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# What is momentum Watch

1. I understand conservation of momentum and equations related to it, but what is the very nature of it? I am speaking in regards to GCSE physics
2. This is a very good question, I hope you are thinking of pursuing physics further? I'm going to try and answer you question as best I can, but if I say something you don't understand/you think might be wrong please just say and I'll try and explain it better!

Okay so in terms of newton's laws: "Newton's second law of motion states that the change in linear momentum of a body is equal to the net impulse acting on it" (Wikipedia). This is another way of saying that momentum is a property that can only be changed by the action of a force. In other words, you have to do work (have you done work/energy yet?) to change momentum, so it can be thought of being a bit like kinetic energy, in that it is a property that cannot be created or destroyed, and any gain in momentum by one object, much equal a loss in momentum by another object.

This is not a rigorous definition, because momentum and energy are actually fundamentally different things. More generally, linear momentum is the manifestation of a much more complex concept called 'translational symmetry of space time.' Now this is something I don't actually know anything about myself (I'm in my second year of doing physics at uni so by no means an expert), however from a quick read of the wikipedia article, I'll try and explain it :P. Basically this means that the laws of physics are the same everywhere, i.e. the rules that govern our universe don't change depending on where you are. However this is far too complex for me to understand so probably will confuse you as well.

3. I think the explanation by Darth_Narwhale is very accurate, detailed and high power, which might be good for the clever GCSE student that ra1500 seems to be.

If I were to answer your question, especially considering you are only doing GCSE (and my highest knowledge is only as a minor subject [Medical Physics] as a medical student), I would (and ONLY could!) put it more simply as below:

Think of momentum as the tendency of a moving object to carry on moving further in the direction it is moving, and if it were to collide with another object, to "push" it in the direction of the first object, or, if the momntum of the second object is in a different direction and is greater than the momentum of the first object, then for the second object to push the first object.

You can imagine that if a rugby player (probably a massive guy) collided into a netball player (probs a much smaller girl!!), it is likely the two together will move in the same direction as the rugby player - this explains why momentum has m (mass) as one item in its calculation.

Secondly, take two identical cars A and B (of the same weight) moving in opposite directions. If car A was moving East at 70mph and car B was moving West at 10mph, and they collide head on, surely it is essy to work out that the two will after the collision be more likely to move Eastwards [because the momentum of the faster car would be greater i.e. the tendency of the two cars to move eastwards will be greater than the tendency for them to move westwards) This explains the second variable v (velocity) in the calculation of momentum.

In a way, momentum is the opposite of inertia. (inertia is the tendency of an object to stay put where it is and not to move).

Hope this helps further.

Mukesh (science tutor)
4. Momentum does not actually exist as a physical force or object. It is simply the product of mass and velocity of an object.

Much like gravity, gravity itself does not exist, it's just the result of mass in space.
5. Answering what momentum is is a very difficult question, especially sticking to GCSEs, so I will try and go a little bit further.

Intuitively, momentum is a sort of measure of how hard it is stop something. The more momentum the object has, the longer you have to apply a force, or the larger that force will be. This is Newton's second law. But this intuition fails when it comes to fields. Fields (like the gravity field or the electric field) and waves can have momentum, yet you can't stop a field. Actually what does it even mean to stop a field?

So basically momentum must have some other definition. Perhaps "the ability to change the motion of other objects" works. A field transfers its momentum to accelerate a particle and waves do the same. Of course this definition is really vague; we need to concretise it. Objects carrying momentum can transfer that momentum to other objects by exerting a force. As long as they are moving they can exert a force, and the heavier they are, the more force they can exert. This prompts one to define momentum as mv. Note that this definition implies conservation of momentum . What I'm trying to say is that momentum is simply a mathematical aide in calculations which does not have a physical representation like distance or speed. As succinctly put by AishaGirl:

(Original post by AishaGirl)
Momentum does not actually exist as a physical force or object. It is simply the product of mass and velocity of an object.
Ultimately, it is just the product of mass and velocity, which is something that just so happens to be useful, because it is conserved so we retain it and give it a name.

If you want to delve deeper, momentum becomes a lot of things. In fact you can show that if energy is to be conserved, "something else" must be conserved, and that "something else" is called the momentum. The reason why energy has to be conserved is that the laws of physics don't change with time, so "something" must remain the same, and that "something" is energy. Based on this definition of momentum, you can show that if you took very small changes in kinetic energy, and divided that by the velocity and then summed them all, the resultant is the momentum. (if you are doing addmaths, what I'm saying here is that the integral of the reciprocal of velocity with respect to the kinetic energy is the momentum.). Then the next question is what is energy? By the way the process I'm describing is totally backwards: historically, the concept of momentum came first, then energy. But conceptually, this is better.

Now energy is something else where are definitions are arbitrary. In fact, the energy and the momentum share very intricate relationships, which ultimately boil down to this: the laws of physics are the same regardless of your velocity (if constant). Thus no law can tell you whether you truly are moving or not, provided you have constant speed. This is known as the principle of relativity, from Galileo. It is a very powerful principle, and ultimately it describes the momentum and the energy, which describes the forces. Therefore, it is at the roots of all laws.

In conclusion, momentum is basically a quantity that is defined as the thing that is conserved because energy is conserved, and energy is this conserved thing that arises from the laws of physics being constant in time, and constant in velocity.
6. (Original post by Darth_Narwhale)

This is not a rigorous definition, because momentum and energy are actually fundamentally different things. More generally, linear momentum is the manifestation of a much more complex concept called 'translational symmetry of space time.' Now this is something I don't actually know anything about myself (I'm in my second year of doing physics at uni so by no means an expert), however from a quick read of the wikipedia article, I'll try and explain it :P. Basically this means that the laws of physics are the same everywhere, i.e. the rules that govern our universe don't change depending on where you are. However this is far too complex for me to understand so probably will confuse you as well
I'll try and explain it.

Consider a quantity L = K - U, where K is the kinetic energy and U the potential energy. A fundamental law in (classical) physics says that for each coordinate, if you differentiate L with respect to the velocity in that coordinate's direction, then differentiate that with respect to time, that's the same as differentiating L with respect to that coordinate.

Because the potential energy does not depend on the speed (in general), the time derivative of dL/dx' is the time derivative of dK/du, which is mx', the momentum. And since the potential is defined as ∫-Fdx, and the since the kinetic energy does not depend on position, dL/dx is just F. So basically:

$\frac {d}{dt} (mv) = F$

where I replaced the x' by v so that the law stands out better. This is Newton's 2nd law. Now suppose that for one coordinate, dL/dx = 0, so that L does not depend on that coordinate. Then the right hand side becomes zero, and you have an equation telling you that the momentum in that direction is conserved. Since the derivative L is zero, this means that L does not change if you moved your entire system along that coordinate. This implies that the system behaves the same no matter the value of that coordinate: the laws of physics are the same under translation.

In fact L is the Lagrangian for classical mechanics, so there is no magnetic field term in there. In the case of a magnetic field, the Lagrangian is no longer K - U, but K - f(v) where f(v) is a function of velocity.
You can extend the approach: symmetry under rotation is angular momentum, and symmetry under time shifts is energy, albeit this one is a bit harder to show.
7. (Original post by dbs1984)
I'll try and explain it.

Consider a quantity L = K - U, where K is the kinetic energy and U the potential energy. A fundamental law in (classical) physics says that for each coordinate, if you differentiate L with respect to the velocity in that coordinate's direction, then differentiate that with respect to time, that's the same as differentiating L with respect to that coordinate.

Because the potential energy does not depend on the speed (in general), the time derivative of dL/dx' is the time derivative of dK/du, which is mx', the momentum. And since the potential is defined as ∫-Fdx, and the since the kinetic energy does not depend on position, dL/dx is just F. So basically:

$\frac {d}{dt} (mv) = F$

where I replaced the x' by v so that the law stands out better. This is Newton's 2nd law. Now suppose that for one coordinate, dL/dx = 0, so that L does not depend on that coordinate. Then the right hand side becomes zero, and you have an equation telling you that the momentum in that direction is conserved. Since the derivative L is zero, this means that L does not change if you moved your entire system along that coordinate. This implies that the system behaves the same no matter the value of that coordinate: the laws of physics are the same under translation.

In fact L is the Lagrangian for classical mechanics, so there is no magnetic field term in there. In the case of a magnetic field, the Lagrangian is no longer K - U, but K - f(v) where f(v) is a function of velocity.
You can extend the approach: symmetry under rotation is angular momentum, and symmetry under time shifts is energy, albeit this one is a bit harder to show.
Thanks, this is a brilliant explanation. We are actually doing a classical dynamics lecture course atm, and have literally just covered this, but your wording is very clear.
8. (Original post by ra1500)
I understand conservation of momentum and equations related to it, but what is the very nature of it? I am speaking in regards to GCSE physics
An older term for momentum was simply "motion". Momentum is the quantity that tells you how much motion there is in a system, in some sense.

Intuitively, a mass of 1 kg moving in a straight line at 1 m/s has less "motion" than a mass of 1 kg moving in a straight line at 10 m/s. However, a mass of 10 kg moving in a straight line at 1 m/s has the same amount of motion as 10 1 kg masses moving in a straight line at 1 m/s - that's because we can imagine the 10 kg mass as 10 1 kg masses stuck together.

However, we also need to take direction into account. Consider the following experiment. You are floating in space with two 1 kg masses. You are in the centre of a large, stationary, rectangular metal box. You push the two masses at the same time with the same speed towards opposite ends of the box. By symmetry, when the masses collide with the walls of the box, you expect it to remain at rest.

From the outside of the box, an observer sees a box at rest, and says that the system has no motion at any time. Inside the box, you see two identical masses moving with the same speed in opposite directions. You must conclude that the two moving masses have no total motion, even though each has an individual motion.

We can also reason about motion using other thought experiments. Imagine you are an observer watching two lumps of clay of mass travel towards each other at speed . When they collide, they stick together to form a lump of mass . By symmetry, it must be at rest relative to you.

Now imagine a lump of clay of mass moving at towards an identical lump of clay. When they collide, they move off relative to you at some speed . However, an observer moving in the same direction as the moving mass sees a mass moving at speed towards a mass moving at speed in the opposite direction i.e. she sees the original situation. So from her POV, the combined mass is stationary after collision, which from your POV means that it moves with speed (the same as the moving observer). If we now tot up the quantities mass x velocity before and after collision, we find that they are the same (.

Arguments like this allow you to conclude that the "motion" in a system is conserved by collisions, if we define motion = mass x velocity (But of course, you need to do experiments to find out if the universe agrees with your reasoning - this is physics, not abstract maths). These days we call it "momentum", of course.
9. (Original post by atsruser)
An older term for momentum was simply "motion". Momentum is the quantity that tells you how much motion there is in a system, in some sense.

Intuitively, a mass of 1 kg moving in a straight line at 1 m/s has less "motion" than a mass of 1 kg moving in a straight line at 10 m/s. However, a mass of 10 kg moving in a straight line at 1 m/s has the same amount of motion as 10 1 kg masses moving in a straight line at 1 m/s - that's because we can imagine the 10 kg mass as 10 1 kg masses stuck together.

However, we also need to take direction into account. Consider the following experiment. You are floating in space with two 1 kg masses. You are in the centre of a large, stationary, rectangular metal box. You push the two masses at the same time with the same speed towards opposite ends of the box. By symmetry, when the masses collide with the walls of the box, you expect it to remain at rest.

From the outside of the box, an observer sees a box at rest, and says that the system has no motion at any time. Inside the box, you see two identical masses moving with the same speed in opposite directions. You must conclude that the two moving masses have no total motion, even though each has an individual motion.

We can also reason about motion using other thought experiments. Imagine you are an observer watching two lumps of clay of mass travel towards each other at speed . When they collide, they stick together to form a lump of mass . By symmetry, it must be at rest relative to you.

Now imagine a lump of clay of mass moving at towards an identical lump of clay. When they collide, they move off relative to you at some speed . However, an observer moving in the same direction as the moving mass sees a mass moving at speed towards a mass moving at speed in the opposite direction i.e. she sees the original situation. So from her POV, the combined mass is stationary after collision, which from your POV means that it moves with speed (the same as the moving observer). If we now tot up the quantities mass x velocity before and after collision, we find that they are the same (.

Arguments like this allow you to conclude that the "motion" in a system is conserved by collisions, if we define motion = mass x velocity (But of course, you need to do experiments to find out if the universe agrees with your reasoning - this is physics, not abstract maths). These days we call it "momentum", of course.
I thought momentum was a fictitious force, much like gravity... https://www.scientificamerican.com/a...titious-force/

I don't know I think whether certain forces are actually real or not is a philosophical debate. Correct me if I'm wrong though.
10. (Original post by AishaGirl)
I thought momentum was a fictitious force, much like gravity...
Momentum isn't a force at all, so this doesn't make much sense, I'm afraid. A force is what acts on a body when its momentum is changing, and the size of the force in Newtons is equal to the rate of change of that body's momentum.

I don't know I think whether certain forces are actually real or not is a philosophical debate. Correct me if I'm wrong though.
I'm not sure I really understand the question. However, generally in physics "real" forces are produced by interactions between bodies e.g. due to the EM field, and they come in Newton III force pairs. "Fictitious" forces have to be invented when a system is viewed from a non-inertial frame e.g. one that is accelerating.

For example, consider a ice hockey puck mass tied to a post in the ice with a spring, and made to move in a circle on the ice. From the POV of an observer who is not rotating relative to the pole, we say that the puck is accelerating with magnitude v^2/r towards the pole and consequently a force acts on it towards the pole of size mv^2/r. The force is due to the stretching of the spring, and we call it centripetal force.

From the POV of an observer rotating at the same rate as puck and in the same sense, then the puck is not moving at all. However, she can see that the stretched spring is pulling on it, so by Newton II it should accelerate. To ensure that Newton's laws still make sense in her rotating frame, she has to invent a new force of the same magnitude as the force, in the opposite direction, to "balance" the force of the spring. This appears (to the rotating observer) to be pulling the puck away from the pole, and we call it centrifugal force.

However, the centrifugal force doesn't really exist - there is no interaction with a phyiscal body that is causing it - it is merely invented to make sure Mrs Rotating can make Newton's law work from her POV.
11. (Original post by Darth_Narwhale)
Thanks, this is a brilliant explanation. We are actually doing a classical dynamics lecture course atm, and have literally just covered this, but your wording is very clear.
Thanks for the compliment. I'm hoping to teach physics some day, so comments like those show I'm on the right track.

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