Final answer:
The question explores the maximum approach distance of a spaceship to a black hole from where it can still escape, which involves understanding event horizons, time dilation, and spacetime metrics. The exact point depends on complex relativistic calculations that consider the spaceship's maximum proper acceleration and the black hole's mass.
Step-by-step explanation:
The question involves a spaceship approaching a black hole and inquires about the limiting distance at which it can still escape the black hole's gravitational pull. When the spaceship falls freely towards the black hole, it initially behaves as if approaching any massive body, accelerating towards it. However, as it gets closer to the event horizon, the effects of the strong gravitational field become significant.
From an outsider's view, time for the astronaut inside the spaceship would appear to slow down due to the intense gravity of the black hole. This is a consequence of general relativity wherein stronger gravity causes time dilation. Once the astronaut crosses the event horizon, there can be no return, as all signals and movements are confined within the black hole.
The astronaut would experience extreme tidal forces due to the gravitational gradient, stretching them in a process often referred to as spaghettification. The distance at which this becomes lethal depends on the mass of the black hole.
To determine the point where a spaceship with a maximum proper acceleration of a0 can no longer accelerate away from a black hole, one needs not just the value of a0 but also an understanding of the physics of black holes and the calculation of escape velocities considering relativistic effects. However, such calculations would require complex knowledge of spacetime metrics and are beyond the scope of typical schoolwork problems.