Special Relativity- Muons in a Storage Ring

Vickitty

Another question, in which I believe I've gotten the same wrong answer two different ways now.

Muons have a mass m = 105 MeV/c^2. They are accelerated to a kinetic energy of 2 TeV in a storage ring with radius r = 2 km. A student speculates that since muons have a lifetime of only T = 2x10^-6 s, they can only go at most cT = 3x10^8m/s*2x10^-6s = 600 m, which means they can't even make a single loop around the storage ring. Is the student right? Calculate the number of loops that the muon can actually make, because of time dilation, before you calculate the total distance the muon can complete.

I know the student is wrong, because of the special relativity theories. I keep getting stuck at trying to figure out the velocity, though. I've been using the formula Ekin = Etotal - Erest, where Etotal = (mc^2)/(1 - v^2/c^2)^1/2, and Erest = mc^2, and then solving for v, but I keep getting 195.16 m/s, which can't be right.

Thank you for any help!

turin

E_kinetic = E_total - E_rest = γ E_rest - E_rest = ( γ - 1 ) E_rest = ( γ - 1 ) m c²

=>

γ = 1 + { E_kinetic / ( m c² ) } = 1 / √{ 1 - β² }

=>

β = √{ 1 - ( 1 - ( E_kinetic / ( m c² ) )² ) }

=>

v = √{ 1 - ( 1 - ( E_kinetic / ( m c² ) )² ) } c ~ c.

You should calculate the time dilation from the γ in the first step.