Question

A body of mass (m) falls from rest through a medium which exact a frictional drag force (bv) proportional to a applied force (f).1. Find it's velocity in terms of time t 2. Find it's terminal velocity

Answer 1

$$ma=m\frac{dv}{dt}=mg-bv+f$$

$$\frac{dv}{dt}=g-\frac{b}{m}v+\frac{f}{m}$$

$$bdt=\frac{dv}{\frac{mg}{b}+\frac{f}{b}-v}$$

$$v(t)=\frac{mg+f}{b}+c\exp{(-bt)}$$

$$v(0)=0=\frac{mg+f}{b}+c$$

So,

$$v(t)=\frac{mg+f}{b}(1-\exp{(-bt)})$$

Terminal velocity:

$$v(\infty)=\frac{mg+f}{b}$$