In his address to Congress on 25th of May 1961, John F Kennedy committed for the first time to land a man on the Moon and bring him safely back before the end of the decade.

But the feasibility studies for the original Apollo program had ended less than two weeks before, and those where done for a project of just orbiting the moon. As an example, General Electric estimated a Moon Orbit flight could be done in 1968.

I don't think you can safely conclude from those studies that it would take roughly one more year to add the landing segment to the mission, so where did Kennedy get his end of decade deadline?

• Shades of the Millennium debate: When did the referenced decade end? – DJohnM Jun 2 '18 at 19:41

Actually, then-NASA Administrator James Webb had already in March requested a budget to achieve a moon landing by the end of the decade. Kennedy's administration had declined to support this budget until Yuri Gagarin became the first human to orbit the Earth in April. The shock of this event forced Kennedy's hand, but there was some substantial planning that occurred prior to his announcement:

Firm evidence for Kennedy's essential unwillingness to commit to an aggressive space program came in March 1961 when the NASA Administrator, James E. Webb, submitted a request that greatly expanded his agency's fiscal year 1962 budget so as to permit a Moon landing before the end of the decade. While the Apollo lunar landing program had existed as a longterm goal of NASA during the Eisenhower administration, Webb proposed greatly expanding and accelerating it. Kennedy's budget director, David E. Bell, objected to this large increase and debated Webb on the merits of an accelerated lunar landing program. In the end the president was unwilling to obligate the nation to a much bigger and more costly space program. Instead, in good political fashion, he approved a modest increase in the NASA budget to allow for development of the big launch vehicles that would eventually be required to support a Moon landing.

A slow and deliberate pace might have remained the standard for the U.S. civil space effort had not two important events happened that forced Kennedy to act. The Soviet Union's space effort counted coup on the United States one more time not long after the new president took office. On 12 April 1961 Soviet Cosmonaut Yuri Gagarin became the first human in space with a one- orbit mission aboard the spacecraft Vostok 1. The chance to place a human in space before the Soviets did so had now been lost. The great success of that feat made the gregarious Gagarin a global hero, and he was an effective spokesman for the Soviet Union until his death in 1967 from an unfortunate aircraft accident. It was only a salve on an open wound, therefore, when Alan Shepard became the first American in space during a 15-minute suborbital flight on 5 May 1961 by riding a Redstone booster in his Freedom 7 Mercury spacecraft.

Close in the wake of the Gagarin achievement, the Kennedy Administration suffered another devastating blow in the Cold War that contributed to the sense that action had to be taken. Between 15 and 19 April 1961 the administration supported the abortive Bay of Pigs invasion of Cuba designed to overthrow Castro. Executed by anti-Castro Cuban refugees armed and trained by the CIA, the invasion was a debacle almost from the beginning. It was predicated on an assumption that the Cuban people would rise up to welcome the invaders and when that proved to be false, the attack could not succeed. American backing of the invasion was a great embarrassment both to Kennedy personally and to his administration. It damaged U.S. relations with foreign nations enormously, and made the communist world look all the more invincible.

While the Bay of Pigs invasion was never mentioned explicitly as a reason for stepping up U.S. efforts in space, the international situation certainly played a role as Kennedy scrambled to recover a measure of national dignity. Wiesner reflected, "I don't think anyone can measure it, but I'm sure it [the invasion] had an impact. I think the President felt some pressure to get something else in the foreground." T. Keith Glennan, NASA Administrator under Eisenhower, immediately linked the invasion and the Gagarin flight together as the seminal events leading to Kennedy's announcement of the Apollo decision. He confided in his diary that "In the aftermath of that [Bay of Pigs] fiasco, and because of the successful orbiting of astronauts by the Soviet Union, it is my opinion that Mr. Kennedy asked for a reevaluation of the nation's space program."

Two days after the Gagarin flight on 12 April, Kennedy discussed once again the possibility of a lunar landing program with Webb, but the NASA head's conservative estimates of a cost of more than \$20 billion for the project was too steep and Kennedy delayed making a decision. A week later, at the time of the Bay of Pigs invasion, Kennedy called Johnson, who headed the National Aeronautics and Space Council, to the White House to discuss strategy for catching up with the Soviets in space. Johnson agreed to take the matter up with the Space Council and to recommend a course of action. It is likely that one of the explicit programs that Kennedy asked Johnson to consider was a lunar landing program, for the next day, 20 April 1961, he followed up with a memorandum to Johnson raising fundamental questions about the project. In particular, Kennedy asked Do we have a chance of beating the Soviets by putting a laboratory in space, or by a trip around the moon, or by a rocket to go to the moon and back with a man? Is there any other space program that promises dramatic results in which we could win? While he waited for the results of Johnson's investigation, this memo made it clear that Kennedy had a pretty good idea of what he wanted to do in space. He confided in a press conference on 21 April that he was leaning toward committing the nation to a large- scale project to land Americans on the Moon. "If we can get to the moon before the Russians, then we should," he said, adding that he had asked his vice president to review options for the space program. This was the first and last time that Kennedy said anything in public about a lunar landing program until he officially unveiled the plan. It is also clear that Kennedy approached the lunar landing effort essentially as a response to the competition between the U.S. and the U.S.S.R. For Kennedy the Moon landing program, conducted in the tense Cold War environment of the early 1960s, was a strategic decision directed toward advancing the far-flung interests of the United States in the international arena. It aimed toward recapturing the prestige that the nation had lost as a result of Soviet successes and U.S. failures. It was, as political scientist John M. Logsdon has suggested, "one of the last major political acts of the Cold War. The Moon Project was chosen to symbolize U.S. strength in the head-to-head global competition with the Soviet Union." Lyndon Johnson probably understood these circumstances very well, and for the next two weeks his Space Council diligently considered, among other possibilities, a lunar landing before the Soviets. As early as 22 April, NASA's Deputy Administrator Hugh L. Dryden had responded to a request for information from the National Aeronautics and Space Council about a Moon program by writing that there was "a chance for the U.S. to be the first to land a man on the moon and return him to earth if a determined national effort is made." He added that the earliest this feat could be accomplished was 1967, but that to do so would cost about \$33 billion dollars, a figure \\$10 billion more than the whole projected NASA budget for the next ten years. A week later Wernher von Braun, director of NASA's George C. Marshall Space Flight Center at Huntsville, Alabama, and head of the big booster program needed for the lunar effort, responded to a similar request for information from Johnson. He told the vice president that "we have a sporting chance of sending a 3-man crew around the moon ahead of the Soviets" and "an excellent chance of beating the Soviets to the first landing of a crew on the moon (including return capability, of course.)" He added that "with an all-out crash program" the U.S. could achieve a landing by 1967 or 1968.

After gaining these technical opinions, Johnson began to poll political leaders for their sense of the propriety of committing the nation to an accelerated space program with Project Apollo as its centerpiece. He brought in Senators Robert Kerr (D-OK) and Styles Bridges (R-NH) and spoke with several Representatives to ascertain if they were willing to support an accelerated space program. While only a few were hesitant, Robert Kerr worked to allay their concerns. He called on James Webb, who had worked for his business conglomerate during the 1950s, to give him a straight answer about the project's feasibility. Kerr told his congressional colleagues that Webb was enthusiastic about the program and "that if Jim Webb says we can a land a man on the moon and bring him safely home, then it can be done." This endorsement secured considerable political support for the lunar project. Johnson also met with several businessmen and representatives from the aerospace industry and other government agencies to ascertain the consensus of support for a new space initiative. Most of them also expressed support.

Air Force General Bernard A. Schriever, commander of the Air Force Systems Command that developed new technologies, expressed the sentiment of many people by suggesting that an accelerated lunar landing effort "would put a focus on our space program." He believed it was important for the U.S. to build international prestige and that the return was more than worth the price to be paid. Secretary of State Dean Rusk, a member of the Space Council, was also a supporter of the initiative because of the Soviet Union's image in the world. He wrote to the Senate Space Committee a little later that "We must respond to their conditions; otherwise we risk a basic misunderstanding on the part of the uncommitted countries, the Soviet Union, and possibly our allies concerning the direction in which power is moving and where long-term advantage lies." It was clear early in these deliberations that Johnson was in favor of an expanded space program in general and a maximum effort to land an astronaut on the Moon. Whenever he heard reservations Johnson used his forceful personality to persuade. "Now," he asked, "would you rather have us be a second-rate nation or should we spend a little money?"

In an interim report to the president on 28 April 1961, Johnson concluded that "The U.S. can, if it will, firm up its objectives and employ its resources with a reasonable chance of attaining world leadership in space during this decade," and recommended committing the nation to a lunar landing. In this exercise Johnson had built, as Kennedy had wanted, a strong justification for undertaking Project Apollo but he had also moved on to develop a greater consensus for the objective among key government and business leaders.

[...]

Johnson asked NASA to provide for him a set of specific recommendations on how a scientifically-viable Project Apollo, would be accomplished by the end of the decade. What emerged was a comprehensive space policy planning document that had the lunar landing as its centerpiece but that attached several ancillary funding items to enhance the program's scientific value and advance space exploration on a broad front:

1. Spacecraft and boosters for the human flight to the Moon.

2. Scientific satellite probes to survey the Moon.

3. A nuclear rocket.

4. Satellites for global communications.

5. Satellites for weather observation.

6. Scientific projects for Apollo landings.

Johnson accepted these recommendations and passed them to Kennedy who approved the overall plan.

The last major area of concern was the timing for the Moon landing. The original NASA estimates had given a target date of 1967, but as the project became more crystallized agency leaders recommended not committing to such a strict deadline. James Webb, realizing the problems associated with meeting target dates based on NASA's experience in space flight, suggested that the president commit to a landing by the end of the decade, giving the agency another two years to solve any problems that might arise. The White House accepted this proposal.

President Kennedy unveiled the commitment to execute Project Apollo on 25 May 1961 in a speech on "Urgent National Needs," billed as a second State of the Union message. He told Congress that the U.S. faced extraordinary challenges and needed to respond extraordinarily. In announcing the lunar landing commitment he said:

If we are to win the battle that is going on around the world between freedom and tyranny, if we are to win the battle for men's minds, the dramatic achievements in space which occurred in recent weeks should have made clear to us all, as did the Sputnik in 1957, the impact of this adventure on the minds of men everywhere who are attempting to make a determination of which road they should take. . . . We go into space because whatever mankind must undertake, free men must fully share.

Then he added: "I believe this Nation should commitment [sic] itself to achieving the goal, before this decade is out, of landing a man on the moon and returning him safely to earth. No single space project in this period will be more impressive to mankind, or more important for the long-range exploration of space; and none will be so difficult or expensive to accomplish."

Source:

Although called2voyage's answer is entirely correct, it omits the immediate source of the "before the end of this decade" language.

That phrase came from a report written for Vice-President Johnson by Secretary of Defense Robert McNamara and NASA Administrator James E. Webb. It was titled "Recommendations for Our National Space Program: Changes, Policies, Goals" and dated May 8, 1961.

Johnson was the chairman of the National Space Council, an organization of senior government officials established by the same law that created NASA itself. After the Gagarin launch less than a month earlier, Kennedy asked Johnson and the Council to identify any "space program which promises dramatic results in which we could win."

https://www.visitthecapitol.gov/exhibitions/artifact/memo-president-john-f-kennedy-vice-president-lyndon-johnson-april-20-1961

In the report, McNamara and Webb said "We recommend that our National Space Plan include the objective of manned lunar exploration before the end of this decade" and asked Johnson to pass this recommendation along to President Kennedy, which he did that very same day, May 8.

https://history.nasa.gov/SP-4106/app-a.htm

After further meetings and discussions, Kennedy announced the plan in an address to Congress and the nation on May 25, 1961.

https://history.nasa.gov/SP-350/ch-2-1.html

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It was a political calculation. NASA's lunar agenda can be traced back to Wernher von Braun's dream, influenced by Hermann Oberth and ultimately Jules Verne. Von Braun's work on the German V2 was at least partially in service to that dream. As a result, the requirements of a lunar landing would have been understood well enough by 1960 or so for NASA to make estimates as to when it could be achieved. "The end of the decade" deadline would have been based on estimated US and Soviet capability - a timeframe believed to be beyond Soviet capability, but likely achievable by the US if given adequate priority and budgetary support. The Soviets did have their own active lunar program during the 1960s and only abandoned it when it was clear that the race was lost. If the American deadline had been set later or with any less priority, the Soviets could well have gotten there first, a major blow to American pride.