Watching football yesterday, it occurred to me that falling is an area where football players appear to have been trained to react differently from normal people. I think the natural impulse is to do like you did and throw one’s arms out to try to break the fall. But if a ball carrier does that, he’s likely to fumble. So they hug the ball and just go down like trees.
I guess wearing a helmet might help. 
Now you’ve sparked a memory, Tom: My maternal grandfather was a telephone lineman, starting waaaaaaaaaaaaay back when the grand resort hotels in the White Mountains of New Hampshire were first being wired for service.
Yes, I’m THAT old.
Anyway, linemen learned how to fall as part of their safety training. That came in handy many decades later, when Grampa was in his eighties. He fell down a flight of stairs but, other than some bruises, was unharmed because his old training – tuck into a ball and roll into the fall – came back to him as he went down.