The Corbomite Maneuver - S1-E11
Revealing mistake: In Kirk's quarters, as he touches the desk top comm panel, it noticeably slides a little rather than being fixed in place.
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.
Revealing mistake: As Kirk fights Finney in the Engine room, in the long shots, Kirk's stunt double is easy to pick out from the close shots.
Tomorrow is Yesterday - S1-E20
Revealing mistake: As the camera looks at the pilot in the cockpit, there are supports on either side of and behind him as well as no clouds where there should be, showing he is on a sound stage set.
What Are Little Girls Made Of? - S1-E8
Revealing mistake: As the big android holds Kirk up by the wall, you can see the skin wig worn by the actor wrinkle up.
A Taste of Armageddon - S1-E24
Revealing mistake: It is easy to tell where the matte painting and live action area meet when Fox and the assistant appear.
Revealing mistake: The shot of the castle on Rigel VII seems to be what Pike sees through his eyes, but in the next shot he himself is in the shot.
Revealing mistake: When the Talosians place Christopher Pike and Vina into the "picnic" illusion (in the countryside on Earth), Pike wanders around marveling at how real it all seems. Well, "real" except for the fact that Pike's body is casting 4 distinct shadows in 4 different directions on the ground, the result of studio set lighting.
The Return of the Archons - S1-E22
Revealing mistake: The rocks pelting Kirk and the landing party when all hell breaks loose bounce when hitting the actors, the brick walls, and the concrete sidewalk.
Revealing mistake: As Trelane takes a swipe at Kirk, he hits a tree branch. As it breaks away, the building material it was made from is revealed, rather than it being wood.
Revealing mistake: In Trelane's castle, Kirk and Sulu are supposed to be frozen, but they slightly waver.
The Man Trap - S1-E2
Revealing mistake: On the planet, as Kirk and Spock try to get to Professor Crater, Crater fires at and destroys the center support of the arch. As it settles, the parts are clearly hollow. (00:34:21)
Space Seed - S1-E23
Revealing mistake: Just before Kirk and Khan fight in Engineering, Khan bends Kirk's phaser pistol in half, but the small hand phaser on top is still functional.
Revealing mistake: Just before Kirk finds himself in Trelane's "courtroom", look behind Kirk on the bridge above Spock's work station. The viewscreen with the planet on it is actually a piece of paper that's coming away from the wall.
Shore Leave - S1-E16
Revealing mistake: During the scene with Sulu and the samurai, each actor casts three shadows. Plus as the samurai rushes Sulu, the "sky" appears black behind him.
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.