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A few days ago I had a discussion with a friend who wants to become a physics teacher (in Germany). He told me that from a pedagogical/didactial point of view it seems to be a good idea to introduce momentum at a very early stage in school but without mentioning any formulas, I didn't really understand it, in particular this leads me to the following question:

Suppose you know that momentum is "something" which can be transferred between bodies, that the total amount of that something is conserved and that a momentum of a body is greater if it is heavier or if it is faster, but you don't even know that $p = mv$.

Can one know anything new about nature or solve any problems in every day life, if one has such a vague, only qualitative notion of the concept of momentum in mind?

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every day life : Marbles, Pool, real Billiard, wreckage ball, bumper cars, car accidents, firing a gun, bullet hit.... – Georg Oct 25 '11 at 11:49
@Georg But can you solve any interesting problems involving those examples if you have only such a qualitative knowledge? – martin Oct 25 '11 at 11:52
@martin if they can't still understand formulas, then a qualitative but 'science compatible' could be better than nothing, moreover if you for example decide to not introduce momentum, or whatever, then gravity, energy, and even quantum physics would be introduced the same by friends, internet or television, without formulas (but perhaps without sense neither) – HDE Oct 25 '11 at 12:20

Yes, of course. What you said implies, even without the $p=mv$ formula, that the momentum for a single object depends on its mass and velocity. Because you have said that the momentum is conserved, it guarantees that the speed won't change (assuming we know that the mass is conserved as well). It's called Newton's first law and it follows just from the qualitative comment about the momentum and momentum conservation.

You may also use the "very limited" description of the conserved momentum to say things about collisions etc. When you shoot into a big ball, the ball will get moving. It's a qualitative conclusion but I can deduce it from your limited information simply because the initial momentum was nonzero, so the final one must also be nonzero.

I don't know what's the point of hiding $p=mv$ but even without this formula, the momentum conservation law is, much like any conservation law, nontrivial. According to Noether's theorem (which is probably not understood by any person from the set of those who don't know $p=mv$, but let me mention it, anyway), the existence of the momentum conservation law is inseparably equivalent to a symmetry of the laws of Nature, namely the translational invariance. And this is a very nontrivial insight, indeed.

In fact, Noether's theorem doesn't even give you any "explicit" formula such as $p=mv$ in the most general case: one needs to know some specific things about mechanics to know or derive this formula. So it's helpful to know at least one simplest example such as $p=mv$, but it is not true that the momentum must always be something like $p=mv$. The momentum density (the stress-energy tensor) of the electromagnetic field looks very differently than $p=mv$.

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I don't see many cases for collisions where such a "very limited" knowledge about momentum gives any insights. Even if you have the formula, in most cases you need to consider the energy as well to get predictions. – martin Oct 25 '11 at 13:57
I'm curious if your discussion depends on a form of momentum any more specific than $p=f(m,v)$. If I could introduce any form for this function, then that would result in very bizarre physical behavior. Should we not at least require it be monotonic regarding velocity? Do we really want to allow collision physics to be a mulch-variable problem with multiple solutions? – Alan Rominger Oct 25 '11 at 21:08

I am a student teacher in Canada (yes, I have gone back for my teacher certification at age 55) and we have exactly the same philosophy here. I find it unbearable! Someone decided that the reason students have "trouble" with physics is that they don't have a solid understanding of the basic concepts before they start doing formulas. The problem with this theory is that the best way to develop that "solid understanding is to work through caluculations using the formulas! It's true that a bad teacher can do a hundred pointless formula calculations without explaining what is going on. But on account of that bad teacher, why should I have my hands tied behind my back? That's what it feels like trying to explain momentum without being able to write down p=mv.

The worst thing about this "philosophy" (I think its leading proponent is someone called Arons) is that we aren't even supposed to start with the classic examples of billiard balls. We don't even talk about the center-of-mass reference frame. We are supposed to make it "meaningful" to the students by talking about examples they can "relate" to, like jumping on a trampoline or catching a football. It's just unbearable.

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+1 Not sure I agree with (or maybe just "follow") your point, but good for you, going into teaching. – Mike Dunlavey Oct 26 '11 at 15:56

It is very useful to introduce momentum without formulas, especially ones that involve mass, as these can be a bit misleading. When my teacher said photons carry momentum, I said "wait a minute! Didn't we define $\vec{p}=m\vec{v}$" Photons have no ?? It was only later that I appreciated the fact that momentum, Energy, space and time are fundamental, while mass is not.

Even from a pedagogical point of view, I find it useful to introduce abstract thinking at a very early age. I think students can handle these concepts and it does make the discourse more exciting.

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In addition to what Lubos said, which is the main point, there is also the intuition for stresses that comes with an appreciation of conservation laws.

If you understand that momentum flows locally in a chain, you can see why the weakest link breaks first. You can get a qualitative insight into why thicker things are stronger--- they have a bigger cross section over which to distribute momentum flow. You will also never be confused by action/reaction pairs.

Feynman, in the lectures, introduces momentum simultaneously with Newton's laws. This is pedagogically correct, and the law can be appreciated even by children under 10 years of age.

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Kids intuitively understand weight and speed, and you don't need algebra just to multiply them. After all, they've all watched contact sports.

What may be a new concept to them is the idea that you can identify a set of objects and talk about the total weight and speed of the set.

So, for example, you could have two billiard balls at opposite ends of a trough, and roll them together so they bounce apart. Each one has momentum that changes direction in the collision, but the momentum of the aggregate remains zero throughout.

Similarly if one is stationary and the other one hits it, the center of mass of the pair continues moving unchanged by the collision.

That's how I would approach teaching momentum, by collisions. Whether to write down formulas is another matter.

P.S. Another cool demonstration I've heard of is - the instructor sits on a wheeled cart on a gymnasium floor, holding a tank of compressed air. Then s/he opens the valve and is propelled across the floor. (However, this is dangerous if the tank gets loose or the air hits somebody.) Guaranteed, it's a teachable moment (so to speak). Alternatively, s/he could just throw heavy stuff off the cart, and be propelled that way. Kids could have a race that way. That gets their interest.

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