but remember in TOS no one knows about time travel,
Um, what?
Captain's log. Using the lightspeed breakaway factor, the Enterprise has moved back through time to the twentieth century. We are now in extended orbit around Earth, using our ship's deflector shields to remain unobserved. Our mission, historical research. We are monitoring Earth communications to find out how our planet survived desperate problems in the year 1968. (~Assignment: Earth)
And it was discovered early on in the show:
SULU: Captain, my velocity gauge is off the scale.
SPOCK: Engine power went off the scale as well. We're now travelling faster than is possible for normal space.
KIRK: Checked elapsed time, Mister Sulu.
SULU: My chronometer's running backwards, sir.
KIRK: Time warp. We're going backward in time. Helm, begin reversing power. Slowly.
SULU: Helm answering, sir. Power reversing.
SPOCK: We're back to normal time, Captain.
KIRK: Engines ahead. Warp one.
SULU: Warp one, sir.
KIRK: Mister Spock.
SPOCK: Yes, sir.
KIRK: The time warp. What did it do to us?
SPOCK: We've regressed in time seventy one hours. It is now three days ago, Captain. We have three days to live over again.
KIRK: Not those last three days.
SPOCK: This does open some intriguing prospects, Captain. Since the formula worked, we can go back in time, to any planet, any era.
KIRK: We may risk it someday, Mister Spock. Resume course to our next destination, Mister Sulu.
SULU: Course laid in, sir.
KIRK: Steady as she goes. (~The Naked Time).