General Leslie Groves, head of the U.S. Armys Manhattan Project which produced the atomic bombs, had predicted it would take two bombs to get the Japanese to surrender one to stun them and a second to demonstrate the first wasnt a fluke or a one-of-a-kind. As of August 9, 1945, General Groves was right. Hiroshima had been destroyed by an atomic bomb, and it had apparently made no difference.
A second mission had been planned for August 11, but an incoming weather system made the planners decide to move it up by two days. This mission would use the Fat Man atomic bomb, the same type as was tested in the Trinity Test in New Mexico less than a month before. The bomb was carried by the B-29 bomber named Bockscar (sometimes written as Bock's Car), piloted by Major Charles Sweeney. The bombing mission's target was Kokura; it only went to its secondary target, Nagasaki, because the target point could not be positively identified through Kokura's smoke and haze.
This seems like a clear application of the concept that overwhelming force is the quickest path to peace, and the way to save the most lives. But, as is often the case, theres more to the story than that and it provides further support for this concept.
Japanese physicists were involved in nuclear studies in the 1930s, just as European and American physicists were. By 1940, the Japanese had determined that they had access to more than enough uranium in Korea and Burma to make an atomic bomb. An atomic bomb project was started in April 1941, but the Japanese project determined by late 1944 that it could not produce a bomb in time to affect the war.
The knowledge they built up during their atomic bomb project was put to use in August of 1945. The story is told that physicists sampled the debris after the Hiroshima bombing, and reported that the city had been destroyed by an atomic bomb built of uranium. To the Japanese authorities, that meant it was probably one-of-a-kind because they knew uranium was so difficult to enrich sufficiently that they cant possibly have another. Given that, the Nagasaki bombing surprised them. They sampled debris again in Nagasaki, as they had in Hiroshima, and reported that a plutonium bomb had been used. This was a shock to the authorities, because it meant to them that the U.S. could have a nearly unlimited number of such bombs, depending on a production rate they had no way to know. (The next plutonium bomb was already on its way to the B-29 base on Tinian Island; the planners had reportedly targeted Tokyo for this bomb.) Suddenly the choice the Japanese authorities faced was very stark, indeed surrender or incineration.
Whether you think General Groves was right, and it would have taken the use of two atomic bombs to bring about a surrender regardless of their type, or whether you think the key factor was the realization that there could be more such bombs on the way whichever view you hold, it seems clear that two atomic bombs were necessary to end the war without a ground invasion of the Japanese home islands. Overwhelming force proved to be the quickest path to peace.
Today is the anniversary of Bockscar's flight to Kokura and Nagasaki, the anniversary of the day World War II in the Pacific Theater really began to end.
More information on the atomic bombs and their use in 1945 to end World War II can be found in Nuclear History, as well as in a number of other sources. Separate parts of the Nuclear History article deal with Hiroshima, Nagasaki, and the earlier Trinity nuclear test in New Mexico.