1. A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\)

A particle \(P\) of mass \(m\) is held at rest at a point \(A\) on the plane.
The particle is then projected with speed \(u\) up a line of greatest slope of the plane and comes to instantaneous rest at the point \(B\). The coefficient of friction between the particle and the plane is \(\frac { 1 } { 7 }\)

1. Show that the magnitude of the frictional force acting on the particle, as it moves from \(A\) to \(B\), is \(\frac { 4 m g } { 35 }\) Given that \(u = \sqrt { 10 a g }\), use the work-energy principle
2. to find \(A B\) in terms of \(a\),
3. to find, in terms of \(a\) and \(g\), the speed of \(P\) when it returns to \(A\).