On May 5, 1961, 37-year-old Alan Shepard blasted off from Cape Canaveral on a historic flight that marked the moment the U.S. caught up to Russia in the Space Race. Here, rare and never-seen photographs from LIFE of that amazing era.

President John Kennedy pins NASA's Distinguished Service Medal on Shepard's chest, May 8, 1961. Two weeks later, JFK famously declared: "I believe that this nation should commit itself to achieving the goal, before this decade is out, of landing a man on the moon and returning him safely to the Earth."...