Suppose we are in a spacecraft that is moving away from the Earth along a straight line with a constant acceleration of $1G$, and that this acceleration can be sustained indefinitely.
Accelerating at $1G$ seems to suggest at first glance that we should move 10m/s faster every second relative to the Earth, but does not sound right since that would take us a finite amount of time to reach the speed of light if that were true. What velocity and time are we talking about? What does constant acceleration mean and in what frame, and what is our speed relative to the Earth at any given time $t$?
