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Physics/Michelson morley experiment

Question
We know that velocity of light is c in absolute frame. In the michelson morley experiment , the velocity of light that travelled along the direction of velocity of earth (v) was taken to be c'=c-v  which  is velocity with respect to earth's reference frame. while the light ray perpendicular to motion of earth was just taken to be c , whereas it should have been 'root under c^2-v^2 , if we added the vectors properly. why it is then taken to be only c instead ?

When an object is in motion (let's call it the x-direction) and the medium it's traveling within is in motion perpendicular to the x-axis (let's say the y-direction), then the velocity of the object along the x-axis is unchanged. The DISTANCE it has to travel is greater than if the medium was not in motion, and its TOTAL velocity is greater, but its velocity ALONG THE X-AXIS is unchanged. The effect is that there is a shift in the grand total time for that object to travel two lengths perpendicular to the medium, as opposed the time for to one length with the medium and one length against the medium.

This URL
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/mmhist.html
sums it up pretty well.

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