Difference between revisions of "Falling Body with Air Resist"

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mdv/dt = mg - kv
 
mdv/dt = mg - kv
  
put into standard order:
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* put into standard form:
  
 
mdv/dt + kv = mg
 
mdv/dt + kv = mg
  
dv/dt + vk/m = g
+
'''dv/dt + vk/m = g'''    <----Standard form
  
find u = e<sup>∫k/mdt</sup> = e<sup>kt/m</sup>
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* find u = e<sup>∫k/mdt</sup> = e<sup>kt/m</sup>
  
multiply by u:
+
* multiply by u:
  
 
e<sup>kt/m</sup>dv/dt + e<sup>kt/m</sup>vk/m = ge<sup>kt/m</sup>
 
e<sup>kt/m</sup>dv/dt + e<sup>kt/m</sup>vk/m = ge<sup>kt/m</sup>
  
d/ds(se<sup>kt/m</sup>) = ge<sup>kt/m</sup>
+
* use product rule:
  
integrate both sides:
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d/dt(ve<sup>kt/m</sup>) = ge<sup>kt/m</sup>
  
se<sup>kt/m</sup> + C<sub>1</sub> = gm/ke<sup>kt/m</sup> + C<sub>2</sub>
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* integrate both sides:
 +
 
 +
ve<sup>kt/m</sup> + C<sub>1</sub> = mg/ke<sup>kt/m</sup> + C<sub>2</sub>
 +
 
 +
* solve for v:
 +
 
 +
v = mg/k - (C<sub>1</sub> - C<sub>2</sub>)/e<sup>kt/m</sup>
 +
 
 +
v = mg/k - Ce<sup>-kt/m</sup>
 +
 
 +
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<sup>-kt/m</sup>
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 +
or
 +
 
 +
v(t) = (mg/k)(1-e<sup>-kt/m</sup>)
 +
 
 +
when t goes to infinity, e<sup>-kt/m</sup> goes to 0 so the terminal velocity = mg/k
 +
 
 +
The value of k depends on mass and shape and it can be determined by measuring terminal velocity:
 +
 
 +
average terminal velocity of an 80kg person = 148mph = 66m/s so k = 80kg(9.8m/s<sup>2</sup>)/66 m/s = 11.8 kg/sec

Latest revision as of 18:09, 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

The value of k depends on mass and shape and it can be determined by measuring terminal velocity:

average terminal velocity of an 80kg person = 148mph = 66m/s so k = 80kg(9.8m/s2)/66 m/s = 11.8 kg/sec