I have these weird brainfarts with physics, where I can't intuitively comprehend even the simplest of things.

Such as this one.
Prove that v = v_rel + v_cm.
I just can't see the logical link that would make me add the different velocities together to get the velocity you see in a stationary frame of reference.
This is just one of those millions of things that I give up on and memorize them instead of understanding.