Question

Two events observed in a frame of reference ? have positions and times given by (?1, ?1) and (?2, ?2), respectively.

a) Frame ?′ moves along the ?-axis just fast enough that the two events occur at the same position in ?′. Show that in ?′, the time interval between the two events is given by

′ √ 2 Δ?2 Δ?= (Δ?)−(?),

where Δ? = ?2 − ?1 and Δ? = ?2 − ?1. Note that this shows that if Δ? > ?Δ?, there is no frame ?′ in which the two events occur at the same point. Note that the interval Δ?′ is the proper time interval. (4)

b) Show that if Δ? > ?Δ?, there is frame of reference ?’ in which the two events occur simultaneously. Find the distance between the two events in ?′; express your result in terms of Δ?, Δ?, and ?. This distance is often referred to as the proper length. (4)

c) Two events are observed in a frame of reference ?′ to occur simultaneously at points separated by a distance of 2.50 m. In a 2nd frame ? moving relative to ?′ along the line joining the points in ?′, the two events appear to be separated by 5.00 m. What is the time interval between the events as measured in ?? (2)

Answer 1