6.03c Momentum in 2D: vector form

1. Two particles, \(P\) and \(Q\), of masses \(3 m\) and \(2 m\) respectively, are moving on a smooth horizontal plane. They are moving in opposite directions along the same straight line when they collide directly.

Immediately before the collision, \(P\) is moving with speed \(2 u\).
The magnitude of the impulse exerted on \(P\) by \(Q\) in the collision is \(\frac { 9 m u } { 2 }\)

1. Find the speed of \(P\) immediately after the collision. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
   Given that the speed of \(Q\) immediately before the collision is \(u\),
2. find the value of \(e\).

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

A smooth uniform sphere \(A\) has mass 0.2 kg and another smooth uniform sphere \(B\), with the same radius as \(A\), has mass 0.4 kg . The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision, the velocity of \(A\) is \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - 4 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) At the instant of collision, the line joining the centres of the spheres is parallel to \(\mathbf { i }\) The coefficient of restitution between the spheres is \(\frac { 3 } { 7 }\)

1. Find the velocity of \(A\) immediately after the collision.
2. Find the magnitude of the impulse received by \(A\) in the collision.
3. Find, to the nearest degree, the size of the angle through which the direction of motion of \(A\) is deflected as a result of the collision.

1. A smooth uniform sphere \(P\) has mass 0.3 kg . Another smooth uniform sphere \(Q\), with the same radius as \(P\), has mass 0.2 kg .

The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision the velocity of \(P\) is \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(( - 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At the instant of collision, the line joining the centres of the spheres is parallel to \(\mathbf { i }\).
The kinetic energy of \(Q\) immediately after the collision is half the kinetic energy of \(Q\) immediately before the collision.

1. Find
   1. the velocity of \(P\) immediately after the collision,
   2. the velocity of \(Q\) immediately after the collision,
   3. the coefficient of restitution between \(P\) and \(Q\),
      carefully justifying your answers.
2. Find the size of the angle through which the direction of motion of \(P\) is deflected by the collision.

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

A smooth uniform sphere \(P\) has mass 0.3 kg . Another smooth uniform sphere \(Q\), with the same radius as \(P\), has mass 0.5 kg . The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision the velocity of \(P\) is \(( u \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(u\) is a positive constant, and the velocity of \(Q\) is \(( - 4 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) At the instant when the spheres collide, the line joining their centres is parallel to \(\mathbf { i }\).
The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 3 } { 5 }\) As a result of the collision, the direction of motion of \(P\) is deflected through an angle of \(90 ^ { \circ }\) and the direction of motion of \(Q\) is deflected through an angle of \(\alpha ^ { \circ }\)

1. Find the value of \(u\)
2. Find the value of \(\alpha\)
3. State how you have used the fact that \(P\) and \(Q\) have equal radii.

4. \begin{figure}[h] \end{figure} Two smooth uniform spheres, \(A\) and \(B\), have equal radii. The mass of \(A\) is \(3 m\) and the mass of \(B\) is \(4 m\). The spheres are moving on a smooth horizontal plane when they collide obliquely. Immediately before they collide, \(A\) is moving with speed \(3 u\) at \(30 ^ { \circ }\) to the line of centres of the spheres and \(B\) is moving with speed \(2 u\) at \(30 ^ { \circ }\) to the line of centres of the spheres. The direction of motion of \(B\) is turned through an angle of \(90 ^ { \circ }\) by the collision, as shown in Figure 3.

1. Find the size of the angle through which the direction of motion of \(A\) is turned as a result of the collision.
2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(B\) in the collision.

5. \begin{figure}[h] \end{figure} A smooth uniform sphere \(S\) of mass \(m\) is moving with speed \(U\) on a smooth horizontal plane. The sphere \(S\) collides obliquely with another uniform sphere of mass \(M\) which is at rest on the plane. The two spheres have the same radius. Immediately before the collision the direction of motion of \(S\) makes an angle \(\alpha\), where \(0 < \alpha < 90 ^ { \circ }\), with the line joining the centres of the spheres. Immediately after the collision the direction of motion of \(S\) makes an angle \(\beta\) with the line joining the centres of the spheres, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\).

1. Show that \(\tan \beta = \frac { ( m + M ) \tan \alpha } { ( m - e M ) }\) Given that \(m = e M\),
2. show that the directions of motion of the two spheres immediately after the collision are perpendicular.

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.]

A particle \(A\) has mass 3 kg and a particle \(B\) has mass 2 kg .
The particles are moving on a smooth horizontal plane when they collide directly.
Immediately before the collision, the velocity of \(A\) is \(( 3 \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the velocity of \(B\) is \(( - 6 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) Immediately after the collision the velocity of \(A\) is \(\left( - 2 \mathbf { i } + \frac { 2 } { 3 } \mathbf { j } \right) \mathrm { ms } ^ { - 1 }\)

1. Find the total kinetic energy of the two particles before the collision.
2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the impulse exerted on \(A\) by \(B\) in the collision.
3. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) immediately after the collision.

7. \begin{figure}[h] \end{figure} A smooth uniform sphere \(A\) of mass \(m\) is moving with speed \(U\) on a smooth horizontal plane. The sphere \(A\) collides obliquely with a smooth uniform sphere \(B\) of mass \(3 m\) which is at rest on the plane. The two spheres have the same radius. Immediately before the collision, the direction of motion of \(A\) makes an angle \(\alpha\), where \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), with the line joining the centres of the spheres. Immediately after the collision, the direction of motion of \(A\) is perpendicular to its original direction, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\).

1. Show that the speed of \(B\) immediately after the collision is $$\frac { 1 } { 4 } ( 1 + e ) U \cos \alpha$$
2. Show that \(e > \frac { 1 } { 3 }\)
3. Show that \(0 < \tan \alpha \leqslant \frac { 1 } { \sqrt { 2 } }\)

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

A smooth uniform sphere \(A\) has mass \(2 m \mathrm {~kg}\) and another smooth uniform sphere \(B\), with the same radius as \(A\), has mass \(3 m \mathrm {~kg}\). The spheres are moving on a smooth horizontal plane when they collide obliquely.
Immediately before the collision the velocity of \(A\) is \(( 3 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - 5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At the instant of collision, the line joining the centres of the spheres is parallel to \(\mathbf { i }\).
The coefficient of restitution between the spheres is \(\frac { 1 } { 4 }\)

1. Find the velocity of \(B\) immediately after the collision.
2. Find, to the nearest degree, the size of the angle through which the direction of motion of \(B\) is deflected as a result of the collision.

5 Two particles \(A\) and \(B\) are on a smooth horizontal floor with \(B\) between \(A\) and a vertical wall. The masses of \(A\) and \(B\) are 4 kg and 11 kg respectively. Initially, \(B\) is at rest and \(A\) is moving towards \(B\) with a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). \(A\) collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\). \includegraphics[max width=\textwidth, alt={}, center]{bf86ac88-0fd1-4d49-a705-9b8d06fbac2a-3_209_803_1658_630}

1. Show that immediately after the collision the speed of \(B\) is \(\frac { 4 } { 15 } u ( 1 + e )\). After the collision between \(A\) and \(B\) the direction of motion of \(A\) is reversed. \(B\) subsequently collides directly with the vertical wall. The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 } e\).
2. Given that there is a second collision between \(A\) and \(B\), find the range of possible values of \(e\).

2 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface. Particle \(A\) has mass 2 kg and velocity \(\left[ \begin{array} { r } 3 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). Particle \(B\) has mass 3 kg and velocity \(\left[ \begin{array} { r } - 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). The two particles collide, and they coalesce during the collision.

1. Find the velocity of the combined particles after the collision.
2. Find the speed of the combined particles after the collision.

5 Two smooth spheres, \(A\) and \(B\), of equal radii and different masses are moving on a smooth horizontal surface when they collide. Just before the collision, \(A\) is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the line of centres of the spheres, and \(B\) is moving with speed \(3 \mathrm {~ms} ^ { - 1 }\) perpendicular to the line of centres, as shown in the diagram below. \begin{figure}[h] \end{figure} Immediately after the collision, \(A\) and \(B\) move with speeds \(u\) and \(v\) in directions which make angles of \(90 ^ { \circ }\) and \(40 ^ { \circ }\) respectively with the line of centres, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-4_392_1102_1155_392}

1. Show that \(v = 4.67 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
2. Find the coefficient of restitution between the spheres.
3. Given that the mass of \(A\) is 0.5 kg , show that the magnitude of the impulse exerted on \(A\) during the collision is 2.17 Ns , correct to three significant figures.
4. Find the mass of \(B\).

2 Two particles of equal mass are moving on a horizontal surface when they collide.
Immediately before the collision, their velocities are \(\left[ \begin{array} { l } 2 \\ 4 \end{array} \right] \mathrm { ms } ^ { - 1 }\) and \(\left[ \begin{array} { c } 6 \\ - 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\) As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle, immediately after the collision.
Circle your answer.
[0pt] [1 mark] \(\left[ \begin{array} { l } 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) \(\left[ \begin{array} { l } 4 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) \(\left[ \begin{array} { l } 8 \\ 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\) \(\left[ \begin{array} { l } 8 \\ 6 \end{array} \right] \mathrm { ms } ^ { - 1 }\)

20 J
25 J
50 J
100 J 2 Two particles of equal mass are moving on a horizontal surface when they collide.
Immediately before the collision, their velocities are \(\left[ \begin{array} { l } 2 \\ 4 \end{array} \right] \mathrm { ms } ^ { - 1 }\) and \(\left[ \begin{array} { c } 6 \\ - 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\) As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle, immediately after the collision.
Circle your answer.
[0pt] [1 mark] \(\left[ \begin{array} { l } 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) \(\left[ \begin{array} { l } 4 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) \(\left[ \begin{array} { l } 8 \\ 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\) \(\left[ \begin{array} { l } 8 \\ 6 \end{array} \right] \mathrm { ms } ^ { - 1 }\) 3 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A ball of mass of 0.75 kg is thrown vertically upwards with an initial speed of \(12 \mathrm {~ms} ^ { - 1 }\) The ball is thrown from ground level. 3

1. Calculate the initial kinetic energy of the ball. 3
2. The maximum height of the ball above the ground is \(h\) metres.
   Jeff and Gurjas use an energy method to find \(h\) Jeff concludes that \(h = 7.3\) Gurjas concludes that \(h < 7.3\) Explain the reasoning that they have used, showing any calculations that you make.
   4 Wavelength is defined as the distance from the highest point on one wave to the highest point on the next wave. Surfers classify waves into one of several types related to their wavelengths.
   Two of these classifications are deep water waves and shallow water waves.
   4
   1. The wavelength \(w\) of a deep water wave is given by $$w = \frac { g t ^ { 2 } } { k }$$ where \(g\) is the acceleration due to gravity and \(t\) is the time period between consecutive waves. Given that the formula for a deep water wave is dimensionally consistent, show that \(k\) is a dimensionless constant. 4
   2. The wavelength \(w\) of a shallow water wave is given by $$w = ( g d ) ^ { \alpha } t ^ { \beta }$$ where \(g\) is the acceleration due to gravity, \(d\) is the depth of water and \(t\) is the time period between consecutive waves. Use dimensional analysis to find the values of \(\alpha\) and \(\beta\) 5 A car, of mass 1000 kg , is travelling on a straight horizontal road. When the car travels at a speed of \(v \mathrm {~ms} ^ { - 1 }\), it experiences a resistance force of magnitude \(25 v\) newtons. The car has a maximum speed of \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on the straight road.
      Find the maximum power output of the car.
      Fully justify your answer.

7 Two snooker balls, one white and one red, have equal mass. The balls are on a horizontal table \(A B C D\) The white ball is struck so that it moves at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to \(A B\) The white ball hits a stationary red ball.
After the collision, the white ball moves at a speed of \(0.8 \mathrm {~ms} ^ { - 1 }\) and at an angle of \(30 ^ { \circ }\) to \(A B\) After the collision, the red ball moves at a speed \(v \mathrm {~ms} ^ { - 1 }\) and at an angle \(\theta\) to \(C D\) When the collision takes place, the white ball is the same distance from \(A B\) as the distance the red ball is from CD The diagram below shows the table and the velocities of the balls after the collision. \includegraphics[max width=\textwidth, alt={}, center]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-08_595_1370_1121_335} Not to scale After the collision, the white ball hits \(A B\) and the red ball hits \(C D\) Model the balls as particles that do not experience any air resistance.
7

1. Explain why the two balls hit the sides of the table at the same time.
   7
2. Show that \(\theta = 17.0 ^ { \circ }\) correct to one decimal place.
   7
3. \(\quad\) Find \(v\) 7
4. Determine which ball travels the greater distance after the collision and before hitting the side of the table. Fully justify your answer.
   7
5. State one possible refinement to the model that you have used. \(8 \quad\) In this question use \(g\) as \(9.8 \mathrm {~ms} ^ { - 2 }\) A rope is used to pull a crate, of mass 60 kg , along a rough horizontal surface.
   The coefficient of friction between the crate and the surface is 0.4 The crate is at rest when the rope starts to pull on it.
   The tension in the rope is 240 N and the rope makes an angle of \(30 ^ { \circ }\) to the horizontal.
   When the crate has moved 5 metres, the rope becomes detached from the crate.

4 Two particles \(A\) and \(B\), of masses \(m \mathrm {~kg}\) and 1 kg respectively, are connected by a light inextensible string of length \(d \mathrm {~m}\) and placed at rest on a smooth horizontal plane a distance of \(\frac { 1 } { 2 } d \mathrm {~m}\) apart. \(B\) is then projected horizontally with speed \(v \mathrm {~ms} ^ { - 1 }\) in a direction perpendicular to \(A B\).

1. Show that, at the instant that the string becomes taut, the magnitude of the instantaneous impulse in the string, \(I \mathrm { Ns }\), is given by \(I = \frac { \sqrt { 3 } m v } { 2 ( 1 + m ) }\).
2. Find, in terms of \(m\) and \(v\), the kinetic energy of \(B\) at the instant after the string becomes taut. Give your answer as a single algebraic fraction.
3. In the case where \(m\) is very large, describe, with justification, the approximate motion of \(B\) after the string becomes taut.

3 Two smooth circular discs \(A\) and \(B\) are moving on a horizontal plane. The masses of \(A\) and \(B\) are 3 kg and 4 kg respectively. At the instant before they collide

• the velocity of \(A\) is \(2 \mathrm {~ms} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the line joining their centres,
• the velocity of \(B\) is \(5 \mathrm {~ms} ^ { - 1 }\) towards \(A\) along the line joining their centres (see Fig. 3).

\begin{figure}[h] \end{figure} Given that the velocity of \(A\) after the collision is perpendicular to the velocity of \(A\) before the collision, find

1. the coefficient of restitution between \(A\) and \(B\),
2. the total loss of kinetic energy as a result of the collision.

1. Two smooth spheres \(A\) and \(B\) are moving on a smooth horizontal plane when they collide obliquely. When the spheres collide, the line joining their centres is parallel to the vector \(\mathbf { j }\), as shown in the diagram below.

Immediately before the collision, sphere \(A\) has velocity ( \(6 \mathbf { i } - 3 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) and sphere \(B\) has velocity \(( - 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Sphere \(A\) has mass 6 kg and sphere \(B\) has mass 2 kg . \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-02_595_972_753_534} Immediately after the collision, sphere \(B\) has velocity \(( - 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).

1. Find the velocity of \(A\) immediately after the collision.
2. Calculate the coefficient of restitution between \(A\) and \(B\).
3. Find the angle through which the direction of motion of \(B\) is deflected as a result of the collision. Give your answer correct to the nearest degree.
4. After the collision, sphere \(B\) continues to move with velocity \(( - 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) until it collides with another sphere \(C\), which exerts an impulse of \(( - 20 \mathbf { i } + 18 \mathbf { j } )\) Ns on \(B\). Find the velocity of \(B\) after the collision with \(C\).
   \section*{PLEASE DO NOT WRITE ON THIS PAGE}

11 A smooth sphere of mass 2 kg has velocity \(( 24 \mathbf { i } - 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and is travelling on a horizontal plane, where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in the horizontal plane. The sphere strikes a vertical wall. The line of intersection of the wall and the plane is in the direction \(( 4 \mathbf { i } + 3 \mathbf { j } )\).

1. Show that the acute angle between the path of the sphere before the impact and the direction of the wall is \(\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right)\).
2. After the impact, the velocity of the sphere is \(( 7.2 \mathbf { i } + 15.4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
   1. the coefficient of restitution between the sphere and the wall,
   2. the magnitude of the impulse exerted by the sphere on the wall.

11 \includegraphics[max width=\textwidth, alt={}, center]{adf5bd3c-5408-421d-b7d5-dea2d0f0185b-6_438_951_255_559} A smooth sphere \(P\) of mass \(3 m\) is at rest on a smooth horizontal table. A second smooth sphere \(Q\) of mass \(m\) and the same radius as \(P\) is moving along the table towards \(P\) and strikes it obliquely (see diagram). After the collision, the directions of motion of the two spheres are perpendicular.

1. Find the coefficient of restitution.
2. Given that one-sixth of the original kinetic energy is lost as a result of the collision, find the angle between the initial direction of motion of \(Q\) and the line of centres.

Small smooth spheres \(A\) and \(B\), of equal radii and of masses 4 kg and 2 kg respectively, lie on a smooth horizontal plane. Initially \(B\) is at rest and \(A\) is moving towards \(B\) with speed \(10 \text{ ms}^{-1}\). After the spheres collide \(A\) continues to move in the same direction but with half the speed of \(B\).

1. Find the speed of \(B\) after the collision. [2]

A third small smooth sphere \(C\), of mass 1 kg and with the same radius as \(A\) and \(B\), is at rest on the plane. \(B\) now collides directly with \(C\). After this collision \(B\) continues to move in the same direction but with one third the speed of \(C\).

1. Show that there is another collision between \(A\) and \(B\). [3]
2. \(A\) and \(B\) coalesce during this collision. Find the total loss of kinetic energy in the system due to the three collisions. [5]

Two particles \(P\) and \(Q\), of masses \(m\) kg and \(0.3\) kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected at a speed of \(5\) m s\(^{-1}\) directly towards \(Q\). After \(P\) and \(Q\) collide, \(P\) moves with a speed of \(2\) m s\(^{-1}\) in the same direction as it was originally moving.

1. Find, in terms of \(m\), the speed of \(Q\) after the collision. [2]

After this collision, \(Q\) moves directly towards a third particle \(R\), of mass \(0.6\) kg, which is at rest on the plane. \(Q\) is brought to rest in the collision with \(R\), and \(R\) begins to move with a speed of \(1.5\) m s\(^{-1}\).

1. Find the value of \(m\). [2]

Three particles \(A\), \(B\) and \(C\) of masses 5 kg, 1 kg and 2 kg respectively lie at rest in that order on a straight smooth horizontal track \(XYZ\). Initially \(A\) is at \(X\), \(B\) is at \(Y\) and \(C\) is at \(Z\). Particle \(A\) is projected towards \(B\) with a speed of \(6\text{ ms}^{-1}\) and at the same instant \(C\) is projected towards \(B\) with a speed of \(v\text{ ms}^{-1}\). In the subsequent motion, \(A\) collides and coalesces with \(B\) to form particle \(D\). Particle \(D\) then collides and coalesces with \(C\) to form particle \(E\) and \(E\) moves towards \(Z\).

1. Show that after the second collision the speed of \(E\) is \(\frac{15-v}{4}\text{ ms}^{-1}\). [3]
2. The total loss of kinetic energy of the system due to the two collisions is 63 J. Use the result from (a) to show that \(v = 3\). [3]
3. It is given that the distance \(XY\) is 36 m and the distance \(YZ\) is 98 m.
   1. Find the time between the two collisions. [4]
   2. Find the time between the instant that \(A\) is projected from \(X\) and the instant that \(E\) reaches \(Z\). [1]