Measuring an Event — Question 3

Question 3

A radar echo assigns both coordinates to one reflection event

An observer at the origin sends a vacuum pulse at $t_-=2.0\,\mu\mathrm s$ and receives its immediate reflection at $t_+=10.0\,\mu\mathrm s$. The reflector lies at positive x. Use $c=3.00\times10^8\,\mathrm{m/s}$ and Einstein-synchronized laboratory clocks.

1. Derive the time and position assigned to the reflection.

2. Evaluate those coordinates and distinguish the reflection event from the two events on the observer worldline.

Question 3 - Solution

A radar echo assigns both coordinates to one reflection event

(a) Method and calculation

$$x_E=c(t_E-t_-)=c(t_+-t_E).$$$$\boxed{t_E=(t_++t_-)/2},\quad\boxed{x_E=c(t_+-t_-)/2}.$$

(b) Interpretation and further calculation

$$\boxed{t_E=6.0\,\mu\mathrm s},\quad\boxed{x_E=1200\,\mathrm m}.$$$$E_-=(0,2.0\,\mu\mathrm s),\quad E_+=(0,10.0\,\mu\mathrm s).$$

Physical check

The radar assignment uses one clock and two light paths; the target need not be stationary if reflection is immediate.