The technology is a long, long way off to be able to accelerate at one G for more than a few minutes. Let's just say though that in the distant future there will be energy sources sufficient to accelerate at one G for months at a time.
If you could maintain this kind of acceleration for around one year, you would be going around 99% of the speed of light.
If you were going on a journey of 20 light-years' distance, in the first year you would cover about 0.5 light-years. Due to Lorentz contraction, over the period of acceleration, the distance will appear to shrink to only one tenth the original distance. So, the destination will now appear to be only 1.95 light years away.
If the destination looked 20 light years away at the start of the trip, but only looks 1.95 after one year, then it will appear to the passengers on this ship that they have covered some 18 light years in one year. Actually, less than a year for them since by the end of it, time will be moving only one tenth as fast as at the beginning.
It would all be spoiled if they looked back at their starting point: That 1/2 light year will have also contracted to 1/20th of a light year and so it will seem to them that they are making woefully slow progress getting away from the starting point.
What would be really odd is if they decided to decelerate over the course of the next year: The whole time they would be moving toward the destination and yet it would look as if it went from 1.95 LY distant to 19 LY distant over the one year period. The starting point too would go from looking .05 LY to 1.0 LY. Both would appear to be getting more distant! Odd stuff.
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