# Explain Momentum

It's a shame for me to be in Year 12 and still haven't understood the concept of Momentum. This is what I think, and I know I'm wrong. But, it is a good place to start for you to explain: "Since the unit of the momentum is kg/ms^-1, it is a measure how much mass is being moved at the speed of 1ms^-1. By the way, this is not a homework or anything. I'm really eager to learn about this concept. Thanks In Advance.

• It's a way to quantify motion. – mcodesmart Dec 6 '13 at 1:22
• Elaborate, please. – The DON Dec 6 '13 at 1:23

One way to think of it is "flux", i.e., the amount of stuff flowing or moving through somewhere. If I wanted to quantify the rate at which fish are escaping from a pen, I could say that $x\,\mathrm{kg}$ fish swim past the opening in the pen in $1s$, or that the flux is $x\, \frac{\mathrm{kg}}{\mathrm{s}}$. Faster fish or heavier fish would contribute more to the flux, or rate of escape, so to each fish I can assign the quantity $m v$. Analogous quantities to think of are electric current, for example.

• Oh.. Thank You.. Can you also explain Conservation of Momentum as well please? By the way, I'm only in year 12. So, try to explain in simple terms. – The DON Dec 6 '13 at 2:50
• Conservation of momentum is like a continuity equation for fluids/currents. However, these are just ways to visualize things. The meat is in symmetry, which is WetSavannaAnimal's answer. – lionelbrits Dec 6 '13 at 16:38

Lionel's answer affords good intuition. For me, another notion that really motivates all this is the notion of symmetry and the proof (in a particular sense) that for every symmetry we can see in nature, there must be a conserved quantity.

This idea is embodied in Noether's theorem. At high school it wouldn't be reasonable to expect you to have the background to understand this fully, unless you really study this stuff for pleasure. But I believe you can grasp the gist of what Noether's theorem tells us and look forward to the day when you may have the background to understand how to prove this and in exactly what sense the theorem holds.

Basically, if physical laws do not change whenever we "impart a continuous transformation" on them, then there is a conserved quantity.

Let's talk about shifting our time origin. The details of the way we measure time are largely human constructs. If I were to say to you that the result of my experiment depends on the time of day it is done, then mostly you would think this absurd (unless, for example, the experiment depends on gathering sunlight): the simple act of shifting the origin and redefining what time we take to be $t=0$ cannot influence physics: Nature doesn't care how we set our watches!

So this is a continuous symmetry of many laws of physics: the results these laws give us cannot change if we slide our time axis along a bit. Noether's theorem then tells us that there must a quantity that does not change that corresponds to this continuous symmetry. There is indeed, and that it the quantity called energy. Energy conservation, although ultimately proven by experimental results, is somewhat an expected result given Noether's theorem.

Now for momentum. Again, our laws of physics cannot give us different results if we humans simply decide to shift our $x,\,y,\,z$ co-ordinates. So we have three new continuous symmetries for laws of physics: the laws must tell us the same thing after we have slidden our co-ordinate origin by any distance in any direction. Noether's theorem foretells that there is one conserved quantity for each direction of "sliding". There are three: the three Cartesian vector components of momentum.

So momentum is the thing that is conserved because our laws must not change if we slide our co-ordinate origins around! Experimentally it is found that there is a vector conserved like this: it is $\sum_j m_j\,\vec{v}_j$ for any system of masses $m_j$ moving with velocity $\vec{v}_j$.

As an aside: we can go further: angular momentum (the "spinning analogue" of momentum which you'll likely learn about in a year or two) is conserved because our physical laws must give us the same outcomes after we rotate our co-ordinate axes to a new orientation.

• Thank You! Loved your answer! Can you also explain Conservation of Momentum, please? Why is the momentum of two objects wide apart is same as the total of the momentums when we get in contact? – The DON Dec 6 '13 at 2:53
• @TheDON Given Newton's second and third laws, we can cast the laws of motion of a whole system into a form (called the Lagrangian form) that does not change when we slide the co-ordinate origin around and that lets us apply Noether's theorem to it. The theorem foretells that there is a vector quantity $\sum_j m_j\,\vec{v}_j$ i.e. the total momentum which cannot change. You can also derive this from first principles for a system of masses $m_j$ with velocities $\vec{v}_j$: Newton's second law simply says that ${\rm d}/{\rm d} t\,(\sum_j m_j\,\vec{v}_j)$ is the nett force on the system .... – WetSavannaAnimal Dec 6 '13 at 3:12
• ...because you know that the forces between the masses must cancel out by Newton's third law. If mass 1 exerts force $\vec{F}_{12}$ on mass 2, then mass 2 exerts force $-\vec{F}_{12}$ on mass one. So this quantity $\sum_j m_j\,\vec{v}_j$ cannot change if there is no nett force on the system. Noether's theorem would thus seem to be a heavy handed way to go about things, but I like the deep motivation that "symmetries beget conserved quantities". One expects physical laws to imply conserved quantities otherwise physics becomes weirdly experimenter-dependent. Noether's theorem demystifies. – WetSavannaAnimal Dec 6 '13 at 3:17
• Very interesting! Does this mean that in spacetime that energy and momentum are the same thing, energy just being the time component of momentum? Oh, wait... (epiphany happening) and the conversion factor is related to the speed of light? – Wossname Dec 6 '13 at 3:44
• @Wossname Kind of: you can combine the two into the conserved quantity called the momentum four-vector. Energy and momentum are different components of this entity and they will transform by a Lorentz transformation under boosts (shifting from one inertial frame to another moving at a constant velocity relative to the first). – WetSavannaAnimal Dec 6 '13 at 3:51