Question

In its own reference frame, the half-life of a muon is 1.52us. Of course, what the muon is measuring is proper time, not necessarily the time that somebody in an inertial frame would say that it took to decay. Suppose we speed up some muons by sending them around and around in a circle. (There are ways to make things go around in circles at high speed by using magnetic fields.) The lifetime of the muons (as measured in the inertial frame of the lab) is an entire second. How fast are they going? HINT: The speed is going to be very close to 1, so you should write the speed in the form 1 6 (b) At this speed, how many times per second does it go around in the ring in the lab? Assume that the radius of the ring is 4 meters.

Answer 1

given, half life in its own refernce frame, To = 1.52 us

let its speed be v and it is moving in a circle

now, life time measured by inertial observer , T = 1 s

hence

T = To/sqroot(1 - v^2/c^2)

1 - v^2/c^2 = 1.52^2*10^-12

v = 0.99999999999884479c ( where c is speed of light)

v = (1 - 1.1552*10^-12)c

b. now, length of path in lab frame = 2*pi*R

R = 4 m

hence

l =8*pi

so, 8*pi/t = v

hence

time period of one revolution, t = 8.377*10^-8 s

number of revolutions per second = 1/t = 11936620.73187