Classical physics is based on observations that humans can make with very low technology: polished metals and glasses, relative velocities up to a few thousand meters per second, timekeeping accurate to milliseconds in one location. Those observations lead to approximations which "feel" correct because humans spend most of their lives in those circumstances.
But at the edges of those circumstances, we see discrepancies not predicted by classical physics. That lets us know that classical physics is an approximation. The truth is that even in your car + thrown object example, the velocities do not actually add linearly -- it's just that the difference between the linear approximation and the reality is tiny.
One analogy that might be helpful: consider every object in the universe to be moving at c, but for most objects that you encounter, the vast majority of that motion is forward in time rather than space. As you add energy to the object, it transfers that velocity from time-motion to space-motion. Because c is so large compared to our usual experience, we have to look really hard to discover the change in time-motion... but it's real. GPS uses satellites that are in a sufficiently different frame from the surface of the Earth, and requires sufficient time precision, that relativistic time difference calculations are required.
Reading this entire thread (I didn't expect this many replies this detailed!), I think it finally clicked with me after all these years.
So, for me, in a vehicle moving at 90% of the speed of light, the time itself slows down such that I observe the photons coming out of my lightbulb still at the speed of light, right? Basically, it would adjust the t in v=s/t because the speed and the distance are to remain constant. And because me and my spaceship or whatever have mass, there will always be a bit of difference between my speed and the speed of light to accommodate this adjustment. That's just so weirdly backwards to think about.
Now, I wonder about redshift and universe expansion deduced from it. Wavelength is a frequency, right? And frequency, by definition, is how many times something (wave period) happens per unit of time. So maybe the universe isn't expanding after all, maybe it's just that time runs faster in the parts of the universe where this light comes from, and the photons just keep oscillating with the frequency that was "correct" in whichever reference frame they were emitted? Besides, the idea that universe expands seems silly tbh. There must be a more sensible explanation. Maybe time slows down over time (?!) and it just ran faster when the universe was younger?
If "time slows down over time" then we would still observe the light at its original frequency - why would the energies of the states of hydrogen atoms in distant stars change over time, but the energies of the photons emitted long ago not change?
How does this hypothesis account for the CMB?
Wikipedia has a nice summary of alternative cosmological theories (and why they are often dismissed by experts):
But at the edges of those circumstances, we see discrepancies not predicted by classical physics. That lets us know that classical physics is an approximation. The truth is that even in your car + thrown object example, the velocities do not actually add linearly -- it's just that the difference between the linear approximation and the reality is tiny.
One analogy that might be helpful: consider every object in the universe to be moving at c, but for most objects that you encounter, the vast majority of that motion is forward in time rather than space. As you add energy to the object, it transfers that velocity from time-motion to space-motion. Because c is so large compared to our usual experience, we have to look really hard to discover the change in time-motion... but it's real. GPS uses satellites that are in a sufficiently different frame from the surface of the Earth, and requires sufficient time precision, that relativistic time difference calculations are required.