What does p = mv (momentum) really mean?

[Mr Davis 97]

What does p = mv really mean? For example, why is there a physical law based upon the product of mass and velocity? Why does a human made operation, multiplication, give a quantity called momentum? If we define momentum as the product of m and v, why can't we define by some other operation, like division? For example, if m = 5, and v = 3, what is it about adding 5 to itself three times that gives this quantity called momentum in 15 kg m/s?

 

[pixatlazaki]

Momentum is significant because it (firstly) is a conserved quantity. This is not just true of linear momentum, but angular momentum as well. Noether's theorem shows that if a system is symmetric under a certain transformation, there is a corresponding conservation law. In the case of $\mathbf{p} = m \mathbf{v}$ (which I may add is a classical approximation), it turns out that the sum of total momenta a isolated closed system does not change in time.

I would highly recommend reading more about Noether's theorem. I think it will satisfy many of your curiosities and it happens to be quite elegant!

 

[dkotschessaa]

What does p = mv really mean?

It means, mathematically, that if you increase m or increase v, then you will increase p. If v is negative, you are going in the opposite direction. This information might be important in certain problems.

For example, why is there a physical law based upon the product of mass and velocity? Why does a human made operation, multiplication, give a quantity called momentum?

You will have an easier time with this stuff if you realize that momentum isn't a "thing" that exists, but a quantity that was invented, and defined mathematically, to describe how things behave. (People also have this problem when it comes to "energy" which is just a mathematical quantity, not some stuff that's floating around).

If we define momentum as the product of m and v, why can't we define by some other operation, like division? For example, if m = 5, and v = 3, what is it about adding 5 to itself three times that gives this quantity called momentum in 15 kg m/s?

Classical physics attempts to use mathematics in a way that describes the behavior. Think about when you roll your shopping cart through the parking lot, or start riding around on it(I totally do this). You will have "more momentum" if there's more stuff (or you) on the basket, or if you are going faster. So that equation describes what's going on very well.


-Dave K

 

[Andrew Mason]

What does p = mv really mean? For example, why is there a physical law based upon the product of mass and velocity? Why does a human made operation, multiplication, give a quantity called momentum? If we define momentum as the product of m and v, why can't we define by some other operation, like division? For example, if m = 5, and v = 3, what is it about adding 5 to itself three times that gives this quantity called momentum in 15 kg m/s?

Welcome to PF Mr. Davis 97!

We can certainly define some other operation, such as division, and say p = m/j where j is the rate of change of time relative to position. It is just easier for humans to think in terms of velocity (v) which is the rate of change of position with time.

As others have said, physicists are interested in describing the physical world so the quantities that we are interested in must describe something that has some physical significance. Newton observed that the same force applied for same time to objects of different mass resulted in the same change in the quantity of motion for each. 'Quantity of motion' was the term Newton used for the product of velocity and mass. So, Newton concluded that keeping track of this quantity of motion, which we now call p = mv would be very useful.

AM

 

[vjacheslav]

You ask about p := mv
or any definitions?

 

[Dale]

What does p = mv really mean? For example, why is there a physical law based upon the product of mass and velocity? Why does a human made operation, multiplication, give a quantity called momentum? If we define momentum as the product of m and v, why can't we define by some other operation, like division?

There is certainly no reason that you couldn't. We have already used the term "momentum" to refer to the product of mass and velocity, so you would need another word. But if you wish you could define "flubnubitz" as the ratio of mass to velocity.

We have found momentum to be a very useful quantity, for the reasons outlined above, but as far as I know flubnubitz is not a useful quantity.

 

[sophiecentaur]

There are arguments that Momentum is at least as fundamental a concept as mass, velocity or energy etc.. Those things are all related by some handy mathematical operations.
If we happened to live in the absence of significant gravity, we might well be more concerned with momentum than with mass (giving weight force) and we could have started on our study of mechanics with momentum being much higher up our list of familiarity.

 

[MrRobotoToo]

Questions such as "What really is momentum?" or "What really is mass?" etc. are utterly useless as far as the science is concerned. Answering them adds nothing to the predictive or explanatory power of the theories they derive from. All you have to know is that momentum is well-defined both mathematically and operationally. Anything else that might be said about them is simply philosophical sophistry.

 

[sophiecentaur]

Questions such as "What really is momentum?" or "What really is mass?" etc. are utterly useless as far as the science is concerned. Answering them adds nothing to the predictive or explanatory power of the theories they derive from. All you have to know is that momentum is well-defined both mathematically and operationally. Anything else that might be said about them is simply philosophical sophistry.

Very true. Once you try to add a verbal /mechanical description to a quantity, you can end up limiting your understanding and can lose the more general meaning. It's a trait that many people follow, who do not want to get into the Maths. Whether you understand the Maths fully or not (I don't) you need to accept that it is by far the best language we have for this stuff.

 

[vjacheslav]

Well, but operating something not defined looks like little bit Crazy, or?

 

[HallsofIvy]

Where did anyone say anything about "not defined"?

 

[sophiecentaur]

Well, but operating something not defined looks like little bit Crazy, or?

Using the Maths of the relationship between one quantity and others is miles better than describing it a 'a sort of pushing thing you get when it bumps into you'. That's extreme, I know but many people do seem to want something like it. It can never be very satisfactory - if you want to take it further and relate it to other things.

 

[Dale]

momentum is well-defined both mathematically and operationally.

operating something not defined looks like little bit Crazy

?

This is an odd comment, vjacheslav.

 

[gmax137]

Very true. Once you try to add a verbal /mechanical description to a quantity, you can end up limiting your understanding and can lose the more general meaning. It's a trait that many people follow, who do not want to get into the Maths. Whether you understand the Maths fully or not (I don't) you need to accept that it is by far the best language we have for this stuff.

True, but I think you can get an intuitive understanding of these classical concepts thru experience: if you've played football you know there's a difference in tackling a 250 pounder running at you compared to a 135er just standing there. Or, if you have spent the day carrying shingles up to the roof, you know why work is force x distance.

Other concepts, say entropy or electron spin, not so much.

 

[Andrew Mason]

Questions such as "What really is momentum?" or "What really is mass?" etc. are utterly useless as far as the science is concerned. Answering them adds nothing to the predictive or explanatory power of the theories they derive from. All you have to know is that momentum is well-defined both mathematically and operationally. Anything else that might be said about them is simply philosophical sophistry.

Where would physics be if Galileo and Newton and Einstein had not tried to explain what mass is? It might be philosophy (or philosophia naturali as Newton called it) but it is the desire to understand that drives human beings.

Momentum is defined this way because of an underlying physical significance. So to fully understand the physics one has to understand both the definition and why it is defined that way.

AM

 

[A.T.]

Momentum is defined this way because of an underlying physical significance.

It's only significance is its conservation under certain conditions. That's the only reason why it is defined this way.

 

[sophiecentaur]

Talk to someone with no idea of the Maths involved in Physics and you usually hear a pretty poor model of the World. Maths is so crucial to understanding at any but the very superficial level. Imagine trying to have a conversation about Finances without a common knowledge of the Arithmetic of Interest and Profit. The consequence of not using appropriate Maths is constantly being demonstrated by how people are regularly conned into bad deals. The numbers (and the Algebra) always count.
I think many of the preceding comments have been made by people who do, in fact, have an appreciation of the Maths but it is so familiar to them that they are hardly aware of it.

 

[voko]

The word "momentum" is not particularly illuminating. Newton (following Descartes, but not exactly) used the term "quantity of motion", which survives as a standard term for the concept in some languages.

As a "quantity of motion", the product of mass and velocity makes perfect sense. It is clear intuitively that greater velocity means "more motion". Greater (moving) mass equally conveys "more motion".

 

[sophiecentaur]

The word "momentum" is not particularly illuminating. Newton (following Descartes, but not exactly) used the term "quantity of motion", which survives as a standard term for the concept in some languages.

As a "quantity of motion", the product of mass and velocity makes perfect sense. It is clear intuitively that greater velocity means "more motion". Greater (moving) mass equally conveys "more motion".

Also, without using the formal definition, Momentum can easily be confused with Kinetic Energy. Both quantities give an indication of, for instance, the result (damage etc.) of a collision. Of course, it's very useful to have the Maths with a verbal accompaniment but what are we even considering doing without the Maths? Going back to what Galileo was doing with the quantity is a bit pointless, except for historical interest, and you can always go far enough back in history to find 'Science' that's acceptable to any level of appreciation. We have moved on.

 

[Andrew Mason]

It's only significance is its conservation under certain conditions. That's the only reason why it is defined this way.

Momentum is conserved under all conditions that we know of. That is rather significant. The fact that momentum is a quantity that is always conserved in all physical interactions is one reason we are interested in it.

The simple fact that a certain force applied for a certain time interval results in the same change in this quantity for all matter is important too, at least in non-relativistic physics. That is just the consequence of Newton's second law.

AM

 

[Andrew Mason]

Talk to someone with no idea of the Maths involved in Physics and you usually hear a pretty poor model of the World. Maths is so crucial to understanding at any but the very superficial level. Imagine trying to have a conversation about Finances without a common knowledge of the Arithmetic of Interest and Profit. The consequence of not using appropriate Maths is constantly being demonstrated by how people are regularly conned into bad deals. The numbers (and the Algebra) always count.
I think many of the preceding comments have been made by people who do, in fact, have an appreciation of the Maths but it is so familiar to them that they are hardly aware of it.

I would not disagree with what you have said, but physics is obviously more than math, just like economics is more than just finances. In physics and economics we need to have a conceptual model to help us understand what is happening in the real world, in addition to just understanding the math.

AM

 

[Dale]

I would say that the math is the conceptual model. You need the math and you need the mapping between the math and experimental measurements, but you don't need a map between the math and any other non mathematical concepts.

 

[vjacheslav]

All in all, physik are badly need of conserving quantity, in order to obtain little bit equation :)

 

[A.T.]

The fact that momentum is a quantity that is always conserved in all physical interactions is one reason we are interested in it.

It's the only reason.

 

[voko]

It's the only reason.

Interesting. Do you accept that a non-conserved quantity can be of importance? E.g., position, velocity, acceleration?

 

[sophiecentaur]

Interesting. Do you accept that a non-conserved quantity can be of importance? E.g., position, velocity, acceleration?

No. I only care about my momentum. Look out, here I come! (I'm not sure where I am or what the time is, so I might miss you completely).

 

[vjacheslav]

Us as well. Try to compare definition in differ theories
Classic and Quantum
for instance.

 

[A.T.]

Do you accept that a non-conserved quantity can be of importance?

Sure.

 

[voko]

Sure.

I cannot reconcile this with the earlier statements of yours. You admit that a non-conserved quantity such as velocity can be of importance, yet the "only significance" you allocate to momentum "is its conservation under certain conditions".

Do I understand you correctly that a statement like "the magnitude of the vehicle's velocity is X" is infinitely more significant than "the magnitude of the vehicle's momentum is Y", because the latter statement does not deal with conservation of momentum?

 

[A.T.]

...yet the "only significance" you allocate to momentum "is its conservation under certain conditions"...

What application would momentum have if it wasn't conserved?