The Gamesters of Triskelion - S2-E16
Revealing mistake: When the camera goes in on a close up of the phaser, when the trigger is pulled, the indicator light to signal a reaction by the intended target that's usually too small to be seen or hidden by animation is visible.
Continuity mistake: When Sulu first emerges on the Bridge, Kirk confronts him, and standing a couple feet from the rapier. A second later, Kirk's a few inches from it.
Other mistake: In the Transporter room, as the Decontamination field is used, the shuttered lighting casts not only on Spock and Joe, but on the walls surrounding the chamber.
Factual error: In the opening sequence, Spock identifies the alien ship as possessing "ion propulsion" which he says is "unique technology." Scotty is similarly impressed and says, "They could teach us a thing or two!" Kirk later comments that "Advanced ion propulsion is beyond even our capabilities." However, even back in the 1960s, ion propulsion was physically feasible, while Warp propulsion was complete fantasy. Ion propulsion of any kind could never even reach lightspeed and would be incredibly primitive compared to Warp technology. In fact, we in the 21st Century have already developed ion propulsion, but it will probably take many hundreds or thousands of years to develop anything even close to Warp technology.
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: In the Enterprise lab, as a sped up Spock goes to move away from McCoy and Chapel, McCoy's eyes follow his movements rather than looking at where Spock was supposed to be after he "disappeared."
Revealing mistake: As Kirk and company exit through the tunnel, the seams of the wall sections are visible.
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.
Continuity mistake: As there were two screen used models of the Enterprise, when the ship is seen flipping from one universe to the other and back, the nacelles of the Imperial Enterprise have the solid red fronts with what appear to be antennae.
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)
Character mistake: In the scene when the shuttle is finally repaired and lifting off the planet, Spock asks "Fuel status?" and Scotty incorrectly answers "15 pounds PSI, sir. Approximately enough for one orbit." The correct answer would be simply "15 PSI" without the redundancy, as the P in PSI already signifies pounds. As an engineer Scotty would know this. (00:42:00)
Factual error: If this planet is a duplicate, an exact copy, of Earth, then Lake Okeechobee, the largest lake in Florida, is wrong. Seen from space, it is much larger and more distinct than the lake shown on this duplicate.
Other mistake: In the opening, the planet looks to be turning the same as Earth does. On the viewer it is turning correctly. However, it is too quickly to be accurate, assuming it is a copy of Earth. If so, at this higher speed, the days would be shorter. Also, the ship should be moving at such a speed as to see the planet moving in the opposite direction.
Continuity mistake: Kirk has a sunburn on his face and chest (it appears to end partway down his chest-most noticeable in the scene shortly after his identity is revealed to the Klingons). First of all, how does one get a sunburn while flying through space (and one that does not match the pattern of the uniform). If one wants to explain it away as being from a recent Shore Leave or Enterprise tanning bed, the issue still remains that the intensity of the sunburn changes as the episode continues.
Deliberate mistake: When the Klingons post signs for their rules, they're printed in English. The Klingons have their own language, and people on Organia most certainly aren't English speakers. Even if the Universal Translator allows Kirk and Spock to hear in English, it wouldn't change the look of a sign.
Revealing mistake: Chekov covers Mara's mouth with his hand when he assaults her in the corridor. Kirk intervenes and throws Chekov against the opposite wall. Brown streaks form where Chekov's hands hit the wall, most likely from the brown makeup used to create Mara's skin tone. (00:32:20)
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.