The Relativity of Time — Question 2

Question 2

A measured lifetime ratio fixes speed but not direction

A particle species has proper mean lifetime $\tau_0=1.80\,\mu\mathrm s$. Its mean lifetime in a laboratory beam is $4.50\,\mu\mathrm s$. Decay is independent of direction. 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. Infer the beam speed.

2. Find the mean laboratory decay length and explain the remaining directional ambiguity.

Question 2 - Solution

A measured lifetime ratio fixes speed but not direction

(a) Method and calculation

$$\gamma=\tau_{\mathrm{lab}}/\tau_0=2.50.$$$$\boxed{|v|/c=\sqrt{1-\gamma^{-2}}=\sqrt{21}/5\simeq0.916515}.$$

(b) Interpretation and further calculation

$$\ell=|v|\tau_{\mathrm{lab}}=c\tau_0\sqrt{\gamma^2-1}.$$$$\boxed{\ell=270\sqrt{21}\,\mathrm m\simeq1237.30\,\mathrm m}.$$

Physical check

A lifetime gives the magnitude of velocity. A spatial measurement is needed to determine its sign or direction.