Factual error: Mr. Spock states that the asteroid is almost as large as the Earth's Moon. Such big bodies must be predominantly spherical due to their own gravity. However, the asteroid is very elongated, with a very irregularly shaped surface. (00:06:40)
Factual error: When Kirk is given the medicine badge, it's a stretchy elastic/Spandex headband. This is supposedly a completely pre-industrial culture paralleling the early Native American tribes. They have no fabric, no yarn, no spinning wheels - only hides and animal hair, neither of which can stretch a la Spandex. (00:14:40)
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.