Falling Body with Air Resist
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Ftotal = Fgrav - Fair
ma = mg - kv where k units are kg/sec
mdv/dt = mg - kv
put into standard order:
mdv/dt + kv = mg
dv/dt + vk/m = g
find u = e∫k/mdt = ekt/m
multiply by u:
ekt/mdv/dt + ekt/mvk/m = gekt/m
d/dt(vekt/m) = gekt/m
integrate both sides:
vekt/m + C1 = gm/kekt/m + C2
solve for v:
v = mg/k - (C1 + C2)/ekt/m
v = mg/k - Ce-kt/m
If at time 0, the body is at rest then v(0) = 0 = mg/k - C so C = mg/k
therefore v = mg/k - mg/ke-kt/m
or
v(t) = (mg/k)(1-e-kt/m)