Final answer:
The use of the second atomic bomb, "Fat Man," on Nagasaki was spurred by the potential high American casualties from an invasion, the possibility of speeding up Japan's unconditional surrender, and political motivations against the Soviet Union, marking early tensions of the Cold War.
Step-by-step explanation:
President Truman authorized the use of the second atomic bomb, "Fat Man," on Nagasaki for several key reasons. The intense resistance from Japanese forces and a projected high number of American casualties from a potential invasion influenced this devastating decision. An invasion of the Japanese home islands was estimated to cost up to 250,000 American lives. Moreover, Truman possibly sought to leverage the atomic bomb as a political tool against the Soviet Union's growing intransigence in postwar Europe, signaling the onset of the Cold War.
The United States had already destroyed substantial parts of Japanese cities through firebombing, with the recent bombing of Tokyo killing as many as both atomic bombings combined. There was a belief that only the direct military use of the bomb could compel Japan to surrender unconditionally. It is also noted that Truman was not initially aware of the Manhattan Project or the potential of the atomic bomb's power until Roosevelt's death. After the bombing of Hiroshima, there was minimal time given before the second bomb was dropped, which some historians argue could have been a move to assert geopolitical power.
Despite moral controversies, the notion of using the bomb militarily underpinned the extensive investments and efforts of the Manhattan Project since its inception. The decision to deploy these nuclear weapons was made with the apparent goal of swiftly ending World War II and was bolstered by the realization of heavy Allied casualties should a conventional invasion occur. Ultimately, the use of atomic weapons on Hiroshima and Nagasaki led to Japan's surrender and the end of the Second World War.