Okay, I'll just take this apart piece by piece and sell it on as the scrap it is.
(here's guessing this board supports UBB code)
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1. The tremendous height of the Saturn V launch vehicle was attributed to the amount of fuel needed to escape Earth's gravity. Given the fact that the moon's gravity is 1/6th that of the Earth, you'd need something 1/6th the height of the Saturn V to blast off from the moon. So how did the Eagle lift off?
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Bear in mind that one extra pound of weight means 40lb of fuel. More fuel is more weight.
Earth is bigger than the moon. Eagle had to reach Lunar Orbit. SaturnV had to reach Earth orbit (and cope with atmospheric drag) and also then ESCAPE from Earth orbit. Eagle did not.
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2. Once achieving escape velocity when leaving the Earth, the Apollo would need to lose 5/6th of its velocity in order to be captured by the moon's gravity and swing into orbit. The spacecraft is moving at a velocity that prevents the Earth's gravity from pulling it back down - so unless they "put on the brakes" they certainly aren't going to be captured by the moon's gravity. However, all three stages of the Saturn V had been dropped - only the LEM remained.
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LEM + Command Module + Service Module.
Recap basic Newtonian physics. You're making these up and claiming false authority.
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3. How do humans survive at the top of a rocket? The space shuttle is fairly well isolated from the heat of its boosters, but no so with Saturn and its command module. At launch they had to spray lots of water on the launch pad to keep the concrete from cracking. Metal is a great conductor of heat. What kept the astronauts cool? Did they have a zillion air conditioners in the command module?
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What metal? The half a foot of metal that wasn't even solid? And pray explain how the ceramic engine nozzles were going to conduct heat to the metal?
And from a different person:
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4. Since studies estimate that the walls of the Apollo craft would have to be between six to eight inches thick with lead, how did we manage to send our astronauts through the Van Allen radiation belts without them receiving lethal doses of radiation? The Apollo craft walls were merely millimeters thick.
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No studies indicated that. That's a lie. Like I said, this is all made up. The Van Allen belts are thickest at the poles. Apollo escaped pretty close to the equator for this reason. At the poles, they are also more energetic. At the equator, they're mainly beta particles and alpha particles. An inch of aluminium will stop all beta particles. A sheet of paper does the same to alpha particles. Apollo had not millimeters (how can a few mm of metal support it's own weight?) but inches. Like I say, this is all made up.