Continuity mistake: In the briefing room, Hengist and the Prefect are sitting in chairs against the wall at least a foot apart, with bright overhead lighting that casts no shadows on the wall behind them. When we cut to a two-shot of them, however, they're suddenly shoulder-to-shoulder with almost no space between them, and the lighting changes to cast very dark shadows behind them. (00:28:20)
Continuity mistake: When Mr Hengist goes on the rampage in the hearing room, Kirk pushes Yeoman Tankris behind him. However, when we see Kirk punching Mr Hengist in the jaw in the next shot, Tankris is missing, and in the shot after that, she is back again.
Continuity mistake: The ID cards Kirk puts into the computer to verify their identity appear/disappear in close/long shots.
Other mistake: In the scene near the end when Kirk orders Kyle to beam Hengist into space, Kirk and Kyle say their lines in the wrong order. Roughly: Kyle: Don't get excited, Captain. I would have done it. Kirk: Spock, you do it.
Visible crew/equipment: After Kirk and Spock place Hengist on the transporter platform to beam out, when McCoy and Scotty join them two white tape marks are visible on the floor. McCoy stands at one tape mark before Kirk begins to describe beaming it out into open space, and Scotty stands at the other tape mark as Kirk realises he'll be going alone to the cafe.
Visible crew/equipment: During Scotty's testimony, when Hengist agrees to the psycho-tricorder analysis of Scotty's memory Kirk tells Scotty he can step down, and the white tape mark is visible on the floor where Kirk stands. (00:28:20)
Revealing mistake: When Mr Hengist goes on the rampage in the hearing room, you can see that the actor playing Hengist is an obvious stunt double wearing a bald cap, whereas John Fiedler has hair on the back and sides.
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.