Freefall From Specified Height

For an object falling from rest through air with quadratic drag, the motion parameters are usually expressed in terms of the terminal velocity and the characteristic time:

For a sphere of radius r = cm

and density ρ = kg/m3 = gm/cm3

the mass is m = gm.

Assume the object has a drag coefficient C = (the default value is C = 0.5 for a sphere)

and is falling through air with density ρ* = kg/m3 (the default value is 1.29 kg/m3).

Under these conditions, it's terminal velocity will be
vt = m/s = km/hr = mi/hr

and its characteristic time is
τ = s.

If dropped from rest at height h = m = ft

it will reach the ground at velocity
vimpact = m/s = ft/s = km/hr = mi/hr

at time timpact = s.

For comparison, if there were no air friction the object would impact with velocity v = m/s at time t = s.

If dropped from great height so that it spends most of the time at terminal velocity, the time of fall would approach h/vt = s. This number represents a lower bound for the fall time since it assumes that the object falls at the terminal velocity for the whole time and there is always a finite time required to reach the terminal velocity. For heights large enough that the acceleration time is very small compared to the total time, the calculated fall time should approach this limit from above.

Note: A value of g = 9.8 m/s2 is presumed in the above calculation.

Hyperbolic functions

Fluid friction
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