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A key advance was the invention of the first liquid-fuelled rocket in 1926 by the American Robert Goddard in Auburn, Massachusetts. Goddard is considered to be one of the fathers of the modern rocket era.
Rockets can get you going much faster. The first spacecraft to make the run was the Soviet Luna 1 probe. It completed a flyby in 1959. Luna 1 wasn't trying to get into orbit. It was there to whiz past the Moon and take a few pictures as it went. It completed the journey in only 36 hours, traveling at a speed of 10,500 kilometers per hour.