Revealing mistake: In McCoy's video of the Capellans, the weapon thrown at the sapling actually misses it and nails a nearby bush instead. The tree, however, obediently breaks in half and falls over anyway. (00:00:20)
Revealing mistake: In the tent after Kirk and friends give up their weapons, there is a small fire in the holder in the center of the room. Although the flames are slightly off center to the camera's left, the are projections of flame light from off the camera's right offscreen.
Revealing mistake: When the delegation of locals approach Kirk's party, the Red-shirt yells "KLINGON!" and goes to fire. As he moves, you can see the outline of the block the alien weapon will stick in.
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.