Measuring an Event — Question 4

Question 4

Two radar exchanges reconstruct a target worldline

A fixed radar station records emission/return pairs $(2,10)\,\mu\mathrm s$ and $(6,18)\,\mu\mathrm s$ from the same uniformly moving target at positive x. Reflection is immediate. Use $c=3.00\times10^8\,\mathrm{m/s}$ and Einstein-synchronized laboratory clocks.

1. Find the two reflection events using midpoint time and half the round-trip distance.

2. Infer the target velocity and position at $t=0$.

Question 4 - Solution

Two radar exchanges reconstruct a target worldline

(a) Method and calculation

$$(t_1,x_1)=(6\,\mu\mathrm s,1200\,\mathrm m).$$$$(t_2,x_2)=(12\,\mu\mathrm s,1800\,\mathrm m).$$

(b) Interpretation and further calculation

$$\boxed{v=\frac{600\,\mathrm m}{6\,\mu\mathrm s}=1.00\times10^8\,\mathrm{m/s}=c/3}.$$$$x(t)=x_0+vt,\qquad\boxed{x_0=1200-v(6\,\mu\mathrm s)=600\,\mathrm m}.$$

Physical check

Use reflection times, not transmission times, when fitting the target trajectory.