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Spoo said:
Seeing the bullet leave the gun must happen first from anybody's point of veiw. Note that in this case there is a profound difference in using sound and using light to tell when the gun was fired. The speed of sound changes relative to the observer's motion. The speed of light, however, is always the same (by "speed of light" I mean the speed that photons travel at). This is why it's possible to hear the gun being fired after it hits you.
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The
Doppler effect applies to both sound (pitch changes) and light (red shift). It does not change the speed, only the frequency. The speed of sound changes with the rigidness of the medium through which it travels, and
the speed of light changes with the refraction index of the medium through which it travels.
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Now let's say that I start running towards you at the speed of sound just before the gun is fired. I still see the gun go off before the bullet hits you, although I never hear the gun fire.
Finally, let's say I run towards you arbitrarily close to the speed of light. I still see the gun go off first. There is nothing that I can do, as an observer, to witness you being hit by the bullet before the gun goes off.
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These two examples are not orthinagonal. In the sound example, the traveller is accepted as travelling at the speed of sound, but in the light example the traveller is accepted as travelling less than the speed of light. It could be just as valid to switch the roles, and say running towards the target at light speed means you never see the gun go off, but running at some speed arbitrarily close to the speed of sound you still hear the gun go off.