Star Trek: Season 1, Episode 20. "Court Martial"

 Summary:

The Enterprise has been through an ion storm and is in for repairs at Starbase 11. Kirk signs off on a form that stated how Lieutenant Commander Benjamin Finney had been killed, jettisoned in a pod after red alert. His daughter Jaime appears to accuse Kirk of murder. However, Spock's data shows the pod was jettisoned before the red alert, so Kirk is to be court martialed for perjury and negligence.

Analysis with spoilers:

Finney and Kirk didn't get along since Kirk had relieved Finney from watch on the United Starship Republic 1371. After Kirk reprimanded Finney and logged the error for leaving open a circuit atomic matter piles that should have been closed, which could have blown up the ship in five minutes.

Fast forward to the disaster when Finney had his turn at the pod. A weather scan (in space?) showed an ion storm. Kirk issued a yellow alert until ship stresses made Kirk hit red-alert status, and Finney had seconds to respond but failed. However, the computer logged Kirk as ordering the jettison of Finney's pod during the yellow alert.

Since the computer transcript disagrees with Kirk's story, Commodore Stone wants to sweep it under the rug, having Kirk be on a permanent ground assignment, but Kirk insists on a court martial.

Kirk gets doe-eyed for his friend and old lover Areel, a lawyer in the judge's advocate. She advocates Kirk not going to court for the same reasons as Stone offered. Areel turns out to be his prosecutor.

Kirk's defender, Samuel T. Cogley, meanwhile, has moved into Kirk's quarters along with a bunch of books because he doesn't like "homogenized, pasteurized, synthesized" computers, which is good for Kirk since he disagrees with his computer's record.

During the trial, Spock defends his captain, saying that the computer was wrong but does not say why. Dr. McCoy is examined where he is asked that if Finney hates -Kirk, then Kirk may reciprocate that hate.

Spock wins five games in a row against the computer, and he expects that if computers don't make errors, and if he does not either, the best they can get is a draw. Therefore, something is wrong with the machine, and Cogley demands that it be examined. Someone has adjusted the computer and, therefore, the memory of the computer.

This is a rather large leap in logic for Spock since one part of a computer could be changed without affecting another, but this would be the early days of the computer. On the other hand, it's possible that in the future computers are designed to malfunction if they are tampered--a blessing and a curse.

Cogley, quoted below, makes a speech that might have worked well on audiences in his day, but fewer might agree with it today, considering AI.

The only people who could tamper with a computer would be Kirk, Spock, and the records clerk, Finney. Cogley suggests Finney isn't dead but hiding on the ship. So they suggest all crew leave the ship and then listen for unknown heartbeats, as they eliminate the remaining heartbeats with a microphone. Somewhere there's still a heartbeat on B deck.

They seal it and Kirk starts searching. It's unclear why they sent just one man when the ship is losing its orbit. How could one man find another in such a large space in a finite time?

Still, he finds Finney in engineering, but Finney gets the drop, aiming a phaser at Kirk. Finney hates that he was never made captain and blames everyone for that mistake and will make the Enterprise crash from orbit because of it.

Classic Kirk brawl with a torn shirt.

Kirk fixes damages, and the ship returns to orbit.  Unsurprisingly, Cogley takes on Finney's case as it is another man vs. machine case.

Quotes:

1. Commodore Stone: "It's in the transcript. And computer transcripts don't lie"
2. Samuel T. Cogley: "I speak of rights. A machine has none. A man must. My client has the right to face his accuser, and if you do not grant him that right, you have brought us down to the level of the machine. Indeed, you have elevated that machine above us. I ask that my motion be granted. More than that, gentlemen, in the name of humanity, fading in the shadow of the machine, I demand it."

Notes:

1. In the last episode, there was no one to reprogram the computer. In this one, crew members appear capable of that (Finney reprograms the computer and keep records although maybe these two are closely related, but he also knows exactly how to damage a ship to cause it to lose orbit) and engineering repair (Kirk can repair and captain a ship). 
2. Spock's serial number is S179-276SP
3. Spock's commendations: Vulcanian Scientific Legion of Honor, awards of valor, twice decorated by Starfleet Command
4. Personnel officer is an Ensign
5. McCoy gets no serial number (or for some reason, his is not listed in court).
6. McCoy's commendations: Legion of Honor, awards of valor, decorated by Starfleet Surgeons
7. Kirk's serial number is SC937-0176 CEC.
8. Kirk's commendations: Palm Leaf of Axanar Peace Mission, Grankite Order of Tactics, Class of Excellence, Prantares Ribbon of Commendation, Classes first and second, Awards of Valor, -Medal of Honor, Silver Palm with Cluster, Starfleet citation for Conspicuous Gallantry, Karagite Order of Heroism
9. While Kirk has a lot of these, the second to last could actually go either way--positive or negative--depending on the definitions one uses. However, their behavior within the episode suggestions these are all positive awards.