What is momentum? An intuitive understanding.
Something like "force" is intuitive because we equate force with pain. Getting punched at school (as most geeks learn) hurts, and the more of this 'force' that the bully uses, the more it hurts. Similarly, energy has an intuitive experience. The parodied geek character will tell you all about that 200m into the school cross country run.
But what is momentum?
It's dead easy to look it up, and you quickly find a formula that says:
momentum = velocity x mass
This is all just fine, but I don't know about you, but it leaves me a little cold compared to thinking about force and energy. I will try to develop an intuitive understanding. This is important because it will help you get a better feel for other physics, and will allow you to read text and equations faster and with more comprehension.
One of the stumbling blocks / hurdles / puzzles comes to many when being told that a photon has momentum. But if momentum is mass x velocity, and a photon is massless, then how can it have momentum?
The problem here is the definition of mass. In this case, the mass is known as 'relativistic mass'. Unfortunately, the intrinsic property "rest mass" is often confused with "relativistic mass" by casual readers. The famous equation E = mc^2 refers to relativistic mass and in this case, the mass is calculated by combining both rest mass, and that which is derived by considering momentum. However, this is rarely explained and so people view the m incorrectly. Later I will provide an alternative for for E which will make things clearer even though it appears to add complexity.
If we take the equation momentum = velocity x mass and re-arrange it we get:
mass = momentum/velocity
It's conventional to assign the the letter p for momentum. For a photon, its velocity is always the speed of light c.
For a photon, relativistic mass m = p/c
c is a non-zero constant, therefore the relativistic mass of a photon is directly proportional to its momentum. So that answers how a massless particle like the photon can have momentum. The question now is, "how can one influence the momentum of a photon"?
Momentum of a photon is influenced by wavelength.
One of the de Broglie relations equates wavelength to frequency as illustrated in the equations above.
The conclusion is that more energetic photons - that means photons with more cycles per second, have more momentum than less energetic photons. Since the speed of light is constant in all inertial frames of reference, if you run 'with' a light beam, it's wavelength appears to be longer, and if you rush towards it, then it appears shorter. So if you chased a light beam with a solar cell, you would get less energy out than if you ran towards it. This is the Doppler effect and is also known as red shift and blue shift.
Another way to reason about this effect is that momentum is conserved and by running with the light beam, you are using some momentum and it's not available to excite electrons in the solar cell. By running towards it, some of your momentum is added to the solar cell/photon collision.
It's the momentum of the photon which has the ability to excite an atom's electrons into the conduction band and produce electricity.
Now we have a feel for relativistic mass for a photon. It is a measure of its wavelength and hence momentum i.e. energy since E=hf. But we know energy and mass are equivalent via the multiplication with c in m/s.
But you can also see that this relativistic mass is physically different to the rest mass even though the units are the same. The rest mass is an intrinsic property of an object while the relativistic mass is added to the system because it has the same units.
Now I can present the more descriptive (and easier but less publicised) version of E=mc^2 which splits momentum and rest mass into two terms.
This equation makes it much easier to see what is going on in special relativity. M0 is the intrinsic rest mass of an object. The first term expresses the energy contained in the mass when it is at rest, or alternatively, when the observer is travelling in the same frame of reference as the mass. It is invariant.
The second term is the energy contributed to the system from relativistic effects. This energy is from the momentum and speed and it is frame-dependent.
Let's work out the mass of a photon now. There are two clues: a photon has no rest-mass, but it has momentum, and it always travels at c.
Without a rest-mass, the equation reduces to expressing relativistic energy = +/- pv
I'll leave the significance of the possibility that E=-pv for another discussion. This negative sign results from taking the square root. (Many texts ignore or forget to consider it).
Similarly, for a particle that has a rest mass, but has no momentum, the equation is again simplified - this time to a more familiar form.
Again, we don't forget the opportunity that the mass could be negative and this is to discuss another time. Note too that the subscript 0 is left in to make it clear that we are dealing with a mass at rest with respect to our frame of reference.
Finally, the true form of E=mc^2 is really supposed to be using M0 modified by gamma to take in the consequences of relativistic energy for a body that is moving with respect to the observer.
From this, we can draw a graph that shows how the total energy increases when an object approaches c. Note that v cannot exceed c otherwise we are forced to take the square root of a negative number, and that results in a complex root with an imaginary component. As v approaches c, it does so asymptotically, which means gamma tends towards infinity. Even the notion 'to cross infinity' is nonsense, and this is why nothing can exceed the speed of light.
BUT: If you start from the other side, and consider an object that can only slow towards the asymptote, you are now considering a purely theoretical particle known as a tachyon. The square-root and gamma become imaginary. Again, this is really a topic for another time.
click to enlarge
Hopefully, you now have a more intuitive feel for not only momentum, but also for special relativity. Impress your friends by telling them that E=mc^2 is wrong. Einstein did not care for the equation much anyway. If you get used to the full equation, it tells you a lot more about the nature of mass and energy.
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