Question

Unstable pions are produced as a beam in a nuclear reaction experiment. The pions leave the target at a speed of 0.995 c. The intensity of the beam reduces to half its original value as the beam travels a distance of 39 m. Find the half-life of pions (a) in the laboratory frame, (b) in their rest frame.

Answer 1

(a) The intensity of the pion beam reduces to half is original value in one half-life. The half of the pions as measured in the laboratory is
$t_{1/2}=\frac{39m}{0.995c}=\frac{39m}{0.995\times 3\times {10}^{8}m{s}^{-1}}$
$=1.3\times {10}^{-7}s$.
(b) The events-a pion leaving the target and its decaying-occur at the same place in the pion-frame. Thus, the time measured in the pion-frame is the proper time and is the smallest. It is equal to
$t_{1/2}^{\prime }=t_{1/2}\sqrt{1-v^2/c^2}=(1.3\times 10-7s)\sqrt{1-(0.995{)}^2}$
$=1.3\times {10}^{-8}s$.