09 Jun '13 08:25>3 edits
Originally posted by KazetNagorraFor someone falling 9ft the correction is tiny. Sticking with imperial units, the acceleration is given by:
Sounds about right, I think I used 24 ft/s, an acceleration of 9.8 m/s² and then rounded the answer up to get 9 ft. (the formula is correct) In any case it only gives a lower bound since you neglect friction, which is a pretty poor approximation in this case.
a = g - k v*v
where k = drag coefficient/mass. Using a = v * (dv/dx) and assuming a stationary start, I can integrate this to get:
v ^ 2 = (g/k) * (1 - exp(-2kx))
in the limit x -> infinity this gives v_t^2 = g/k. So I can write the above in terms of the terminal velocity to give:
v = v_t * sqrt((1 - exp(-2*g*x/(v_t^2)))
As a quick check, in the limit that v_t is very large the exponential expands to give:
v = v_t * sqrt ( 2*g*x / v_t^2)
=> v^2 = 2*g*x which is the vacuum result.
Wikipedia gives a terminal velocity for a human as 70 m/s, I'll approximate that as 200 ft/s. Putting this as well as your figure of 9 ft into the equation gives:
v = 200 * sqrt( 1 - exp (- 2 * 32 * 9/200*200))
= 200 * sqrt(0.0143) = 23.91, which rounds up to 24 ft/s
Which is the number you started with. I think the error due to air resistance is negligible over this distance. The effect of horizontal speed is going to be more important and could easily make the landing feel more like the 12 ft USArmyParatrooper's c/o quoted.