This is the WAC-Corporal liquid-fuel sounding rocket, the U.S.'s first successful sounding rocket. Developed from 1944 at the Jet Propulsion Laboratory, it could lift 25 pounds of instruments to 20 miles. The motor used nitric acid and aniline and produced 1,500 pounds of thrust. The first rocket was launched in 1945.
However, captured German V-2 rockets soon became available that could carry heavier payloads to higher altitudes. The WAC was thus little used. One was placed on top of a V-2, however, as part of the U.S.'s first experimental two-stage liquid propellant rocket series called Project Bumper. One Bumper in 1949 went up to a record 244 miles. This rocket was donated to the Smithsonian by Caltech in 1959.
This object is not on display at the National Air and Space Museum. It is either on loan or in storage.