A particle which is moving freely under gravity has a perfectly elastic collision with a vertical wall. Show that the path followed after the collision is the mirror image of the path that would have been followed if the wall were absent. A cubical room has a horizontal floor. A particle is projected from a point of the floor at an angle of elevation \(\alpha\), so as to move in a vertical plane parallel to one of the walls. It bounces successively off a wall, the ceiling, and the opposite wall, and then strikes the floor at the point of projection. All the bounces are perfectly elastic. Show that \(1 < \tan\alpha < 2\).
A hollow cylinder of internal radius \(3a\) is fixed with its axis horizontal. There rests inside it in stable equilibrium a uniform solid cylinder of radius \(a\) and mass \(M\). The axes of the cylinders are parallel and no slipping can occur between them. A particle of mass \(m\), with \(m > M\), is now attached to the top of the inner cylinder. Show that this equilibrium position is no longer stable. If the equilibrium is slightly disturbed, show that the particle touches the outer cylinder in the subsequent motion only if \(m \geq 2M\).
An aeroplane flies at a constant air speed \(v\) around the boundary of a circular airfield. When there is no wind it takes a time \(T\) to complete one circuit of the airfield. Show that when there is a steady wind blowing, whose speed \(u\) is small compared with \(v\), the increase in the time required for one circuit is approximately \(3Tu^2/4v^2\).
A uniform cylinder of radius \(a\) and mass \(M\) rests on horizontal ground with its axis horizontal. A uniform rod of length \(2l\) and mass \(m\) rests against the cylinder and has one end attached to the ground by a smooth hinge. The rod makes an angle \(2\alpha\) with the horizontal such that \(a\cot\alpha < 2l\), and it lies in the vertical plane through the centre of the cylinder which is perpendicular to its axis. The coefficients of friction between the cylinder and the rod, and between the cylinder and the ground, both have value \(\mu\). Show that the system is in equilibrium provided that \(\mu > \tan\alpha\). A force \(P\) is now applied at the centre of the cylinder along a line parallel to the rod and directed away from the hinge. Find the smallest value of \(P\) for which the cylinder will move, on the assumption that slipping occurs first between the cylinder and the rod.
A light elastic string of unstretched length \(3l\) passes over a small smooth horizontal peg. Particles \(A\) and \(B\) of masses \(m\) and \(3m\) respectively are attached to the ends of the string. Initially \(B\) is held fixed at a distance \(2l\) vertically below the peg, and the string hangs in equilibrium with \(A\) and \(B\) at the same level. Particle \(B\) is now released. Show that \(A\) moves upwards until it strikes the peg, and that the maximum length of the string during this motion is \(5l\).
\(O\), \(P\), \(Q\), \(R\) are four non-coplanar points. \(A\), \(B\), \(C\), \(D\) are four coplanar points which lie respectively on the straight lines \(OP\), \(PQ\), \(QR\), \(RO\). Let \[\alpha = OA/AP, \beta = PB/BQ, \gamma = QC/CR, \delta = RD/DO.\] Using three-dimensional vectors with origin \(O\), express the position vectors of \(A\), \(B\), \(C\), \(D\) in terms of those of \(P\), \(Q\), \(R\) and the ratios \(\alpha\), \(\beta\), \(\gamma\), \(\delta\). Deduce that \[\alpha\beta\gamma\delta = 1.\]