Question

1) Muons naturally decay with a half-life of 1.52 μs, meaning that if Mo muons are created at t'=0, then vo will survive at time t 1.52 us as measured by their 'internal clock'. The population N actually decays exponentially according to N(t')=Mo exp(-t σ), where σ=2.2,s. Assume that muons are created in a laboratory and accelerated to essentially the speed of light (a wee bit slower, of course!) in a time much shorter than their half-life. The muons then travel in a vacuum beamline to a target 1200 m away. a) If the muon's 'internal clock' runs at the same rate as the laboratory clocks, what fraction of No will have survived by the time they reach the target? b) At what time t, (measured by the muons, internal clock) will 99% of the muons still survive? c) What speed u do the muons have to be traveling so that 99% of them survive when they reach the target? (if v=c-5v, give δν in m/s.) d) What is the corresponding y of the muons with respect to the laboratory?

Answer 1

distance 1200m assuming the speed to be v=c-3*108 m/s, the time it will take by the muonus to over 1200m is: t-d/V-4* 10-6S so, the fraction of No

so,the fraction of No, that will survive will be: N/No-e-t/o-e-4/2.2μ-0.162-16, 2396 this assume as that the half life of muons is same both the lab frame as well as the rest frame of muons. b) here,N/No0.99 taking natural log on both sides gives, T--oln(0.99)-22.11ns c) vd/t where the distance d is now the lorentz contracted distance as measured in muons frame of reference. v-d/t [1200(V1-v/c)l/t vt'-1200-1200v/c v-V1200/It+1200/c')] substitute t-22.11ns to get v-2.94647 10 m/s öv-c-v-5353000m/s

d) The gamma factor is y-1/V(1-v/c2)-5.3174