A Straight Rod Is Moving Along The X Axis Of An Inertial Reference Frame S The Two Endpoints A And B Follow Hyperbolic 1 (176.79 KiB) Viewed 26 times
A straight rod is moving along the x-axis of an inertial reference frame S. The two endpoints A and B follow hyperbolic space-time trajectories, described by the following time dependent x-coordinates in S, 2A = cV t2 + c2/a?, 2b = cV t2 + c2/62, (1) = where c is the speed of light, and a and b are positive constants, with b < a. a) A second inertial frame S' moves along the x-axis with velocity v rela- tive to S. Show that the motion of A and B, when expressed in terms of the coordinates of S', has precisely the same form as in S, x'a = cV t2 + c2/a>, d'b = cV t12 + c2/62. (2) = Hint: In order to demonstrate this it may be convenient to rewrite the above relations in terms of the squared coordinates x2 and t2. b) At time t = 0 the frame S is an instantaneous rest frame of both A and B. Show this and find the distance between A and B measured in S at this moment. The same results are valid for the reference frame S' at time t = 0.
Based on this we may conclude that for any point on the space-time trajectory of A, the instantaneous inertial rest frame of A is a rest frame also for B. Furthermore the distance between A and B, when measured in the instantaneous inertial rest frame, is constant. Explain these conclusions. c) Use the above results to show that the proper accelerations of the A and B are constants, and give the values of these. d) At a given instant t= 0 a light signal with frequency vo is sent from A and is subsequently received at B. What is the velocity of B (measured in S) when the signal is received, and what is the frequency of the signal, measured at B?
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