WARNING: This thread WILL contain unhidden spoilers for this entry. Because this re-watch is in a slightly-subjective chronological order and not in production order, please refrain from talking about the content of other episodes or movies in this thread. If you do, please put that content inside spoiler tags. Some people here may be watching Star Trek for the first time.
#22: Star Trek: The Original Series, season 1, episode 19 "Tomorrow is Yesterday"
Written by D.C. Fontana, directed by Michael O'Herlihy.
Stardates 3113.2 through 3114.1 (October 2267)
(Some events take place during July 1969)
"You're as much a prisoner in time as I am." - Captain John Christopher
An American Air Force base in Omaha scrambles fighters to intercept a UFO that suddenly appeared hovering in the skies overhead. It's the Enterprise. A dark star caused the ship to slingshot backwards through time to 1960s Earth. While the damage to the Enterprise is light and can be easily repaired, one of the fighters makes visual contact with the Enterprise. Captain Kirk orders the fighter to be held in a tractor beam, which causes the fragile aircraft to break apart. Kirk has no choice but to beam its pilot; Captain John Christopher (Roger Perry) aboard and try to explain things to him. In addition to figuring out how to get back to the 23rd Century, the problem now includes how to get Captain Christopher back to Earth and erasing all records of the Enterprise's presence without contaminating the timeline any further than it already has been.
Originally released: 26 January 1967
"Tomorrow is Yesterday" on Memory Alpha
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What did you think?

This is the first real We're Just Having Fun episode. In spite of the serious plot points, the acting and music is so lighthearted. I especially like Kirk just about rolling his eyes while being interrogated by Air Force personnel.
The logic of time travel in this episode is pretty wild, especially the part about beaming people into themselves to erase their memories. But you have to hand it to Star Trek that it remains a scientific constant that slingshotting around a star at warp is a sure-fire, albeit risky, way to time travel.
And on the Remastered version, there's a VERY close fly-by of the Enterprise that lets you see the CG model in great detail. They did such a good job making it look physical and imperfect as though it were built by hand.
I think you put your finger on it-- this isn't a bad ep, per se, but it's hard to take very seriously.
I'd say the idea was improved with the 'Gary 7' episode, but not by all that much. I haven't seen both eps for a long time, and have no real desire to do so.