Halliday, Resnick & Walker — Relativity: Problem 50

50 As in Fig. 37-9, reference frame \( S' \) passes reference frame \( S \) with a certain velocity. Events 1 and 2 are to have a certain temporal separation \( \Delta t' \) according to the \( S' \) observer. However, their spatial separation \( \Delta x' \) according to that observer has not been set yet. Figure 37-19 gives their temporal separation \( \Delta t \) according to the \( S \) observer as a function of \( \Delta x' \) for a range of \( \Delta x' \) values. The vertical axis scale is set by \( \Delta t_a = 12.0 \mu\text{s} \). What is \( \Delta t' \)?