Chapter 9: First-Order Differential Equations - Section 9.4 - Graphical Solutions of Autonomous Equations - Exercises 9.4 - Page 551: 16

$v = (\dfrac{m g}{k})^2$

We start with the force equations: $F=F_p-F_r$ $ma=m -k g \sqrt v $ This implies that $a= g -\dfrac{k\sqrt v}{m}$ and $\dfrac{dv}{dt}=g -\dfrac{k\sqrt v}{m}$ When $\dfrac{dv}{dt}=0 $, so $g -\dfrac{k\sqrt v}{m}=0$ This implies that the terminal velocity is $v = (\dfrac{m g}{k})^2$