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.
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.