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A person moving in the same frame as the moving mirrors would not be able to observe any change in the time it took the photon to travel between the two mirrors. It would not matter if the direction of travel was parallel or perpindicular to the motion of the photon, and this null effect would continue even if the observer was travelling at 99.9999% of the speed of light.

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Q: What will happen if a photon is bouncing in between two mirrors and the apparatus is moving gradually in straight line approaching the speed of light.?
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