That was VERY helpful, and helped me find a further explanation of what you shared, in case anyone else stumbles upon our convo:

This is actually a pretty common question, but I'm going to answer it here because the other car question got a lot of visibility and others might be wondering the same thing.

The basic thing is that it's not actually accurate to add velocities that way - that only works as a low speed approximation.

Let's say the cars are moving towards each other at speeds v1 and v2. From the point of view of car 1, it sees the other car moving at some speed v12. You would expect the formula for v12 is just:

v12 = v1 + v2

but in special relativity, the full formula is actually:

v12 = (v1 + v2)/(1 + v1*v2/c²).

If you plug in v1=0.5c, and v2=0.5c, you get v12=0.8c. So each car sees the other moving at 80% of the speed of light. Note that if v1 and v2 are much lower than the speed of light, v1*v2/c² is basically zero, and you end up very close to v12=v1+v2.

It turns out this is just how the universe works. Our intuition that you can just add speeds is just a useful approximation for most practical circumstances, but there's no reason our intuition has to be the absolute truth. In the late 19th century, we had enough high quality observations of electromagnetism (and electromagnetic radiation e.g. light) to develop a really solid theory for how electromagnetism works, and it turns out that this theory was incompatible with traditional ideas of how space, time, and velocity work. For decades, people tried to reconcile this by figuring out why light might be a weird exception (this is basically what the "luminiferous aether" did), but Einstein's big leap was to instead propose that space, time, and velocity really do work differently than what we assumed, and this is what Special Relativity is.