When an object moves through a medium (either a liquid or a gas), the medium exerts a resistive force $\vec{R}$ on it.

We can model the resistive force in two ways.

$\vec{R}$ is proportional to $v$

$\vec{R}=-b\vec{v}$

$\vec{R}$ is proportional to $v^2$

$R=\frac{1}{2}D\rho A{v}^2$ $D$ is a dimensionless empirical quantity called the drag coefficient $\rho$ is the density of the medium (air most of the time) $A$ is the cross-sectional area of the object $v$ is the speed of the object

Finding terminal velocity $mg-R=ma$ $a=\frac{mg-R}{m}$ At terminal velocity, the acceleration is 0. $mg-R=0$

$v_T=\sqrt{\frac{2mg}{D\rho A}}$