How is speed measured in the LHC In this question: "Without the Michelson-Morley experiment, is there any other reason to think speed of light is the universal speed limit?", it is stated in Emilio Pisanty's answer that

From a purely mechanical perspective, the LHC routinely produces 7 $TeV$ protons, which would speed at about $120c$ in Newtonian mechanics [...]

Is there a way to measure the velocity of a particle beam in the LHC (or other particle accelerators) independent of the energy? If one knows the energy of the particle beam one can either calculate the velocity by $E=\frac{1}{2}mv^2$ or by $E=c^2m\gamma(v)$, but this requires choosing between Galilean and special relativity and can't be used to verify that either of them is true.
 A: If protons in the LHC weren't obeying special relativity, then the accelerator wouldn't work at all.
In the LHC, protons are injected into the ring in "bunches" of a few hundred billion each, with an initial energy of around 450 GeV. These bunches are accelerated by "kicking" them with an electric field when they reach certain points around the accelerator ring. If you do this with a static electric field, then your bunches end up gradually spreading out in the longitudinal direction, to the point where they're no longer usable for data-taking. Instead, an oscillating electric field is applied, whose frequency is tuned to the frequency of passing bunches. The oscillating electric field accelerates slower protons more and faster protons less, so that the bunch is pushed back together as it accelerates.
This is the key point: the frequency of the oscillating electric field is directly dependent on the rate at which bunches pass the acceleration points. If the frequency is wrong, you won't get a stable beam. In turn, the rate at which bunches pass is, in turn, directly dependent on the speed that the bunches are traveling at.
The frequency of the oscillating electric field at the LHC is constant, regardless of proton energy, usually set to 400 MHz (corresponding to one bunch every 25 ns).
Since the LHC is able to accelerate bunches using a constant-frequency oscillating electric field, that means that the protons are traveling at a constant speed regardless of their energy, which is exactly what special relativity predicts at those energies. 
A: I will give a shot at answering the question. From my perspective, the LHC is a direct descendant of Wideroe's proposal for the linac (Archiv fur Electrotechnik, Vol 21 p 387 (1928)). His illustration for the general idea of the linac is:

Ions enter from the left. If the rf voltage across gap I is right, the ion will be accelerated across the gap and then fly into the field-free tube to the right. This tube is of the right length for the ion charge/mass so that when they hit gap II they are nicely in phase to be accelerated, again, by the rf voltage. This continues on and on, steadily increasing the ion voltage using a single rf power supply. 
But, if the ions hit at the "wrong" time, they will find a smaller field across gap I, or even a reversed field, and will not get enough energy to hit gap II at the right time. Only ions in a narrow time window will manage to get to, and across, each gap with the right energy to make it to the next gap. Historically, for the early linacs, if fed with a continuous beam of ions you would get ~1% through the linac. This can be improved by "bunching" the incoming beam to turn the continuous beam into a series of bunches, where each bunch hits gap I at the right time. 
Now a circulating machine is a bit more difficult, but ultimately you still need bunches of ions/electrons hitting the accelerating gaps at just the right time to steadily increase the beam energy. If you look at a current monitor you would see a series of blips as the bunches fly by. Any difference in the charge in a bunch can be seen, so you can track the progress of individual bunches around the machine. And, you need to keep track of them since the timing of each gap has to be kept in synch with the beam or else you lose beam.
