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As a quick option on how to rule it you could compare his running speed to the speed of a fall and use this as a guideline for the damage he takes.
As to RAW there is no damage: If I jump off a cliff and declare to bullrush someone on the ground do I take damage? I should not as I am not falling I am willingly bullrushing someone.
I could even say that I think I had seen indication that there is someone invisible down there and try to bullrush him.
** spoiler omitted **
If you find something wrong in this estimate (wasn't a real calculation with all that rounding and guessing) let me know.
Would perhaps have been easier to do just in feet but I'm used to using meters for calculation.
Your math is incorrect. The formula for determining the final velocity is:
v = square root of (2 * acceleration * distance)
Plugging in the numbers, for a 10 foot fall (1d6 damage), you hit the ground at 17mph, which equates to pretty close to 150'/round:
((17mph * 5280 feet per mile)/3600 seconds per hour)*6 seconds per round
Compare this to the Feather Fall spell which describes falling 60'/round as equivalent to dropping a few feet and does no damage.
So, unless the character was running at 150'/round I'd probably just have them be stunned, maybe take some minor damage depending on how fast they were going.
Also, I am having a hard time imagining a character running head-first into what they think is an open hallway. Headlong maybe, but not head-first.