Operation -- Annihilate! - S1-E30
Revealing mistake: While Spock and Kirk discuss the phaser resistant creatures, Kirk declares that it may be a trap and to move out. When one of the flying parasitic creatures attacks Spock, in his first closeup just as Spock turns around we can see the thin wire leading from the creature to the right side of the screen, and the wire also casts a shadow on Spock's uniform. (00:18:10)
Operation -- Annihilate! - S1-E30
Revealing mistake: When Kirk and company beam down, the trees are brown. In the next shot, the trees are green. Most probably a different camera lens was used for the shots. (00:07:10)
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.