Projectile with plane collision

7. \begin{figure}[h] \end{figure} A light inextensible string of length \(a\) has one end attached to a fixed point \(O\) on a horizontal plane. A particle \(P\) is attached to the other end of the string. The particle is held at the point \(A\), where \(A\) is vertically above \(O\) and \(O A = a\). The particle is then projected horizontally with speed \(\sqrt { 10 a g }\), as shown in Figure 2. The particle strikes the plane at the point \(B\). After rebounding from the plane, \(P\) passes through \(A\). The coefficient of restitution between the plane and \(P\) is \(e\).

1. Show that \(e \geqslant \frac { 1 } { 2 }\) The point \(C\) is above the horizontal plane such that \(O C = a\) and angle \(C O B = 120 ^ { \circ }\) As the particle reaches \(C\), the string breaks. The particle now moves freely under gravity and strikes the plane at the point \(D\).
   Given that \(e = \frac { \sqrt { 3 } } { 2 }\)
2. find the size of the angle between the horizontal and the direction of motion of \(P\) at \(D\).

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 $$4 h e ( e + 1 ) \sin \theta$$

5 A ball is projected from a point \(O\) above a smooth plane which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The point \(O\) is at a perpendicular distance of 1 m from the inclined plane. The ball is projected with velocity \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(70 ^ { \circ }\) above the horizontal. The motion of the ball is in a vertical plane containing a line of greatest slope of the inclined plane. The ball strikes the inclined plane for the first time at a point \(A\).
\includegraphics[max width=\textwidth, alt={}, center]{d8c723df-d10a-4fdf-b5ca-ea12633f999a-14_478_913_571_561}

1. 1. Find the time taken by the ball to travel from \(O\) to \(A\).
   2. Find the components of the velocity of the ball, parallel and perpendicular to the inclined plane, as it strikes the plane at \(A\).
2. After striking \(A\), the ball rebounds and strikes the plane for a second time at a point further up than \(A\). The coefficient of restitution between the ball and the inclined plane is \(e\).
   Show that \(e < k\), where \(k\) is a constant to be determined.
   [0pt] [4 marks] \(6 \quad\) In this question use \(\cos 30 ^ { \circ } = \sin 60 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\).
   A smooth spherical ball, \(A\), is moving with speed \(u\) in a straight line on a smooth horizontal table when it hits an identical ball, \(B\), which is at rest on the table. Just before the collision, the direction of motion of \(A\) is parallel to a fixed smooth vertical wall. At the instant of collision, the line of centres of \(A\) and \(B\) makes an angle of \(60 ^ { \circ }\) with the wall, as shown in the diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{d8c723df-d10a-4fdf-b5ca-ea12633f999a-18_499_1036_721_593} The coefficient of restitution between \(A\) and \(B\) is \(e\).
3. Show that the speed of \(B\) immediately after the collision is \(\frac { 1 } { 4 } u ( 1 + e )\) and find, in terms of \(u\) and \(e\), the components of the velocity of \(A\), parallel and perpendicular to the line of centres, immediately after the collision.
4. Subsequently, \(B\) collides with the wall. After colliding with the wall, the direction of motion of \(B\) is parallel to the direction of motion of \(A\) after its collision with \(B\). Show that the coefficient of restitution between \(B\) and the wall is \(\frac { 1 + e } { 7 - e }\).
   [0pt] [7 marks]

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.