Let the original length of the rod (in a frame fixed to the rod) be l0. Also, let the length of the scale's unit be 1 as measured in the frame fixed to the rod.Suppose that the rod is moving with velocity v relative to the scale.
In the reference frame fixed to the scale:
As seen from the scale, since the rod is moving it would appear smaller. Its length would be
. Thus, as measured by the scale, the length of rod will be
In the reference frame fixed to the rod:
As seen from the frame fixed to the rod, the rod is stationary and has a length of l0 , however, the scale is moving with a velocity v. Thus, each unit of the scale will shrink and will be
. Thus, when measured using the scale the measured length will be,
From (1) and (2) we have,
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