Tomorrow Is Yesterday by D.C. Fontana
What an odd episode.
So little felt familiar, so I have to wonder if I've seen this one at all, or maybe not since the early 70s.
We open on a USAF plane, which probably made a few people wonder if they had the right show.

We find out the Enterprise has been flung back in time following an encounter with a "black star". I found it interesting that we the episode aired in 67 but refers to the "first manned moon shot," so the episode is set in what was then the near future.
Air Force Captain John Christopher is sent to investigate the UFO that showed up out of nowhere. Kirk orders the tractor beam used, but it crushes Christopher's jet and they beam him aboard. As they show him around, you can see that meeting Spock gives him pause, but I like that he's stays composed and doesn't say anything. Uhura has a nice amused look on her face.
Supposedly, Christopher does nothing of note historically (imagine being told that!), so he can't go home because he could disrupt the timeline if he reports on the Enterprise and the future. Then Spock admits he was wrong (!) and Christopher's not-yet-existing son will be important. He already has kids, but glows at the idea of having a boy. Whatev.
Which brings up the interesting sexism/not-sexism that plagues TOS. Christopher sees a woman on the Enterprise and asks about it.
CHRISTOPHER: A woman?
KIRK: Crewman.
Kirk's reply is cool, but then later, in Kirk's quarters, we get an overly affectionate computer (was that sexy voice Majel or someone else?). It calls Kirk "dear" and giggles (per Spock).
CHRISTOPHER: I take it that a lady computer is not routine.
SPOCK: We put in at Cygnet Fourteen for general repair and maintenance. Cygnet Fourteen is a planet dominated by women. They seemed to feel the ship's computer system lacked a personality. They gave it one. Female, of course.
Oy.
Anyway, Christopher tries and fails to escape, Spock finds out they have to send him home, and they decide to pull all the audio and video so he'll be "simply be one of the thousands who thought he saw a UFO." Christopher is okay with this and even sketches out the base for them. Kirk and Sulu go to get the evidence and get stopped by an armed guard. The guard takes their stuff and accidentally sets off an emergency signal, getting himself beamed to the Enterprise. I got quite a kick out of watching him frozen on the transporter pad, scared to move!
Kirk and Sulu get the audio recordings and go for the film from the plane. Kirk gets captured by security (giving a lovely "I'm so innocent" look), but Sulu escapes back to the ship with the evidence. Christopher forces them to take him to get Kirk back and tries to get left on Earth. Spock downs him with a nerve pinch.
Back on the ship, Scotty and Spock have figured out the slingshot maneuver (used many times after this). They start moving backwards in time and beam Christopher back into his jet at the moment he first saw Enterprise, erasing everything that happened after (and giving me a headache). Enterprise goes back to the future.
I'm not sure about the solution, but it's no worse than the technobabble we got in other episodes.
SPOCK: Logically, as we move faster and faster toward the sun, we'll begin to move backward in time. We'll actually go back beyond yesterday, beyond the point when we first appeared in the sky. Then, breaking free will shoot us forward in time, and we'll transport you back before any of this happened.
How is that logical?
There's a lot of humor in the episode and there are a lot of mishaps, so I wonder if this was supposed to be a comedy or maybe the actors just played it that way. Shatner especially seems to have fun with the Air Force security people.
My favorite line:
KIRK: All right, Colonel. The truth is, I'm a little green man from Alpha Centauri. A beautiful place. You ought to see it.
By the way, per Wikipedia, "It was the first Star Trek episode to be written solely by a woman (Dorothy Fontana had previously written the teleplay for the episode “
Charlie X” but the story was credited to Gene Roddenberry)."