I'm completely stuck, I think I have to use Newton's second law but I have no idea where to start, any help would be appreciated!
At time $t=0$ a particle of unit mass is projected vertically upward with velocity $v_0$ against gravity, and the resistance of the air to the particle's motion is $k$ times its velocity. Show that during its flight the velocity $v$ of the particle at time $t$ is:
$$v = \left(v_{0} + \frac{g}{k}\right) e^{-kt} - \frac{g}{k}$$
Deduce that the particle reaches its greatest height when
$$t = \frac{1}{k} \ln\left({1+\frac{kv_{0}}{g}}\right)$$
and that the height reached is
$$ \frac{v_{0}}{k} - \frac{g}{k^2} \ln{\left(1 + \frac{kv_{0}}{g}\right)}$$
Thanks!