Tomorrow is Yesterday - S1-E20
Stupidity: The Enterprise is very concerned about John Christopher going back to earth and changing history, but at the same time he has free access to go anywhere in the ship, and all his questions are being answered.
Stupidity: When Sulu and the others were freezing on the planet's surface, a transporter malfunction interfered with their rescue. Kirk, Spock, etc., did not consider using the Enterprise's shuttle which could have flown to the planet's surface and rescued Sulu and the away crew. (00:30:00 - 00:35:00)
Space Seed - S1-E23
Stupidity: In sick bay, Khan wants to read the Enterprise's technical manuals, explaining to Kirk he was once an engineer. Kirk acquiesces, saying they are available to ANY patient on the viewing screen. Apparently Captain Kirk does not seem to mind total strangers (this time with Napoleonic complexes) reading detailed documents outlining the strengths and weaknesses of a Federation starship.
Chosen answer: If they're in orbit, they're being pulled along by the planet's gravity well, therefore, impulse engines would only be used for minor corrections and would be "on standby" while in orbit, but not active. (Like keeping your car idling without revving the engine and creating plumes of exhaust).
Captain Defenestrator
Thank you for the info.
Movie Nut