Operation -- Annihilate! - S1-E30
Revealing mistake: When Kirk and company beam down, the trees are brown. In the next shot, the trees are green. Most probably a different camera lens was used for the shots. (00:07:10)
Revealing mistake: As the Enterprise passes by the camera in the opening, as Kirk does the Log entry monologue, it's easy to tell where the studio lighting was positioned.
Revealing mistake: When Spock and "good" Kirk are searching engineering for the "bad" Kirk, there is an obvious cut in the shot just before "bad" Kirk jumps down from his hiding place.
Revealing mistake: When the "pleasant" Kirk and Spock go down to the engineering level to get the "animal" Kirk, they split up and "pleasant" Kirk meets the "animal" face to face. As he steps out from behind a screen type wall, you can see that the "animal" Kirk is really Shatner's stunt double helping to fill the double role. This can also be told by the size difference when the two are close together and the shot is from behind the double.
The Conscience of the King - S1-E14
Revealing mistake: As Kirk and Lenore leave the cocktail party to walk around outside, they stop to talk. From Kirk's perspective, he is to the left of Lenore, enough to see what should be in view ahead of him. Only seconds later when she leans in to kiss him does he notice Dr. Leighton lying next to some rocks. She was not obscuring his view before. Why didn't Kirk see this while they first started talking? (00:10:36)
Shore Leave - S1-E16
Revealing mistake: After Rodriguez and Angela run from the strafing plane, they end up stopping next to a tree. Angela falls to the ground without Rodriguez knowing that she was hit by the gunfire. He continues to watch the plane disappear with his head looking upwards. When his head comes back down, rather than looking towards her to talk (where her face would be if she was standing), his eyes immediately look down at her on the ground...as if he'd been expecting that she'd be lying there all along. (00:34:01)
Revealing mistake: When first seen on Gothos, the "frozen" Kirk's eyes blink.
Space Seed - S1-E23
Revealing mistake: During the fight sequence between Kirk and Khan in the engine room, you can see in some shots that it is not William Shatner and Ricardo Montalban fighting, but their stunt doubles.
Shore Leave - S1-E16
Revealing mistake: When Yeoman Barrow places her tunic on a shrub while changing, automobiles can be seen in the background through the brush.
Revealing mistake: After Scotty finally locates Seven on the gantry while he's tampering with the rocket, it cuts to a quick shot of Kirk and Spock, then in the following shot of Roberta when she turns the vault's combination locks, the stunt wire is visible behind her as it moves up to turn the door's wheel handle. (00:35:25)
Revealing mistake: During the fight between Kirk and Tracey, the outline of the knee pads under Kirk's pants are visible, particularly while Kirk is on the floor with his knees bent.
Revealing mistake: When Abraham Lincoln shoves the Klingon away from him during their fight, the Klingon collides with a heavy boulder which moves several inches at impact.
Revealing mistake: When Kirk holds a golden egg, a seam can be seen on it. (00:43:55)
Revealing mistake: As Kirk and the Gorn battle, they briefly confront each other from opposite sides of a dead tree which has truncated limbs. One of the larger tree limbs on this supposedly uninhabited, tiny world has obviously been cut off with a chain saw or similar cutting instrument.
Revealing mistake: The stuntman playing Latimer is obvious when he falls after being hit in his back with the creature's spear.
Revealing mistake: During the televised fight, in the wide angle shots, you can tell it's Spock's stunt double fighting.
The Empath - S3-E12
Revealing mistake: When the Vian fires his weapon at Kirk after the crew finds Gem, Kirk flies backwards. At this point, Kirk is not William Shatner but a stunt double.
A Piece of the Action - S2-E17
Revealing mistake: The drawers of Krako's desk are on the front side, not on Krako's side where they should be.
Revealing mistake: As Charlie watches the pole lowering into a square hole in the deck, the wood grain of the set piece is visible. The wood has been painted gray/silver to look like metal. (00:06:06)
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.