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A and B have relative velocity v as before but now B holds a rod in a direction parallel to the direction of v.
End 1 of the rod passes A at the instant when the flash of light is sent towards the mirror.
End 2 of the rod passes A at the instant when the reflected light returns to the torch.
From A’s point of view the situation is
i) light leaves the torch
A will therefore conclude that the length of the rod is given by = 2vto
From B’s point of view the situation is
So, using the same "light beam clock" observer B measures the length of the same rod to be where
o = 2vt
and, using the time dilation formula, we have
You have probably noticed that the factor (1 - v²/c²)-1/2 appears very often in relativistic equations. To save time, it is usually abbreviated by the letter .