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Are the photoreceptors in our eyes not fast enough to register the fast moving bullet, or is the brain not powerful enough to make sense of something happening that fast?

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It's less a problem of speed and more of raw photon count. Assuming a brightly lit day, the bullet will move so fast that it doesn't reflect enough photons to register against the background. High speed images of bullets usually involve a very bright flash (and other controlled settings) for the camera to pick it up. (Also, a very short flash helps the bullet to appear stationary in the image.

On the other hand, seeing where your bullets are going is pretty useful. Therefore tracer bullets were created, which are essentially a pyrotechnic flare that you substitute in every 5th bullet, typically in machine gun ammunition. Proof that if you make a bullet bright enough, you can register it just fine.

http://en.wikipedia.org/wiki/Tracer_ammunition

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    $\begingroup$ It's not even necessary to use tracers if one is in a shooting range where all the illumination is from behind the shooter and the background is dark. $\endgroup$ – supercat Jan 29 '15 at 18:56

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