A particle moves vertically under gravity and a retarding force proportional to the square of the velocity. If $\displaystyle v$ is its upward or downward speed and we know that $\displaystyle \dot{v}=-g-kv^2$, show that its position at time t is given by

$\displaystyle z = z_0 + \frac{1}{k}ln\cos[\sqrtgk (t_0 -t)]$