Particle-barrier collision with angle

1. \begin{figure}[h] \end{figure} A fixed smooth plane is inclined to the horizontal at an angle of \(45 ^ { \circ }\). A particle \(P\) is moving horizontally and strikes the plane. Immediately before the impact, \(P\) is moving in a vertical plane containing a line of greatest slope of the inclined plane. Immediately after the impact, \(P\) is moving in a direction which makes an angle of \(30 ^ { \circ }\) with the inclined plane, as shown in Figure 1. Find the fraction of the kinetic energy of \(P\) which is lost in the impact.

A smooth sphere of mass 0.3 kg bounces on a fixed horizontal surface. Immediately before the sphere bounces the components of its velocity horizontally and vertically downwards are \(4 \text{ m s}^{-1}\) and \(6 \text{ m s}^{-1}\) respectively. The speed of the sphere immediately after it bounces is \(5 \text{ m s}^{-1}\).

1. Show that the vertical component of the velocity of the sphere immediately after impact is \(3 \text{ m s}^{-1}\), and hence find the coefficient of restitution between the surface and the sphere. [3]
2. State the direction of the impulse on the sphere and find its magnitude. [3]

A smooth uniform sphere \(P\) of mass \(m\) is falling vertically and strikes a fixed smooth inclined plane with speed \(u\). The plane is inclined at an angle \(\theta\), \(\theta < 45°\), to the horizontal. The coefficient of restitution between \(P\) and the inclined plane is \(e\). Immediately after \(P\) strikes the plane, \(P\) moves horizontally.

1. Show that \(e = \tan^2 \theta\). [6]
2. Show that the magnitude of the impulse exerted by \(P\) on the plane is \(mu \sec \theta\). [4]