I'm not one to nitpick tools to grok things but I think this could confuse more than help!
Because everything is relative to something else - and your example of the pendulum on a ship is suggestive of a "real" velocity, which does not exist.
I think a far easier scario to imagine is some ship flying away from Earth, and this ship has a magic button to release an impulse enough to give it a 10% of the speed of light boost in speed.
So what happens the tenth, or hundredth time that button is pushed? For those on the ship they would begin to observe (out the window) length contraction and of course time dilation - if they could somehow see Earth, everything would be in fast forward.
And vice versa, what happens on Earth? The ships observed speed would asymptotically approach the speed of light, but never reach it with its "apparent mass" approaching infinity, and thus the amount of velocity boost from each impulse approaching 0.
If our ship travels 20,000 light years in 40 years (from the perspective of those on the ship) then that would take a "real" 20,000 years from the perspective of those on Earth, who for many centuries would be able to track it moving away. If they somehow had a magic eye to look in the ship, things would seem to be going in extremely slow motion.
It's this nature of velocity (and dilation/etc) always and only being relative to something else that's really at the guts of all of this.
"If our ship travels 20,000 light years in 40 years (from the perspective of those on the ship) then that would take a "real" 20,000 years from the perspective of those on Earth"
This part just breaks my brain. I've been reading about this stuff for decades, and it just does not compute. It's also a little funny because there's basically zero incentive for the folks on earth to send astronauts away at near-light speed, as they'd never realize the benefits of the mission.
Yip the reality that the speed of light is fixed, but the 'speed' of time is variable is just so intuitively absurd.
This is the reason that I find things like the Fermi Paradox no more than mildly interesting.
It implicitly assumes we have a sound understanding of the fundamentals of the universe yet each revolutionary discovery we make soundly refutes that assumption.
And with the vast number of 'known unknowns' and an unknowable number of unknown unknowns, it seems ridiculous to imagine we're anywhere near the end of revolutionary discoveries.
>there's basically zero incentive for the folks on earth to send astronauts away at near-light speed, as they'd never realize the benefits of the mission.
Because everything is relative to something else - and your example of the pendulum on a ship is suggestive of a "real" velocity, which does not exist.
I think a far easier scario to imagine is some ship flying away from Earth, and this ship has a magic button to release an impulse enough to give it a 10% of the speed of light boost in speed.
So what happens the tenth, or hundredth time that button is pushed? For those on the ship they would begin to observe (out the window) length contraction and of course time dilation - if they could somehow see Earth, everything would be in fast forward.
And vice versa, what happens on Earth? The ships observed speed would asymptotically approach the speed of light, but never reach it with its "apparent mass" approaching infinity, and thus the amount of velocity boost from each impulse approaching 0.
If our ship travels 20,000 light years in 40 years (from the perspective of those on the ship) then that would take a "real" 20,000 years from the perspective of those on Earth, who for many centuries would be able to track it moving away. If they somehow had a magic eye to look in the ship, things would seem to be going in extremely slow motion.
It's this nature of velocity (and dilation/etc) always and only being relative to something else that's really at the guts of all of this.