![]() ![]() ![]() So we have 2 d 1 over v 1 plus 2 d over c minus 2 d 1 over c minus 2 d over c all over 2 d over c times 100 percent and these are equal and opposite signs so they make zero and we can factor out the 2 d 1 from this term and this term so we have 2 d 1 times 1 over v 1 minus 1 over c all over 2 d over c. And then multiply both terms by 2 and then separate this into 2 terms by dividing both terms in the numerator by c. Now d 2 is the total distance from the Earth to the Moon minus the thickness of the atmosphere so we can replace d 2 with that and we do that here. So t c is replaced by 2 times the Earth-Moon distance divided by c- this term, and down here as well- and this time here is what's actually going to be measured and it's 2 times the atmosphere thickness divided by the speed through the atmosphere plus the top of atmosphere to the Moon distance divided by c. So the time if there's no atmosphere affecting anything the time at the speed of light would be 2 times the distance from the Earth to the Moon divided by c so we can replace a whole bunch of things in this step here. So t 1 is this distance d 1- the thickness of the atmosphere-divided by the speed of light through the air v 1 and time two is this distance from the top of the atmosphere to the Moon divided by the speed of light in a vacuum and so we can replace each of the time terms with those fractions and the percent error then is going to be the difference in time between this versus what would be measured if the light went at the speed c the entire time divided by the time at speed c times 100 percent. Well let's suppose that we know the exact distance to the Moon is 3.84 times 10 to the 8 meters and we know that the height of the atmosphere is 30.0 kilometers, which is 30.0 times 10 to the 3 meters and the atmosphere has an index of refraction, n 1 which is 1.000293 if the people measuring this time make the assumption that the speed of light goes at the speed c, which is the speed of light in a vacuum, they will not be accounting for this portion of the trip when light is actually going slower through the air and so we want to figure out what percent error needs to be taken into account to account for this slowness through the air? Okay! So the total time it takes the light to make a round trip is 2 times the time it takes to get through the air, which we'll call t 1, plus the time it takes to get from the top of the atmosphere through empty space, which is a vacuum to the Moon. A laser is shining from the Earth onto a mirror mounted on the Moon by some astronauts and then the light is bouncing back to the Earth and being detected sometime later and the time it takes for the light to make this round trip is being measured and used to figure out the distance to the Moon. This is College Physics Answers with Shaun Dychko.
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