Measuring an Event — Question 6

Question 6

End-station arrival differences locate an unknown flash

Synchronized detectors at $x=0$ and $x=L=1200\,\mathrm m$ receive a flash emitted between them at $t_A=5.0\,\mu\mathrm s$ and $t_B=3.0\,\mu\mathrm s$. Use $c=3.00\times10^8\,\mathrm{m/s}$ and Einstein-synchronized laboratory clocks.

1. Derive the source position and emission time from the two arrivals.

2. Calculate them and check that the inferred source lies between the detectors.

Question 6 - Solution

End-station arrival differences locate an unknown flash

(a) Method and calculation

$$t_A=t_E+x/c,\qquad t_B=t_E+(L-x)/c.$$$$\boxed{x=\frac{L+c(t_A-t_B)}2},\qquad\boxed{t_E=\frac{t_A+t_B-L/c}2}.$$

(b) Interpretation and further calculation

$$\boxed{x=900\,\mathrm m},\qquad\boxed{t_E=2.0\,\mu\mathrm s}.$$$$0\le x\le L\ \Longleftrightarrow\ |t_A-t_B|\le L/c;\quad2\le4\,\mu\mathrm s.$$

Physical check

The arrival difference fixes position, while the arrival sum fixes emission time.