Question

The rod A'B' moves with a constant velocity v relative to the rod AB (Fig. 1.91). Both rods have the same proper length l₀ and at the ends of each of them clocks are mounted, which are synchronized pairwise: A with B and A' with B'. Suppose the moment when the clock B' gets opposite the clock A is taken for the beginning of the time count in the reference frames fixed to each of the rods. Determine:

(a) the readings of the clocks B and B' at the moment when they are opposite each other;

(b) the same for the clocks A and A'.

Answer 1

At the instant the picture is taken the coordintes of A, B, A', B' in the rest frame of AB are

 

In this frame the coordinates o f B ' at other times are B': (t, vt, 0, 0). So B ' is opposite to B at time t(B)=l_o/v. In the frame in which B', A' is at rest the time corresponding this is by Lorentz tranformation.