Particle bouncing on inclined plane

5 A particle is projected from a point \(O\) on a smooth plane, which is inclined at \(25 ^ { \circ }\) to the horizontal. The particle is projected up the plane with velocity \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(30 ^ { \circ }\) above the plane. The particle strikes the plane for the first time at a point \(A\). The motion of the particle is in a vertical plane containing a line of greatest slope of the inclined plane. \includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-12_518_839_552_630}

1. Find the time taken by the particle to travel from \(O\) to \(A\).
2. The coefficient of restitution between the particle and the inclined plane is \(\frac { 2 } { 3 }\). Find the speed of the particle as it rebounds from the inclined plane at \(A\). (8 marks)

5 A small smooth ball is dropped from a height of \(h\) above a point \(A\) on a fixed smooth plane inclined at an angle \(\theta\) to the horizontal. The ball falls vertically and collides with the plane at the point \(A\). The ball rebounds and strikes the plane again at a point \(B\), as shown in the diagram. The points \(A\) and \(B\) lie on a line of greatest slope of the inclined plane. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-12_318_636_548_712}

1. Explain whether or not the component of the velocity of the ball parallel to the plane is changed by the collision.
2. The coefficient of restitution between the ball and the plane is \(e\). Find, in terms of \(h , \theta , e\) and \(g\), the components of the velocity of the ball parallel to and perpendicular to the plane immediately after the collision.
3. Show that the distance \(A B\) is given by $$4 h e ( e + 1 ) \sin \theta$$

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.

6 A particle \(P\) of mass 2.5 kg strikes a rough horizontal plane. Immediately before \(P\) strikes the plane it has a speed of \(6.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its direction of motion makes an angle of \(30 ^ { \circ }\) with the normal to the plane at the point of impact. The impact may be assumed to occur instantaneously. The coefficient of restitution between \(P\) and the plane is \(\frac { 2 } { 3 }\). The friction causes a horizontal impulse of magnitude 2 Ns to be applied to \(P\) in the plane in which it is moving.

1. Calculate the velocity of \(P\) immediately after the impact with the plane.
2. \(\quad P\) loses about \(x \%\) of its kinetic energy as a result of the impact. Find the value of \(x\).

11 A particle of mass 0.2 kg is projected so that it hits a smooth sloping plane \(\Pi\) that makes an angle of \(\sin ^ { - 1 } 0.6\) above the horizontal. The path of the particle is in a vertical plane containing a line of greatest slope of \(\Pi\). Immediately before the first impact between the particle and \(\Pi\), the particle is moving horizontally with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between the particle and \(\Pi\) is 0.5 .

1. Find the magnitude of the impulse on the particle from \(\Pi\) at the first impact, and state the direction of this impulse.
2. Find the distance between the points on \(\Pi\) where the first and second impacts occur.
3. Find the time taken between the first and third impacts.