What is the speed of the muons in the storage ring??

Question

the half life of a muon is 1.5 microseconds. muons that have been accelerated to avery high speed and are then held in a circular storage ring have a half life of 7.5 microseconds.

a) what is the speed of the muons in the storage ring?

b) what is the total energy if a muon in the storage ring? (the mass of the muon is 207 times the mass of an electron)

Answer 1

t' = t / √(1-v²/c²)

1/√(1-v²/c²) = 7.5/1.5 = 5

√(1-(v/c)²) = 1/5

1 - (v/c)² = (1/5)² = 1/25 = .04

(v/c)² = 1 - .04 = .96

v = c √.96

v = 0.9797959 c

m' = m / √(1-v²/c²)

Since m'/m = t'/t,

then (m'/m)² = (t'/t)² = 25

m' = 25m

E = mc² = Kinetic and rest mass energy

E = 25(mu) * 207(me) * (3e8)²

E = 2.5e1 * 2.07e2 * 9.11e-31 kg * 9e16 m²/s²

E = 4.2429825e-10 J

I think I did the math right.

Answer 2

7.5/1.5 = 5 = 1/sqrt(1 - v^2/c^2)

so 1 - v^2/c^2 = 1/25; v = 0.9798 c