The Relativity of Time — Question 3

Question 3

Survival through a flight distance depends on beta gamma

Particles with proper mean lifetime $\tau_0=2.00\,\mu\mathrm s$ move at $v=0.80c$ through a laboratory distance $L=800\,\mathrm m$. Their survival law is $P=e^{-\Delta\tau/\tau_0}$. Use $d\tau=dt\sqrt{1-v^2/c^2}$ for an ideal clock, with $c=3.00\times10^8\,\mathrm{m/s}$ when needed.

1. Find the elapsed proper time and survival probability.

2. Compare with the prediction obtained by mistakenly using laboratory flight time in the decay law.

Question 3 - Solution

Survival through a flight distance depends on beta gamma

(a) Method and calculation

$$t=L/v=3.333\,\mu\mathrm s,\qquad\gamma=5/3.$$$$\Delta\tau=L/(\gamma v)=2.00\,\mu\mathrm s,\quad\boxed{P=e^{-1}=0.367879}.$$

(b) Interpretation and further calculation

$$P_{\mathrm{wrong}}=e^{-L/(v\tau_0)}=e^{-5/3}=0.188876.$$$$\boxed{P/P_{\mathrm{wrong}}=e^{2/3}=1.94773}.$$

Physical check

The decay law is a law of the particle clock; flight distance enters through its accumulated proper time.