Continuity mistake: The aging Kirk's hairline has receded considerably. When he ages still more and goes completely gray, however, the receding process inexplicably reverses itself, and he suddenly has much more hair on his forehead than before. (00:34:35 - 00:43:55)
Continuity mistake: The Romulan vessels fire plasma weapons (as seen in "Balance of Terror"), but when they hit the Enterprise they become photon torpedoes.
Factual error: As Kirk, McKoy and Scottie age during the episode, their hair goes from brown to fully grey. Mr. Spock shows this to a lesser extent, but by the end of the episode, he has full streaks of grey. This isn't how grey hair works. It starts from the root and works it way outward over time to the end of the strand. Unless their hair is growing quite rapidly, and then cut to keep it short, they shouldn't have full heads of grey hair.
Audio problem: As Commodore Stocker freezes up on the bridge, before the cut to sick bay, the bridge is shaken by a Romulan blast, but the sound is absent.
Continuity mistake: Mr Checkov has a rather unfortunate hair styling during the beginning of his role in Season 2. In this episode his hair is normal until the end, where in between being given an order and carrying it out he somehow manages to go and have his hair professionally done and made all bouffant. (00:47:15)
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.