Understanding Momentum: Definition, Intuition, and Properties

[annie122]

I know momentum is mass times velocity, and that it is a conserved quantity, but I can't get the intuition of what momentum is, unlike mass and velocity.
Mass relates to how heavy an object is, and velocity is how fast an object is moving.

One idea I had is that momentum is how hard an object hits me, but I'm not entirely sure if that's an okay thing to say.

 

[phinds]

Yeah, momentum is how hard something hits you ... that's not a bad way to think of it.

If it's moving faster, it hits you harder. If it's heavier, it hits you harder. If it's both heavier and faster, you really should get out of the way.

 

[caleb5040]

You could also think of it as "the quantity of motion" of an object.

 

[Thundagere]

The easiest way for me to think of it is "mass in motion." Momentum is just how much "mass in motion."

 

[Neandethal00]

I know momentum is mass times velocity, and that it is a conserved quantity, but I can't get the intuition of what momentum is, unlike mass and velocity.
Mass relates to how heavy an object is, and velocity is how fast an object is moving.

One idea I had is that momentum is how hard an object hits me, but I'm not entirely sure if that's an okay thing to say.

Momentum can also be thought of as Kinetic energy.

KE = p²/2m

p = SQRT(2m*KE)

The higher the KE, the higher the momentum.

 

[Vanadium 50]

The higher the KE, the higher the momentum.

True, for two objects of the same mass.

Momentum can also be thought of as Kinetic energy..

Not true.

 

[ModusPwnd]

Well they each have the same unit... the eV :p

 

[Dadface]

Well they each have the same unit... the eV :p

Not sure what you mean.The eV is a unit of energy,not momentum.

 

[stephenn]

momentum p is indeed mass m x velocity v
p = mv

mass... is not actually the weight... weight is mass x gravity... so mass of something is its weight then deduct any gravity it was weighed in ie earth, moon etc

velocity... though is not actually speed... it is a speed and a direction... 50mph east

 

[Drakkith]

Kinetic energy scales at 1/2 mv^2. When you double your speed you quandruple your kinetic energy. This does not happen with momentum. It scales linearly at p=mv.

 

[TheLil'Turkey]

If 2 objects made of the same material have the same momentum, but different KE, the little one will be harder to stop. If they both have the same KE but different momentum, the big one will be harder to stop.

 

[sophiecentaur]

Well they each have the same unit... the eV :p

? Momentum has the Unit Ns (Newton seconds) but KE has the unit J (Joules). Not the same at all.
Momentum is a Vector quantity - it has a direction associated with it. KE has no direction specified because it is a Scalar, not a Vector.

Momentum is conserved in all collisions. Kinetic Energy is not.

But, as an object speeds up, both its momentum and KE increase - so there is a kind of association between them.

 

[ModusPwnd]

It was kind of a joke; http://en.wikipedia.org/wiki/Electronvolt#MomentumGuess I should take care not to confuse people though.

 

[sophiecentaur]

That article claims that "in high energy Physics, electron-volt is often used as a unit of momentum". I really doubt that, unless they are talking in some very isolated context. It sounds so wrong that I can't take that sentence seriously. Wiki can often be wrong and, more often, be written badly or with insufficient editing.
Whatever the article says, it is important to realize that they are two distinct quantities. I can't think anyone would claim otherwise.

 

[Dadface]

That article claims that "in high energy Physics, electron-volt is often used as a unit of momentum". I really doubt that, unless they are talking in some very isolated context. It sounds so wrong that I can't take that sentence seriously. Wiki can often be wrong and, more often, be written badly or with insufficient editing.
Whatever the article says, it is important to realize that they are two distinct quantities. I can't think anyone would claim otherwise.

I had the same reaction when I first read it but then the paragraph goes on to describe that momentum can be described by eV/c.I'm guessing that if there are people who use the eV as a unit of momentum then it is implied(though not stated) that c is included as above.If so it seems a bit sloppy to me.

 

[sophiecentaur]

Sloppy, yes. But you get this sort of thing with terminology within specialised fields. Very confusing for the outsider. (Perhaps that's why it's used?)

 

[ModusPwnd]

That article claims that "in high energy Physics, electron-volt is often used as a unit of momentum". I really doubt that, unless they are talking in some very isolated context. It sounds so wrong that I can't take that sentence seriously. Wiki can often be wrong and, more often, be written badly or with insufficient editing.
Whatever the article says, it is important to realize that they are two distinct quantities. I can't think anyone would claim otherwise.

We used natural units in just about every physics class we had. Its pretty common.

 

[sophiecentaur]

"Natural", meaning what?

 

[ModusPwnd]

The system of natural units.

http://en.wikipedia.org/wiki/Natural_units

?? I am not sure where the confusion lies. Have you really not heard of natural units before? Your professors are doing you a disservice!

 

[Studiot]

Good King Hal established the only system of Natural Units of any importance and the Americans, God bless their cotton socks, are still using some of them.

 

[sophiecentaur]

The system of natural units.

http://en.wikipedia.org/wiki/Natural_units

?? I am not sure where the confusion lies. Have you really not heard of natural units before? Your professors are doing you a disservice!

HAHA. My professors are nearly all dead by now, I reckon.

I still think that a system of units which draws no distinction between Momentum and Energy is asking for trouble.

But I can't find a reference to Momentum in that link. The link itself seems to make good sense. Is there some confusion somewhere in this thread, perhaps?

 

[ModusPwnd]

There is a distinction, but they are also intimately related. Both energy and momentum are the constituents of the 4-momentum. You boost between the two and turn your energy into momentum and back. In this sense, I think you can think of momentum as energy (or vise versa).

 

[sophiecentaur]

What does "boost" mean, please?

 

[Studiot]

I think modus is referring to the four vector p_μ for a relativistic particle formed from its three spatial momenta and its energy divided by c.

p_μ = the four vector {p_x, p_y, p_z, iE/c}

The scalar product p_μp_μ is invariant

p_μp_μ = p²_x + p²_y+ p²_z - E²/c²

= -m₀²c² = a constant

 

[sophiecentaur]

I have yet to see anything to tell me that momentum and energy are the same, though. I know that mass and energy are equivalent but are they also both equivalent to momentum? I don't think so.

 

[Studiot]

I have yet to see anything to tell me that momentum and energy are the same, though. I know that mass and energy are equivalent but are they also both equivalent to momentum? I don't think so.

I didn't say they were the same.

The equation I quoted leads to another

E² - p²c² = m₀²c⁴

This doesn't say you can trade momentum for energy directly but tells us that a change of mass must also be involved to achieve this.

Incidentally, (someone correct me if this is wrong) I think a boost is the spacetime 4vector analog of acceleration ie what you obtain if you (vector) differentiate 4velocity. This differentiation is a linear transformation.

 

[ModusPwnd]

What does "boost" mean, please?

Boosting is when you change your inertial frame, when you do a lorentz transformation.

 

[sophiecentaur]

Boosting is when you change your inertial frame, when you do a lorentz transformation.

You get into your rocket and turn on the engines, you mean?
That requires energy, surely, which nullifies any implied equation between momentum and kinetic energy.

I got into this thread in response to a bald statement that the two are "the same". If a statement like that is left unchallenged then all sorts of people will go away with the wrong idea. I really can't see the point in posts (from several different people) which just play with words an dissemble about the topic. Some of the points are interesting, of course, and take us into more advanced Physics but I think it is only fair, when making them, to point out their context and that they don't mean that, in a car crash or a game of snooker, momentum and KE are the same.

I have done my share of 'showing off', I know, but it is helpful, when airing some extra knowledge, if people put the new stuff in context and don't imply that 'old' Physics got it wrong - which is what has been happening on this thread.

 

[Islam Hassan]

Is it correct to say that momentum is of no use unless we are talking about collisions of a sort?

Can momentum have any useful applications in a system where objects/particles move but are not liable to collide/interact at all? Need it be defined in the first place in such a system?

IH

 

[sophiecentaur]

Can you think of an example of such a system and how it could be of interest in Physics?

 

[Studiot]

Can momentum have any useful applications in a system where objects/particles move but are not liable to collide/interact at all? Need it be defined in the first place in such a system?

Yes of course, the interaction of a particle (or stream of them) and a field changes the momentum of the particle(s).

There is a simple secondary school experiment which discusses this in terms of the interaction of alpha, beta and gamma rays with a magnetic field.

Do you know this?

 

[sophiecentaur]

Are you suggesting that momentum is not exchanged or changed then under those circumstances? Is that not where the idea of photon interaction comes in? Alpha particle moves to the right - magnet moves to the left. If not then we have reactionless propulsion. (And you could carry your own magnetic field with you yet still get a force)

 

[Studiot]

Are you suggesting that momentum is not exchanged or changed then under those circumstances?

Again I said no such thing.

Indeed quite the opposite.

a field changes the momentum of the particle(s).

However Hassan specifically asked about the situation where the particles

objects/particles move but are not liable to collide/interact at all?

A single or stream of positive or negative particles passing through a magnetic or electric field do not collide or interact with other particles.

They interact with the field (which can have no momentum of its own) however.

Which is what I said.

 

[sophiecentaur]

So where does the momentum come from? If the particle's motion is changed then there has to be some. The concept of a Field is an artifice, because all fields are there because of the presence of some particle(/s) somewhere -just like the Earth is there for a bouncing ball but we 'ignore it'. In all cases there is a change in momentum which may be regarded as being magicked away (not very satisfactory). But if you regard the change in direction as being due to a photon interaction you're back in business. The deviated particle will have launched its own photon i.e. perturbed the Field.

 

[Delta Kilo]

What does "boost" mean, please?

Boost is a Special Relativity term. It just means changing observer's point of view from one coordinate system to another, moving at a constant speed relative to the first one. It is a fancy kind of rotation of 4D space-time coordinate system in X-T plane. During the boost, spatial and time coordinates get mixed up a little, just like X and Y get mixed up when rotating in XY plane.

I got into this thread in response to a bald statement that the two are "the same".

They are indeed the same but in different dimensions :)
Four-momentum 4-vector can be loosely interpreted as a "rate of flow of mass through 4D space-time" with respect to its own "proper time". Thus momentum (p¹,p²,p³) becomes a flow of mass through space and the energy term p⁰ is a "flow of mass through time". Kinetic energy turns out to be Lorentz factor correction to the total relativistic energy in the limit v<<c.

PS: Relativistic Caveat: This is relativistic treatment of momentum. There is no such obvious connection between momentum and KE in Newtonian physics.

 

[sophiecentaur]

I'm fine with that but that Relativistic Caveat should be writ large and clear for the benefit of all snooker players and others.

 

[Islam Hassan]

Can you think of an example of such a system and how it could be of interest in Physics?

Not really no, I admit it was more an epistemological question than a purely physical one. There may be limited, closed systems of such a nature of which I do not know however.

IH

 

[Studiot]

But if you regard the change in direction as being due to a photon interaction you're back in business. The deviated particle will have launched its own photon i.e. perturbed the Field.

And if you don't?


I have been consistently dealing classically with Hassan's query.

Classically momentum is a vector so yes when a beta particle undergoes a change of direction (but not magnitude) due to its interaction with a magnetic field edit: I am considering this in terms of force (the Lorenz force) = rate of change of momentum, not particle exchange.
Yes in modern terms (and strangely in ancient science and religion too) anything any particle does anywhere in the universe affects all other particles in the universe to some extent or other.

But in classical terms we isolate a section of the universe and consider what happens within that microcosm.

That is what I understand Hassan's question to mean viz

Can momentum in a stream of beta particles be changed without the beta particles bumping into themselves or any other material object.

To which the answer is an unequivocal classical yes.

If I have misunderstood that question then I need to think again.

 

[Dadface]

They are indeed the same but in different dimensions :)
Four-momentum 4-vector can be loosely interpreted as a "rate of flow of mass through 4D space-time" with respect to its own "proper time". Thus momentum (p¹,p²,p³) becomes a flow of mass through space and the energy term p⁰ is a "flow of mass through time". Kinetic energy turns out to be Lorentz factor correction to the total relativistic energy in the limit v<<c.

PS: Relativistic Caveat: This is relativistic treatment of momentum. There is no such obvious connection between momentum and KE in Newtonian physics.

Where in your references does it state or imply that they(energy and momentum)"are indeed the same"? Early on in the Wiki article it is given that Po=E/c where P and E are different and with different units.

 

[sophiecentaur]

And if you don't?


I have been consistently dealing classically with Hassan's query.

Classically momentum is a vector so yes when a beta particle undergoes a change of direction (but not magnitude) due to its interaction with a magnetic field edit: I am considering this in terms of force (the Lorenz force) = rate of change of momentum, not particle exchange.
Yes in modern terms (and strangely in ancient science and religion too) anything any particle does anywhere in the universe affects all other particles in the universe to some extent or other.

But in classical terms we isolate a section of the universe and consider what happens within that microcosm.

That is what I understand Hassan's question to mean viz

Can momentum in a stream of beta particles be changed without the beta particles bumping into themselves or any other material object.

To which the answer is an unequivocal classical yes.

"Material Object"? When does anything hit one of those? It's all fields or photons, depending on what's of interest at the time. When a charge accelerates, it radiates EM as a photon / photons, doesn't it? That photon will, eventually, effect the system producing the original field. Where is the difference between that and what you refer to as a particle particle interaction? Afaics, it's only a matter of relative distance.

 

[jetwaterluffy]

Yeah, momentum is how hard something hits you ... that's not a bad way to think of it.

If it's moving faster, it hits you harder. If it's heavier, it hits you harder. If it's both heavier and faster, you really should get out of the way.

No, I wouldn't call that a good analogy. A bullet "hits you harder" than an ocean liner, but an ocean liner has vastly more momentum. A better analogy would be "if you are trapped in front of a wall, momentum is how much will an object will crush you.".

Is it correct to say that momentum is of no use unless we are talking about collisions of a sort?

Can momentum have any useful applications in a system where objects/particles move but are not liable to collide/interact at all? Need it be defined in the first place in such a system?

IH

You could say the same thing about velocity.

 

[sophiecentaur]

No variable is much use except to describe some sort of relationship between two things.

Momentum, like Electrical Resistance, is one of those quantities that involve two other, more familiar, quantities. People wast a lot of time trying to put them into 'more simple' terms: eg "Resistance is how hard you have to try to push a current blah blah".

If you just stick with the mass times Velocity definition (or h/λ) and use it enough times in real situations then you can stop trying to find a chatty descriptions for momentum.

 

[Delta Kilo]

Where in your references does it state or imply that they(energy and momentum)"are indeed the same"? Early on in the Wiki article it is given that Po=E/c where P and E are different and with different units.

Different units for time and space are there for our convenience and historical reason. c is just a conversion factor between units, you'll get the same mess if you measure X in feet and Y in meters. But the dead giveaway is E and p get mixed up during boosts, for example in one coordinate system you see an object with energy E and momentum p=(px,py,pz), but when you look at the same object from another coordinate system, moving with velocity v along x direction, you see linear combinations of those:
E'/c = γE/c - βγp_x, p_x' = γp_x - βγE/c, p_y'= p_y, p_z'= p_z, where β=v/c, γ=1/√(1-β²) are Lorentz factors.
This can also be written as:
E'/c = E/c cosh θ - p_x sinh θ, p_x' = p_x cosh θ - E/c sinh θ, p_y'= p_y, p_z'= p_z
Compare this with spatial rotation in XY plane:
E' = E, p_x' = p_x cos α - p_y sin α, p_y' = p_y cos α + p_x sin α, p_z' = p_z.
This just goes to show that E and px,py,px are not independent quantities but components of the same thing.

 

[Delta Kilo]

No variable is much use except to describe some sort of relationship between two things.

Momentum, like Electrical Resistance, is one of those quantities that involve two other, more familiar, quantities. People wast a lot of time trying to put them into 'more simple' terms: eg "Resistance is how hard you have to try to push a current blah blah".

Not really. Its most important feature, namely momentum conservation law, does not follow from p=mv. Momentum survives in places where the both mass and velocity go south, like momentum of a photon. In fact, Newton mechanics is derived from conservation laws and not the other way around.

For example, you wouldn't say that force is nothing more than just a mass times acceleration, as in F=ma. Force is a useful concept on its own, used in many places where neither m nor a is well-defined, like force of a compressed spring for example. But... F=ma is just a time derivative of p=mv, so it must have the same fundamental significance.

 

[James_Harford]

One way to intuitively disentangle momentum from energy of motion is to consider a moving mass (e.g. vehicle) brought to a stop by a constant force (e.g. brakes). Then time to stop "measures" momentum, and distance to stop "measures" energy.

More specifically, for a given force, different masses with the same momentum take the same time to stop. Different masses with the same energy take the same distance to stop. The concepts are different, but have a pretty symmetry when thus described.

 

[James_Harford]

The concepts are different, but have a pretty symmetry when thus described.

I should mention that the above intuitive description is correct even for relativistic masses.

 

[sophiecentaur]

Newton mechanics is derived from conservation laws and not the other way around.

.

Thats absolutely fine but people ask many questions which are totally in the context of Newtonian Laws. There are perfectly reasonable and helpful answers to those questions which are also in terms of Newton.
I wish people would actually read the questions and interpret them in the context of elementary Science first. Many of the questions are posted by School and College students and they are based on the Newtonian stuff they learn on their courses. Can it really help them if thy are hit with all that advanced stuff, involving Vectors and Relativistic niceties?

How many of the people who supply these confusing answers could, hand on heart, say that their answer would have been any use to them when they were 17 years old and had missed some point that their teacher had made earlier on that day?

When youv'e been told about dimensional analysis as a way of checking that you got an A level Physics formula right you will only be confused when someone glibly tells you that, in another frame, the dimensions are all wrong.

Perhaps we should have a more strict system for the placing of posts so that people who want 'the easy answer' are protected from the 'clever answers' - at least at the start of their thread.

 

[sophiecentaur]

One way to intuitively disentangle momentum from energy of motion is to consider a moving mass (e.g. vehicle) brought to a stop by a constant force (e.g. brakes). Then time to stop "measures" momentum, and distance to stop "measures" energy.

More specifically, for a given force, different masses with the same momentum take the same time to stop. Different masses with the same energy take the same distance to stop. The concepts are different, but have a pretty symmetry when thus described.

That is a good, down-to-earth answer and I'm sure it's the sort of thing the original questioner was after. I hope he's still with us and can read it.

 

[Islam Hassan]

That is a good, down-to-earth answer and I'm sure it's the sort of thing the original questioner was after. I hope he's still with us and can read it.

Agreed 100%, very elegant way to put it, thanks.

IH

 

[Studiot]

Spot on, James.

This is easy to prove with high school Physics.

For energy, E

\begin{array}{l}<br /> E = \frac{{m{v^2}}}{2} = \int_0^x {Fdx} = F\int_0^x {dx} \\ <br /> E = Fx \\ <br /> \end{array}

Where the kinetic energy is lost as work against the Force F.

So if F is the same constant force for Energy E₁ and E₂ then if E₁ = E₂ ; x₁ =x₂

Alternatively the formula also shows that if F is not constant, but has the same variation with distance for two cases then the work done is still the same so the distances are still the same. However we have to find an expression for this variation to perform the integration for actual numbers.

For momentum, p

Force is the rate of change of momentum.

\begin{array}{l}<br /> F = \frac{{dp}}{{dt}} = {\rm{constant}} \\ <br /> {\rm{dp = Fdt}} \\ <br /> \int_0^p {dp} = \int_0^t {Fdt} = F\int_0^t {dt} \\ <br /> p = Ft \\ <br /> \end{array}

We have a similar calculation to the energy one and a similar line of reasoning shows that for a constant force

If p₁=p₂ then t₁=t₂

and also that for a variable force the stopping times are still the same, this in this case so long as the variation with time is the same, the stopping times are still equal. Again we need an expression to perform the numerical calculation.