This is the Talos ship-to-air missile used by the U.S. Navy from 1957 to 1979, and developed by the Bendix Corporation. The Talos had a ramjet main stage and a first-stage, solid-fuel rocket booster that burned for two seconds, then dropped off after it had accelerated the missile to the high speed necessary for the ramjet to operate. The booster is not shown.
Development of the Talos began about 1945 as part of Project Bumblebee, that led to a family of missiles that included the Terrier, Tartar, and Talos. The Johns Hopkins University conducted most of the research for the Talos. This missile was donated to the Smithsonian in 1982 from the Applied Physics Laboratory of Johns Hopkins University.
This object is on display in Rockets & Missiles at the Steven F. Udvar-Hazy Center in Chantilly, VA.