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In an exercise I am asked to calculate the time taken for the light to travel the path $ct=\sqrt{L^2-v(\frac{t}{2})^2}\Rightarrow t=2\frac{L}{\sqrt{c^2-v^2}}$. Where the Pythagorean theorem was used to calculate the slanted path the light must travel with $L$ being the length of the vertical path from the first mirror (at the intersection paths) to the second mirror and $vt$ being the distance earth has traveled until the light reaches the second mirror. Now what I don't understand is this: why can we equate the distance of the vertical path(s) to the slanted path(s) with $ct$ being the vertical path since $c$ is only the speed the light is traveling in the vertical path? Isn't the horizontal component of the light's velocity still $v$ if we assume or don't assume the existence of ether?

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As the earth travels along, the light on the vertical path does not just go perpendicular. The photons travel every direction and the ones traveling diagonally will meet the first mirror further down the way. The photons still travel at the speed of light but follow a longer trajectory. The same is true for the photons returning.

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