Have been hit by cars from behind twice in the last year.
Neither was all that fast, though.
Both times I felt perfectly fine (albeit shaken) at the time, and for the rest of the day, and then had neck pain for a couple of weeks afterwards.
Didn't put any claims in, though.
The bloke that hit me was tailgating and didn't react fast enough when the car in front of me braked suddenly. The guy behind slammed on the anchors, and clipped me as he went into a ditch. Caused thousands of pounds of damage to his own car, and left a small insignificant scrape on my bumper - and was clearly shaken and a bit bashed about by the experience. I just made sure he was alright and left it at that.
The bird that hit me a couple of months later just rammed into me while I was sat in traffic a hundred yards from a busy roundabout. It's always got a long tailback there, and it's almost perfectly straight road for a mile before the point where she hit me, so I've no idea what she was doing. I suspect either mobile phone or make up. She hit me perfectly straight on, and shunted me forward, but the car in front was able to inch out of my path. No real damage to either car. I gave her a stern bollocking about paying attention when driving and left it at that.
TBH I'm not sure how I'd fare with a claim, as I've had a lot of problems with my neck and my lower back since my teens anyway (have seen doctor for neck, and had physio for the back - though nothing in the last decade). That'd possibly count against any claim I made.
And I was able to get to work after both accidents, so it probably would've been a bit clear it was a money-grab if I'd made claims on either.
Was tempted, though, and most folk told me I should claim on both. Money for nothing.
No thanks. I'm not American. I'll probably put a claim in if I'm properly injured, but if I can walk away from it then I'll be the sterotypical Brit, pat the other party on the back, tut a little, and get on with it.