This is the Falcon, or GAR-1 (Guided Aircraft Rocket), the first of a large family of U.S. air-to-air solid-fuel guided missiles and the first guided air-to-air missile to enter operational service. GAR-1 was radar-guided. The missile had a range of up to five miles and speed of about Mach 2.8. Designed to destroy enemy bombers, it was carried on the F-89-H, F-102A, and F-106A interceptor aircraft.
Development of the Falcon began in 1947 and it became operational with the U.S. Air Force in 1956. In 1962 GAR-1 was re-designated AIM-4 (Air Interceptor Missile). Several more AIM models later appeared. NASM acquired the missile in 1958 from its developer, the Hughes Aircraft Co.
This object is not on display at the National Air and Space Museum. It is either on loan or in storage.