Continuity mistake: When the ship approaches the galaxy edge Yeoman Smith is standing behind the captain's chair, but when Kirk orders Mitchell to, "Neutralize warp...hold this position," and also "Address intercraft," in Mitchell's closeups Smith has vanished - even in the shadows being cast, though she reappears in the next shot.
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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.
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.
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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.
Continuity mistake: In Janice Rand's quarters, Charlie mentally throws Kirk and Spock against the wall. After they collide with it and slide down the face of the wall, there is a huge indention and crack where Spock hit. After a camera cut to Charlie, when the camera looks at Kirk and Spock again, there is no damage to the wall.
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Continuity mistake: The bundle of clothes Kirk steals from the balcony changes several times while he and Spock face the policeman. The red shirt is not visible, then is hanging down on the right, then isn't. It then disappears from view until they reach Edith's basement, where Kirk puts it on. Two belts, two jackets and Spock's cap also seem to have added themselves to the bundle somehow. They're not there when Kirk first dismantles the tangle of clothes in the basement.
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Character mistake: Spock expresses amazement that the natives of this planet speak Earth English, however, he and Kirk were both on the bridge when Uhura tapped into the planet's news broadcast and already know that they speak Earth English, and there's no reason to tell McCoy, because he heard it too. Commenting on it would be illogical.