Question

Muons are short-lived particles that have a rest energy of 105.6 MeV. In an accelerator, muons are accelerated to a kinetic energy of 410 MeV.

A) Determine the momentum of these muons in units of MeV/c.

B) The mean lifetime of muons that are at rest in the laboratory is 2.20 μs Determine the mean lifetime of these accelerated muons.

Answer 1

A): Rest energy = 105.6 MeV

⇒ mc²  = 105.6 MeV

or m  = 105.6 MeV/c²

kinetic energy = 410 MeV

   ⇒   (1/2)mv² = 410 MeV

or (105.6 MeV/c²) v² = 2 x 410 MeV

or v² = (820MeV)/(105.6 MeV/c²) = 7.765c²

or v = 2.787c

therefore momentum = mv = (105.6 MeV/c²)x 2.787c = 294.30 MeV/c

B): mean life time of muon, t₀ = 2.3 μs

therefore mean life time of accelerated muons, t = t₀/[1 - (v/c)²]^1/2

= 2.2/[1-(2.787c/c)]^1/2

= 2.2/(1-2.787)^1/2

= 2.2/(-1.787)^1/2

= 2.2/(1.787)^1/2(-1)^1/2

= 1.645/(-1)^1/2​​​​​​​ is indeterminate