This launcher tube is was used with one of the larger Panzerfaust (literally Tank Fist) anti-tank grenades, probably the Panzerfaust 30 Gretchen (for Gross = large) (the 30 indicates optimum range in meters, about 100 ft). Thesegrenade launchers were not rockets, as the charge remained in the disposable tube. The Panzerfaust 30 Gretchen was produced in 1943-44.
The Smithsonian's artifact is cut open lengthwise and was acquired from the U.S. Army Ordnance Museum in 1989.
This object is not on display at the National Air and Space Museum. It is either on loan or in storage.