Answer to Question #202253 in Classical Mechanics for Glory

Question

A body of mass m falls from rest through a medium which exerts a frictional drag force bv proportional to a frictional force F. Expand the velocity in power series up to t^4, at what time does the body reach 1/e of it terminal velocity, find it displacement, expand the displacement in power series up to t^4

Answer 1

$F=mg-bv,$

$m\frac{dv}{dt}=mg-bv,$

$\frac{dv}{dt}=\frac{mg-bv}{m},$

$\frac{dv}{mg-bv}=\frac{dt}{m},$

$v=\frac{mg}{b}(1-e^{-\frac bm t}).$