6 A stone, of mass \(m\), falls vertically downwards under gravity through still water. At time \(t\), the stone has speed \(v\) and it experiences a resistance force of magnitude \(\lambda m v\), where \(\lambda\) is a constant.

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = g - \lambda v$$
2. The initial speed of the stone is \(u\). Find an expression for \(v\) at time \(t\).
   [0pt] [6 marks] \(7 \quad\) A uniform ladder, of weight \(W\), rests with its top against a rough vertical wall and its base on rough horizontal ground. The coefficient of friction between the wall and the ladder is \(\mu\) and the coefficient of friction between the ground and the ladder is \(2 \mu\). When the ladder is on the point of slipping, the ladder is inclined at an angle of \(\theta\) to the horizontal.
3. Draw a diagram to show the forces acting on the ladder.
4. Find \(\tan \theta\) in terms of \(\mu\).