This Robert Goddard P-series rocket is likely the one that jammed in the launch tower on 10 October 1941 and failed to lift-off. The series was so designated because they contained his propellant pumps. They were also his largest and last liquid-fuel rockets and were tested at Roswell, New Mexico, during 1938-1941.
It is probably the same rocket that was launched twice, 9 Aug. 1940 and 8 May 1941, and repaired after each flight. In both cases, the rocket reached a very low velocity and only 300 ft. (about 90 m) in the first test and 250 ft. (about 75 m) in the second. The tests were discontinued because Goddard moved in 1942 to Annapolis, Maryland, to undertake wartime work for the Navy.
The Daniel and Florence Guggenheim Foundation for the Promotion of Aeronautics donated it to the Smithsonian in 1950 after a nation-wide tour of Goddard rocket artifacts.
This object is on display in Rockets & Missiles at the Steven F. Udvar-Hazy Center in Chantilly, VA.