Revealing mistake: When Kirk enters Spock's quarters at the beginning of the movie, he hears Spock's voice calling out to him from a darkened area. As he walks into Spock's quarters, pay attention to the top right area of the screen. You'll notice the top of the walls and no ceiling. That lets you know that they were filming on a sound stage in a set with no ceiling. (00:14:50)
Factual error: According to "The Doomsday Machine", full impulse drive is one-quarter the speed of light. In the first two movies, Enterprise used thrusters as opposed to impulse drive to leave Spacedock, confirming the notion that impulse drive is far too fast to leave such a (comparatively) small structure. Styles, however, orders Excelsior to one-quarter impulse, which is 18,750 km/s. In one second, she will travel half again Earth's diameter. From the time he gives the order to the time we see Excelsior clear spacedock's doors is approximately 40 seconds. Even allowing 30 seconds to go from rest to one quarter impulse, spacedock must be 13-15 times bigger than Earth! That's some serious engineering. (00:23:45)Grumpy Scot
Captain Spock: My father says that you have been my friend. You came back for me.
Kirk: You would have done the same for me.
Captain Spock: Why would you do this?
Kirk: Because the needs of the one... Outweigh the needs of the many.
Captain Spock: [begins to remember] I have been and ever shall be your friend.
Kirk: Yes. Yes, Spock.
Captain Spock: The ship... Out of danger?
Kirk: You saved the ship. You saved us all. Don't you remember?
Captain Spock: Jim... Your name is Jim.
Trivia: At the beginning of the movie Kirk rescues McCoy from a jail of some sort. Before leaving Kirk asks McCoy how many fingers he's holding up and does the Vulcan hand thing. However, Shatner had severe difficulties putting his fingers into place, they just wouldn't hold into position. So the crew wound up wrapping fishing line around Shatner's fingers, he would put them into position out of the camera and the shot quickly jumped while his fingers where still in place. If you look closely you see the line around his fingers. (Note that this problem occurred several times during Star Trek, actually, Nimoy and Leonard seem to be the only people doing this without a problem.)
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