Where No Man Has Gone Before - S1-E4
Revealing mistake: After the ship enters the galactic barrier, when Kirk orders "navigation on automatic" there are flashes of light, and in a closeup of the first console just to the left of the main viewscreen, we can see the special effects material spread across the panel below the controls, for the fire to burn when it explodes. Then it cuts to the wideshot as a second console (near Spock's station) blows up, but in the background that first console is intact - no fire/damage whatsoever (it explodes again in a following wideshot). (00:10:00)
Where No Man Has Gone Before - S1-E4
Revealing mistake: When Kirk & Spock are watching Gary read the pages on the monitor one after the other, look closely: it's the same two pages back and forth.
Where No Man Has Gone Before - S1-E4
Revealing mistake: The readers in Sick Bay are supposed to be display screens, a type of CRT, for reading material. Mitchell turns his off to rest, but the words are still plainly visible.
Answer: Kirk was getting his physical and Dr. McCoy probably turned off communications, because if he hadn't, Kirk would have left and headed straight for the bridge, leaving McCoy irritated.