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 = 2vt_o

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

1.	The length of a rod as measure by an observer who is at rest relative to it is called the proper length, o. (In this example, B gives the "proper length".)
2.	Other inertial observers will measure improper lengths for the same object
3.	Improper length < proper length, hence the term "length contraction". (This effect is often called the Fitzgerald contraction after the Irish mathematician who predicted it using a different theory.)
4.	This contraction only affects the length of a rod held parallel to the direction of the relative motion.

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 g.

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