Was thinking about interstellar travel and the ability to provide artificial gravity by using a smooth acceleration and deceleration across the journey, changing from acceleration to deceleration at the halfway mark.

If we ignore relativistic effects, with smooth acceleration of 9.81 ms^-2, you’d be going 3.1e8 ms^-1 after the first year (3.2e7 s), if I’m not making a mathematical blunder. That’s more than the speed of light at 3.0e8.

My main question, and the one that I initially came here to ask, is: if their ship continues applying the force that, under classical mechanics, was enough to accelerate them at 9.81 ms^-2, would the people inside still experience Earth-like artificial gravity, even though their velocity as measured by an observer is now increasing at less than that rate?

A second question that I thought of while trying to figure this out myself as I wrote it up, is… My understanding is that a trip taken at the speed of light would actually feel instantaneous to the traveller, while taking distance/speed of light to a stationary observer. In the above scenario, would the final time taken, as measured by the traveller, be the same as if they were to ignore the speed that they are travelling at according to an outside observer, and instead actually assume they are undergoing continuous acceleration?