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This would not be possible for a couple of reasons. First let us set up the following scenario. We will shoot a laser from the Earth to the Moon which will take 1.2 seconds to arrive. You will be observing from 240,000 miles away in a spaceship that is at a right angle to the laser beam. This will make the Moon the same apparent size as it is on Earth. In theory, you should see a laser beam begin from Earth and quickly get longer and longer until it reaches the Moon in 1.2 seconds. The problem is that when we see a laser beam, we are not actually seeing the beam itself, only a small part of the beam reflecting off particles in the laser beam's path. Since space is a vacuum, there are no particles for the beam to reflect off, so we see nothing unless the laser beam is pointed directly at us, which in this case is not. The other problem is that when the laser light is reflected toward us from the particles, it is also scattered and would be much too faint to observe from that distance, even with a telescope.

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Q: How far away would a beam of light have to be to be able to watch it travel?
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