Please note: this reply does not bring up the various issues associated with the description of "most powerful man in the world", and assumes a mostly military and economic standpoint. I'd also like to point out that the President isn't truly the most powerful man, as he requires the help of the rest of the government to wield the power of the U.S., especially of Congress, which issues tariffs (if I'm not mistaken) and declarations of war.

I'm not particularly well-versed in early American history, but I feel that it's safe to discount most of the first century of American history- the British Empire was the real global power player at that point. However, there was never a violent usurpation of power between the two, it was a largely peaceful, gradual exchange (this study supports that statement). With this in mind, one can look to World War I and World War II as the U.S. really arriving on the world stage militarily and economically- both wars devastated Europe economically and militarily, but the U.S. left them better off than Europe did, due to the lack of fighting on U.S. soil, when compared to Europe. After WWII, two superpowers emerged: the U.S. and the USSR. Throughout the Cold War, both sides saw large increases in nuclear stockpiles (look at the graph on this page). Furthermore, both superpowers had similar military alliances in place, such as NATO and the Warsaw Pact. The complications of international relations at the time, as well as the various other economic, military, and social events during those times, make it relatively difficult to determine which nation was more "powerful". However, with the dissolution of the USSR in 1991, the situation changed. I would, based on these facts, say that U.S. truly became the most powerful nation sometime around 1991 or 1992.

I've never tried to answer before, but here goes. I would say it started around the time of Teddy Roosevelt's Presidency. The US effectively bullied Colombia into letting America work on the Panama Canal, by fanning the flames of a revolt in modern Panama.

After the Canal's completion, the Great White Fleet made it's voyage more or less around the world (I believe mostly Pacific though). That was a demonstration of the United States' new naval capabilities brought about by the two - ocean access via the Panama Canal.

Fast forward to Wilson's Presidency, the US distinguished itself in WWI as powerful though not the most powerful. Although, we came out of the war in better shape than most of involved Europe. However Wilson's failure to convince his own people of the importance of the League of Nations probably prevented him from taking a more central position in global affairs.

But finally, the end of WWII and the next decade to follow was the final bit of setup. The US ends the war victorious and as the USSR began aggressive expansion of communism the US takes the biggest military role (at least of the democratic side). That said, as the Korean War intensifies and Vietnam begins the Commander in Chief of the Armed Forces has troops all over in a worldwide struggle against communism plus the atomic bomb. Also, the newfound US influence in Japan and West Germany gave added weight to the US's presence in Europe and Asia.

To conclude, it took around 40 years of build up which came to a head in the Cold War.