First, I enjoyed initial premise, even though it was obvious that more was going on. Why are these folks walking around in a gravity field, be it real or artificial? Gravity plating? Naw. Acceleration? At one G? For 51 years?
See also: Space travel using constant acceleration
Regardless of how it is achieved, if a propulsion system can produce 1 g of acceleration continuously from departure to destination, then this will be the fastest method of travel. If the propulsion system drives the ship faster and faster for the first half of the journey, then turns around and brakes the craft so that it arrives at the destination at a standstill, this is a constant acceleration journey. This would also have the advantage of producing constant gravity. This is, however, largely unfeasible with current technology because of the difficulty in maintaining acceleration the closer one gets to the speed of light. This is illustrated by the definition of force: F=dp/dt. This is also a part of Newton's second law of motion.
From the planetary observer perspective the ship will appear to steadily accelerate but more slowly as it approaches the speed of light. The ship will be close to the speed of light after about a year of accelerating and remain at that speed until it brakes for the end of the journey.
So obviously something else is happening....
The ship never left Earth. It's a big simulation rig, and the "crew" has been living a lie.
Cool concept... but I just don't buy that they could have pulled it off.
Do we really believe that the crew doesn't understand basic physics? That they don't know they should already be at their destination?
Do we really believe that they could fake a launch, in 1963, so convincingly that the crew wouldn't figure it out?
How could they have faked the view of Earth from the space ship? In color? In 1963?
Then there are the anachronisms.
One of the cool things on the show is the retro styling. The use of magnetic tape on big spools in the control room. Big clunky switches. Raising pigs for protein.
But then we're to believe that a crew of 600 invented MRI? (Because that the explanation of how the experiment is funded... spin off technology, including MRI.)
Memory cards? That play on black and white cathode ray tube displays?
They repeatedly use the word "upload" when the first recorded use of the word is in 1980?
Then we learn there's a man on the inside!
Then we learn about the witch child!
Then I lose interest....