Two events observed in a frame of reference S have positions and times given by (x1, t1) a...

Two events observed in a frame of reference S have positions and times given by (x1, t1) and (x2, t2) respectively. (a) Frame S' moves along the x-axis just fast enough that the two events occur at the same position in S'. Show that in S', the time interval Δt1 between the two events is given by

where Δx = x2 – x1 if and Δt = t2 − t1 Hence show that if Δx > e Δt, there is no frame S' in which the two events occur at the same point. The interval Δt' is sometimes called the proper time interval for the events is this term appropriate? (b) Show that if Δx > c Δt, there is a different frame of reference S’ in which the two events occur simultaneously. Find the distance between the two events in S' express your answer in terms of Δx, Δt, and c. This distance is sometimes called a proper length is this term appropriate? (c) Two events are observed in a frame of reference S' to occur simultaneously at points separated by a distance of 2.50 m. In a second frame S moving relative to S' along the line joining the two points in S', the two events appear to be separated by 5.00 m. What is the time interval between the events as measured in S? [Hint: Apply the result obtained in part(b).]