Falling Body with Air Resist: Difference between revisions

Revision as of 06:58, 5 May 2020

F_total = F_grav - F_air

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 = e^kt/m

multiply by u:

e^kt/mdv/dt + e^kt/mvk/m = ge^kt/m

d/dt(ve^kt/m) = ge^kt/m

integrate both sides:

ve^kt/m + C₁ = gm/ke^kt/m + C₂

solve for v:

v = mg/k - (C₁ + C₂)/e^kt/m

v = mg/k - Ce^-kt/m

@@ Line 17: / Line 17: @@
 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>
+d/dt(ve<sup>kt/m</sup>) = ge<sup>kt/m</sup>
 integrate both sides:
-se<sup>kt/m</sup> + C<sub>1</sub> = gm/ke<sup>kt/m</sup> + C<sub>2</sub>
+ve<sup>kt/m</sup> + C<sub>1</sub> = gm/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>