This might be an ill-formed question, but maybe you can help clear up my misunderstanding (I think this question was getting at similar concept).
On this website we see a calculator for a rocket to travel half-way to a location under a constant acceleration and then travel the last half under a constant deceleration (opposite direction, same magnitude).
So, let's create a specific example.
- Rocket accelerates at $10 \; m/s^2$ for $50 \; lightyears$ and then decelerates at $10m/s^2$ for $50 \; lightyears$.
- In doing so, it travels from star $A$ to star $B$, where star $B$ is $100 \; lightyears$ from star $A$.
- The rocket starts and ends its journey with $0$ velocity relative to the stars which are at $0$ velocity relative to each other.
From the special theory of relativity, we are able to calculate the following.
- The time for the journey as measured on star $A$ is $101.9 \; years$.
- The time for the journey as measured from the rocket is $8.9 \; years$.
- The maximum velocity is $99.9956 \; %$ of $c$.
So far so good. Obviously, the rocket experiences spatial contraction and time dilation so it does not measure its own velocity as faster than the speed of light relative to, say, a series of milestones along the way.
However, if I were on the rocket and continuously measured the distance to star B ($d_{RB}$) and the time on the rocket ($t$), then I would see the distance shrink from $100 \; lightyears$ to $0 ;\ lightyears$ (including both my changing position and the spatial contraction, but both endpoints would be only from the changed position) and the time increase from $0 \; years$ to $8.9 \; years$. I could, for example, plot the distance and the time on a graph and it would be a continuous graph with an average gradient of $\frac{100}{8.9}c$. So that would mean that the planet was approaching faster than the speed of light in my frame of reference (an observer at star B would see me approach over the course of $101.9 \; years$, so the same restriction wouldn't apply in reverse).
Is this just that looks can be deceiving when spacetime is dilating? For example, there's nothing special about star $B$. If I experienced any amount of spatial contraction, then a distant object would appear to race towards me at a velocity proportional to its distance from me. So spatial contraction is just weird and you could correct for it when doing measurements? Or what?