Determine the motion for all time

Question

In the frame $F=[0,\hat{k}]$, a particle of mass $m$, whose trajectory $[0,\infty)\xrightarrow{\rm r}\mathbb{R}$ is

$r=z\hat{k}$

moves in response to a force

$F=-mg\hat{k}-\lambda\mid\dot{r}\mid\dot{r}$

where $g,\lambda > 0$. The particle starts at rest from a point $r_{0}$ satisfying $\hat{k},r_{0}>0.$

Determine the motion for all time

Answer 1

Let $\dot z=v$.

Since $v<0\,$, one solves the initial value problem $$m \dot v=-mg+ \lambda v^2 \qquad v(0)=0$$ Separating the variables, one obtains $$v=-\sqrt{\frac {mg}\lambda} \tanh \left(\sqrt{\frac {\lambda g}m}\,t \right)$$Integrating, one finds $$z=z_0-\frac m \lambda \ln \cosh \left(\sqrt{\frac {\lambda g}m}\,t \right)$$ Note that $$\lim_{t \rightarrow \infty} v=-\sqrt{\frac {mg}\lambda}$$ the so called terminal velocity .