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This is a funny question, but worth answering. The distance between the moon and the Earth is 384,400 km. The speed of light is 299792.458 km/s. It will take 1.3 seconds (Approx.) for my laser beam to reach the moon. Will I be able to see the spot of my laser beam over the moon?

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The XKCD webcomic went into some detail about this subject here. –  Nic Mar 20 '13 at 12:00
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Absolutely not. Even though a laser beam is very narrow, it does spread out, and by the time it reaches the moon it will be a couple of hundred meters wide. The moon is not very reflective (with an albedo it only reflects 1/8th of the light falling on it). Thet reflected light is spread out further by the roughness of the lunar surface so that, by the time it arrives back on earth, it probably covers the entire earth, and the amount of light falling on your eye is minuscule.

The Apollo astronauts left special reflectors on the moon. Scientists used those to reflect powerful laser beams and measure the distance to the moon with very great accuracy, but your laser does not have anywhere near enough power for that.

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Not with the human eye, but with a powerful laser, reflectors on the moon and a telescope then yes, you can indeed reflect light off the moon and detect it.

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