Impulse on inclined plane

2 A particle \(P\) of mass 3.5 kg is moving down a line of greatest slope of a rough inclined plane. At the instant that its speed is \(2.1 \mathrm {~ms} ^ { - 1 } P\) is at a point \(A\) on the plane. At that instant an impulse of magnitude 33.6 Ns , directed up the line of greatest slope, acts on \(P\).

1. Show that as a result of the impulse \(P\) starts moving up the plane with a speed of \(7.5 \mathrm {~ms} ^ { - 1 }\). While still moving up the plane, \(P\) has speed \(1.5 \mathrm {~ms} ^ { - 1 }\) at a point \(B\) where \(A B = 4.2 \mathrm {~m}\). The plane is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The frictional force exerted by the plane on \(P\) is modelled as constant.
2. Calculate the work done against friction as \(P\) moves from \(A\) to \(B\).
3. Hence find the magnitude of the frictional force acting on \(P\). \(P\) first comes to instantaneous rest at point \(C\) on the plane.
4. Calculate \(A C\).

2 \includegraphics[max width=\textwidth, alt={}, center]{cc74a925-f1c8-4f59-a421-b46444cae5ec-3_442_636_255_715} \(B\) is a point on a smooth plane surface inclined at an angle of \(15 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 0.45 kg is released from rest at the point \(A\) which is 2.5 m vertically above \(B\). The particle \(P\) rebounds from the surface at an angle of \(60 ^ { \circ }\) to the line of greatest slope through \(B\), with a speed of \(u \mathrm {~ms} ^ { - 1 }\). The impulse exerted on \(P\) by the surface has magnitude \(I\) Ns and is in a direction making an angle of \(\theta ^ { \circ }\) with the upward vertical through \(B\) (see diagram).

1. Explain why \(\theta = 15\).
2. Find the values of \(u\) and \(I\).

1. 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 \(v\).

1. 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 \(e > 0\)
2. Show that \(v ^ { 2 } = u ^ { 2 } \left( \sin ^ { 2 } \alpha + e ^ { 2 } \cos ^ { 2 } \alpha \right)\)
3. Show that the kinetic energy lost by \(P\) in the impact is $$\frac { 1 } { 2 } m u ^ { 2 } \left( 1 - e ^ { 2 } \right) \cos ^ { 2 } \alpha$$
4. Hence find, in terms of \(m\), \(u\) and \(e\) only, the kinetic energy lost by \(P\) in the impact.

A small smooth ball of mass \(m\) is falling vertically when it strikes a fixed smooth plane which is inclined to the horizontal at an angle \(\alpha\), where \(0° < \alpha < 45°\). Immediately before striking the plane the ball has speed \(u\). Immediately after striking the plane the ball moves in a direction which makes an angle of \(45°\) with the plane. The coefficient of restitution between the ball and the plane is \(e\). Find, in terms of \(m\), \(u\) and \(e\), the magnitude of the impulse of the plane on the ball. (11)