According to the theory of special relativity, the speed of light in a vacuum is an absolute constant, denoted by "c," which is approximately 299,792,458 meters per second. This speed is independent of the motion of the source or the observer.
In your scenario, if you have a tube moving at 1/10th the speed of light (0.1c) and you fire a photon from the back to the front end of the tube, the photon will still travel at the speed of light (c) relative to the tube. This is because the speed of light is always measured as c regardless of the motion of the source or the observer.
From the perspective of an observer inside the tube, the photon will appear to be moving at the speed of light in that reference frame, which is consistent with the fundamental postulate of special relativity. The observer inside the tube would not measure the photon's speed as exceeding the speed of light.
Special relativity predicts that velocities do not simply add up linearly. When velocities approach the speed of light, the addition of velocities follows a relativistic formula. In this case, the addition of 0.1c (the velocity of the tube) and c (the velocity of light relative to the tube) does not result in a sum of 1.1c. Instead, the relativistic addition of velocities yields a velocity that is still equal to c.
In summary, the photon would travel at the speed of light relative to the tube, not exceeding the speed of light. The addition of velocities in relativistic scenarios does not lead to velocities greater than the speed of light.