TL;DR: in case of an accident, frame yourself in the image of a medical doctor or nurse and do what they would do. "Social" things need to step back there a little bit, I feel, until it is clear that there is no life-threatening, urgent medical issue (whether you can actually decide that, as a layman, aside). Also, keep in mind that people in shock are different than normal people; things like light body contact, eye contact, soothing voice etc. might just be more helpful than what words you use, exactly. Imagine yourself as a parent helping a little child (without the hugging/picking up part...).
someone just fell down 8 meters
Funny that you should mention that.
When bouldering recently, some lone stranger fell and hit her chin on the rock in a hard fall (same effect as a brutal "upper cut" in boxing with the complete body weight behind it). It was obvious that it was bad - not necessarily life-threatening, but at least a slight concussion/whiplash or even some face damage.
I think the problem rather is: what do you want to know?
You want to know if there is a urgent life-threatening issue first (breathing...); then, whether there is pain involved; then, whether they need help at all.
While she lay slightly dazed, I knelt down close to her in a way that she did not need to move her head to see me (but not touching) and did the following:
- Made sure we had good eye contact and that she was actually aware of me. There was no blood and no obvious damage, she was generally moving, so the worst fears were dispelled quickly. Our eye contact was much more intense than would be appropriate of (different-sex) strangers just meeting on a street, but that was exactly correct at this moment. It was clear that there was fear in here eyes, and it helped her to literally see someone close.
- If it had been impossible to make eye contact (eyes closed or not reacting), I would probably, depending on the situation, have calmy talked to her while making obvious but very light body contact (like 2-fingers-to-shoulder or whatever was appropriate). Think "lay on hands" instead of "poking".
- She obviously smacked her teeth together substantially on the way down and was probably afraid to find out whether teeth (or bits of the tongue...) were missing. I asked her if she was bleeding or there were loose teeth; she said no.
- I asked her where it hurts, she indicated the general head area, which was expected as she also hit the back of her head when hitting the floor mat.
- I held her eyes the whole time in a conscious effort to make clear to her that I "got her". It was noticeable that that was a good thing as she held the contact as well and was not made (any more) uncomfortable by that.
- I made sure to talk in an earnest but very calm, unhurried way, also giving her enough time to feel around her own body. This also seemed to work well.
- I again asked very calmly if she had pain or if something felt wrong/numb, which likely put her in a state of analyzing the damage instead of being overwhelmed by fear; it prompted her to tell of a brutal headache; but I told her that I see no obvious damage in her mouth (no blood, no obvious teeth damage).
And so on (at that point, which took maybe 30-60 seconds, she slowly turned "normal" again, the immediate shock went away and other people came as well to help). There are plenty of things you can do and say. I don't even feel that asking "are you ok" in lieu of a "hello" would have been bad in any way.
I would probably tend to use "are you ok" if I happen upon somebody in obvious distress where I didn't see what happened, to see if there actually is a problem - in which case the question is not cynical or sarcastic, at all.