This is the U.S. Navy's solid-fuel Subroc (Submarine Rocket), the first guided missile capable of underwater launch, guided airborne trajectory, and underwater detonation. It was therefore very complex and underwent an extensive development program from 1955 until it became operational in 1965. Subroc could either be used as an underwater-to-air, underwater-to-underwater, or surface-to-underwater weapon and carried a nuclear warhead.
It was fired from a standard torpedo tube, then rose to the surface, flew for 25-50 miles before re-entering the water with its rocket motor already ejected, then homed in on an enemy submarine. It went out of service in 1987. This object was donated to the Smithsonian in 1966 by the U.S. Naval Ordnance Lab.
This object is on display in Rockets & Missiles at the Steven F. Udvar-Hazy Center in Chantilly, VA.