Difference between revisions of "Falling Body with Air Resist"

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Line 27: Line 27:
 
solve for v:
 
solve for v:
  
v = mg/k - (C<sub>1</sub> + C<sub>2</sub>)/e<sup>kt/m</sup>
+
v = mg/k - (C<sub>1</sub> - C<sub>2</sub>)/e<sup>kt/m</sup>
  
 
v = mg/k - Ce<sup>-kt/m</sup>
 
v = mg/k - Ce<sup>-kt/m</sup>

Revision as of 07:12, 5 May 2020

Ftotal = Fgrav - Fair

ma = mg - kv where k units are kg/sec

mdv/dt = mg - kv

put into standard form:

mdv/dt + kv = mg

dv/dt + vk/m = g <----Standard form

find u = e∫k/mdt = ekt/m

multiply by u:

ekt/mdv/dt + ekt/mvk/m = gekt/m

use product rule:

d/dt(vekt/m) = gekt/m

integrate both sides:

vekt/m + C1 = mg/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)

when t goes to infinity, e-kt/m goes to 0 so the terminal velocity = mg/k