Thank you all, for trying to help me “get” it. I understand the very straightforward definitions of “work” and “force”, and Pasta got me to a breakthough by pointing out that kinetic energy also involves velocity squared, which is precisely the idea that I’m trying to wrap my brain around.
So I found this in the Wikipedia article on Kinetic Energy:
By dropping weights from different heights into a block of clay, Willem 's Gravesande determined that their penetration depth was proportional to the square of their impact speed.
Aha! So if I drop two equal weights, and one has triple the speed at impact, it will penetrate nine times as far? Okay, now I have to digest this a bit… Thanks!