The model rocket seen in Charlie's office (on the table and on a shelf behind his desk) is a semi-scale model of a real rocket. It's a Nike-X (specifically a Nike Zeus B (XLIM-49A)), an Army rocket. The model (identified by the decal markings) was probably made from a model rocket kit produced by Estes in the '70s and '80s (starting in '75) which could actually fly. The launch pad which it is displayed on (on the shelf) is an Estes PortaPad.
When Ed McMahon was announcer on The Tonight Show Starring Johnny Carson (1962), he wanted to show a clip from this movie on the show to help promote the film. When he explained he could not because the clip was damaged when he was a guest on the show hosted by rival talk show host Merv Griffin, his boss Johnny Carson ribbed him a little bit about being on another talk show.
Closing credits: The characters and incidents portrayed and the names used herein are fictitious and any similarity to the name, character or history of any person is entirely coincidental and unintentional.