- A particle \(P\) of mass \(m\) is falling vertically when it strikes a fixed smooth inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(0 < \alpha \leqslant 45 ^ { \circ }\)
At the instant immediately before the impact, the speed of \(P\) is \(u\).
At the instant immediately after the impact, \(P\) is moving horizontally with speed \(v\).
- Show that the magnitude of the impulse exerted on the plane by \(P\) is \(m u \sec \alpha\)
The coefficient of restitution between \(P\) and the plane is \(e\), where \(e > 0\)
- Show that \(v ^ { 2 } = u ^ { 2 } \left( \sin ^ { 2 } \alpha + e ^ { 2 } \cos ^ { 2 } \alpha \right)\)
- Show that the kinetic energy lost by \(P\) in the impact is
$$\frac { 1 } { 2 } m u ^ { 2 } \left( 1 - e ^ { 2 } \right) \cos ^ { 2 } \alpha$$
- Hence find, in terms of \(m\), \(u\) and \(e\) only, the kinetic energy lost by \(P\) in the impact.