3. \begin{figure}[h] \end{figure} Figure 1 shows part of the end elevation of a building which sits on horizontal ground. The side of the building is vertical and has height \(h\). A small stone of mass \(m\) is at rest on the roof of the building at the point \(A\). The stone slides from rest down a line of greatest slope of the roof and reaches the edge \(B\) of the roof with speed \(\sqrt { 2 g h }\) The stone then moves under gravity before hitting the ground with speed \(W\).
In a model of the motion of the stone from \(\boldsymbol { B }\) to the ground

• the stone is modelled as a particle
• air resistance is ignored

Using the principle of conservation of mechanical energy and the model,

1. find \(W\) in terms of \(g\) and \(h\). In a model of the motion of the stone from \(\boldsymbol { A }\) to \(\boldsymbol { B }\)
   • the stone is modelled as a particle of mass \(m\)
   • air resistance is ignored
   • the roof of the building is modelled as a rough plane inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 3 } { 4 }\)
   • the coefficient of friction between the stone and the roof is \(\frac { 1 } { 3 }\)
   • \(A B = d\)
   Using this model,
2. find, in terms of \(m\) and \(g\), the magnitude of the frictional force acting on the stone as it slides down the roof,
3. use the work-energy principle to find \(d\) in terms of \(h\).