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Why did apollo 11 go to the moon?
Short answer
Because the United States had publicly committed itself to it. On 25 May 1961 Kennedy asked Congress to back landing a man on the moon and returning him safely to the earth before the decade was out, weeks after Gagarin's flight and the Bay of Pigs; Apollo 11 was the mission that met the deadline.

A decision taken in 1961
Apollo 11 flew because a president had staked American prestige on it eight years earlier. Yuri Gagarin became the first human in space on 12 April 1961, greatly embarrassing the United States, and the Bay of Pigs fiasco in mid-April added pressure of its own. On 25 May Kennedy told Congress: "I believe that this nation should commit itself to achieving the goal, before this decade is out, of landing a man on the moon and returning him safely to the earth." He asked for 531 million dollars in fiscal '62 and an estimated seven to nine billion dollars additional over the next five years.
Why the Moon
The target was picked for being hard enough to matter and open enough to win. Space achievement, Kennedy argued, might hold the key to our future on earth, and the Moon was an area of space exploration in which the United States actually had a potential lead: the rockets and spacecraft needed did not yet exist on either side, so Soviet advantages in existing boosters counted for less. A nearer goal would have been reached by both powers at once; a harder one would have missed the decade.
What the deadline left out
The pledge was about landing and returning safely, not about exploring, and Apollo 11 was built to that wording. The crew stayed 21 hours and 36 minutes on the surface, made one excursion and brought back 21.55 kg of rock and soil — enough to open lunar geology, far less than later crews would collect. Science was real but secondary; the mission existed to settle the question the speech had posed. By the time Eagle landed, the goal had outlived its author and much of the Cold War alarm that produced it.
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