This is the famous Voyage to the Moon Ride at Story Land in Glen NH. Sadly the ride closed in 2008 and was converted into a different attraction. There are videos of this ride that exist on youtube and are well worth the watch.
The history of this ride are as follows:
"After a year of study and preparation, Story Land announced in 1981 that it would add a new, original, state-of-the-art dark ride. It would be housed in a fabric-covered dome 50 feet (15 m) tall and 124 feet (38 m) wide. Bob Morrell had seen a similar structure in a Florida cow field and was determined to add one to the park. The interior of the white fabric roof was painted black, and two of the creative collaborators on Heritage New Hampshire, artisans Peter Stone and David Norton, were hired to create the storyline and scenes inside. The dome took shape in 1981. The skeletal steel of the huge cannon was ready in winter 1982. "Space Fantasy", later renamed "Voyage to the Moon", officially opened in June 1983 after nearly three years of engineering and construction. Governor John Sununu shared the first five-minute voyage with Bob Morrell, riding in one of the bullet-shaped cars built by Bradley and Kaye of California. Each car had speakers to broadcast sound effects triggered by an antenna underneath, computer-synchronized with lighting effects. The cars moved slowly through the cannon on a chain-driven conveyance into the darkened dome, slowly winding back down through a series of fantasy scenes such as a candy land and a polka dot sea, each populated by friendly, animated, imaginary creatures."