# Relativistic muons traveling in a circle

I know the rest mass of a muon (106 MeV) and rest frame lifetime (2.2 $$\mu s$$).

Suppose we accelerate muons that travel through a circular ring. At a specific energy, I want to know how many times a muon will travel around the ring. Therefore, I need to find the velocity of the muon, so I know how fast it will travel around the ring, as well as calculate a time-dilated lifetime to see how many times it will go around the ring before decaying.

However, I am not sure I am calculating the velocity correctly from the energy. Let's say the muon is accelerated to an energy of 2 TeV. Do I simply use $$E = \gamma mc^2$$ or some other formula? Is 2 TeV the kinetic energy or total energy?

Once I found the velocity from the above, I will use it to calculate the time-dilated lifetime of the muon. Then, I will divide the circumference of the circle by the velocity. Then, I will divide the muon lifetime by this time to find out how many times the muon can travel the circle before it decays. Is this all, or am I missing some relativistic effect? (I.e. Does traveling in a circle complicate this problem anymore than if the muon was traveling in a straight line?)

As you say, the muon's mass is 106 MeV. Suppose that you're not certain whether an energy of $$\rm 2\,TeV = 2\,000\,000\,MeV$$ is the total energy $$\gamma mc^2$$ or the kinetic energy $$(\gamma-1)mc^2$$. Either way you have $$\gamma \approx 2\times10^4$$, you're solidly in the ultrarelativistic regime, and the speed is experimentally indistinguishable from $$c$$.
My favorite application of this is at Jefferson Lab in the US, where there are two electron linacs arranged in a racetrack configuration, feeding beams with different energies to four experiments at once. The electrons which have just been injected in the accelerator at kinetic energy $$50\rm\,MeV$$ are interleaved with electrons on their fifth pass around the accelerator, with energy $$12\rm\,GeV$$. The low-energy and high-energy beams don't overtake each other, because the $$50\rm\,MeV$$ electrons with $$\gamma=100$$ are already ultra-relativistic and traveling at speed that's not practically different from $$c$$.
If you're thinking about storage rings for relativistic muons, you are going to read about the $$g-2$$ experiment sooner or later.