In short, we can designate v as the relative velocity of the two
systems.
Furthermore, the principle of relativity teaches us that, as judged
from K, the length of a unit measuring-rod which is at rest with
reference to K1 must be exactly the same as the length, as judged from
K', of a unit measuring-rod which is at rest relative to K. In order
to see how the points of the x-axis appear as viewed from K, we only
require to take a " snapshot " of K1 from K; this means that we have
to insert a particular value of t (time of K), e.g. t = 0. For this
value of t we then obtain from the first of the equations (5)
x' = ax
Two points of the x'-axis which are separated by the distance Dx' = I
when measured in the K1 system are thus separated in our instantaneous
photograph by the distance
eq. 34: file eq34.gif
But if the snapshot be taken from K'(t' = 0), and if we eliminate t
from the equations (5), taking into account the expression (6), we
obtain
eq. 35: file eq35.gif
From this we conclude that two points on the x-axis separated by the
distance I (relative to K) will be represented on our snapshot by the
distance
eq.
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