensive, velocity is intensive. But remember it can only make sense after the addition of the two systems reaches equilibrium. The velocity will be in the end the result of the center-of-mass formula. Relativity is straight-forward as long as you use quantities such as rest mass, four-momentum and four-velocity. – Void Aug 14 '14 at 22:15
• @Void Thank for your comment. Regarding equilibrium, I thaought I was saying precisely that in my longest paragraph, just before I state: That should make velocity an intensive property. ... What else is wrong since you approve in a comment without voting? I mean, everyone has been so far trying to convince me that velocity is not an intensive property. – babou Aug 14 '14 at 22:26
• That's pretty convincing to me, now I read it. Perhaps what I meant to say with my original comment should better have been put "the notion of extensive-ness doesn't make sense for vector quantities", but momentum is evidently extensive, and then there is no good argument why velocity wouldn't be intensive. – leftaroundabout Aug 15 '14 at 1:25

Here's a point of view from thermodynamics that might be useful.

Typically, the intensive quantities (in the form they're usually defined) arise as derivatives of the total (internal) energy $U$ by some particular extensive quantity. Thus:

• Temperature $T=\frac{\partial U}{\partial S}$, the derivative with respect to the entropy

• Pressure $P=-\frac{\partial U}{\partial V}$, the derivative with respect to the volume (with the opposite sign convention to the other quantities, but this isn't important)

• Chemical potential $\mu_i=\frac{\partial U}{\partial N_i}$, the derivative with respect to particle number of a given species

• Electrical potential $\phi=\frac{\partial U}{\partial Q}$, the derivative with respect to charge

and so on. These can be summarised as the "fundamental equation of thermodynamics:

$$dU = TdS - PdV + \sum_i \mu_i dN_i + \dots$$

Now, the question is, can we add a term along these lines that involves speed, or rather velocity? The answer is yes, because

$$v_x = \frac{\partial U}{\partial p_x},$$

where $v_x$ is component of a particle's velocity in the $x$ direction, and $p_x$ the $x$ component of its momentum. Similar equations apply for the $y$ and $z$ directions of course.

The above equation is quite general, and applies in relativistic situations. However, if we restrict ourselves to a simple Newtonian particle in one dimension, for which $U = V(x) + \frac{1}{2}mv^2$ (where $V(x)$ is the potential energy) then it's easy to see that it's true in this special case, since we may rewrite the energy as $V(x) + \frac{1}{2m}p^2$.

Therefore, if the situation warrants it, we can add a $vdp$ term to the fundamental equation of thermodynamics, and thus velocity (in some direction) may be seen as the intensive quantity corresponding to the momentum in the same direction, which is extensive. (Note that it does make sense to add, rather than average, the momenta of thousands of particles.)

In fact this is often done when applying non-equilibrium thermodynamics to fluid mechanics, and to atmospheric dynamics in particular. The difference in horizontal velocities between the prevailing wind and the stationary Earth surface may be seen as the driving force that causes a flux of (horizontal) momentum downwards from the atmosphere into the Earth, leading to a tiny amount of frictional heating. This is exactly analogous to the way in which a difference in temperatures drives a flow of heat, or a difference in electric potentials drives a flow of electrons.

• Thanks. First answer that seems based on a clear definition, rather than hand waving. But I have a consistency problem: it leave open my complementary question. Isn't it the case that $V=\frac{\partial U}{\partial P}$? If that is correct, then volume should be an intensive property too. Sorry if I am completely off the mark. - - - - - - - - - I never thought of flux of momentum caused by a difference of potential. Nice. - - - - (+1, though I am rather incompetent to have an informed assessment of the answer). – babou Sep 18 '14 at 22:23

This is a very interesting question. Velocity can be considered as either an intensive or an extensive property, depending on whether we are inquiring about the parts of a single system, or considering relations among separate systems.

Velocity must be an intensive property, for consider: If I and my passenger and my books are traveling in my car, and if my car has the property of moving at 60 mph relative to an observer resting at the side of the road, then not only do I share in that property, but so does my passenger, every single one of my books, and every elementary particle in that car. Hence, the velocity of the car is an intensive property with respect to all of the parts of my car.

The velocity of the car will not change if I throw one of my books out the window, although its momentum will change- especially if it happens to be a hefty book, such as J. A. Wheeler's Gravitation. Whereas velocity is intensive, momentum is extensive.

The initial question seems to address the problem of adding the velocities of separate systems (of passengers, books, particles, etc.) Suppose that I toss my book out the window of my car (doing 60 mph), and it happens to land in the bed of a truck speeding past me in the same direction at 40 mph. Now my book is going 100 mph with respect to the rest observer. If my hapless book then bounces out of the bed of the truck, and lands on a train going 180 mph (with respect to the rest observer) in the opposite direction, then my book is traveling -240 mph relative to me and my passenger.

We can watch my book bounce among distinct moving systems moving at different velocities (along the same dimension with respect to the rest observer) all day long. So velocity is an extensive property with respect to separate systems.

Now here's another interesting question: how does the velocity of light serve to limit the addition of speeds of separate systems?

• We agree on the first part. But if I can rephrase the last party about the wandering book, temperature is an intensive property: given a system of ice, with a lobster on top, if I separate the lobster, both parts of my system keep their temperature. However, if I throw the lobster in boiling water (the passing truck), it will cook and become red, because it changes its temperature. I have added to its temperature the temperature difference between the ice and the boiling water. But I am not sure this has anything to do with the issue. Is this set-up a system at equilibrium? – babou Sep 18 '14 at 21:48