A body falls through a medium I'm hating these variable resistance questions. A body of mass $m$ falls from rest in a medium that produces a resistance of magnitude $m\cdot k \cdot v$. where $k$ is a constant, where the speed of the particle is $v$. Show that when the body has reached a speed $V$ it will have fallen for a time $$\frac{1}{k} \ln \left(\frac{g}{g-k\cdot V}\right) \; .$$ Help much appreciated.

Well the force on the particle due to gravity is $mg,$ where $g$ is the acceleration due to gravity, so the net downward force on the particle is $m(g-kv).$ So its acceleration is $g-kv.$ So you need to solve $dv/dt = g-kv$ given the initial conditions. That is

$$\int_0^V \frac{ \text{d}v}{g-kv} = \int_0^T \text{d}t.$$