Oblique collision of spheres

2. \begin{figure}[h] \end{figure} Two smooth uniform spheres \(A\) and \(B\) have masses \(3 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively and equal radii. The spheres are moving on a smooth horizontal surface. Initially, sphere \(A\) has velocity \(( 5 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and sphere \(B\) has velocity \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). When the spheres collide, the line joining their centres is parallel to \(\mathbf { j }\), as shown in Figure 1.
The coefficient of restitution between the two spheres is \(e\).
The kinetic energy of sphere \(B\) immediately after the collision is \(85 \%\) of its kinetic energy immediately before the collision. Find

1. the velocity of each sphere immediately after the collision,
2. the value of \(e\).

7. Two smooth uniform spheres \(A\) and \(B\), of mass 2 kg and 3 kg respectively, and of equal radius, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of \(A\) is \(( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Show that, at the instant when \(A\) and \(B\) collide, their line of centres is parallel to \(- \mathbf { i } + \mathbf { j }\).
2. Find the velocity of \(B\) immediately after the collision.
3. Find the coefficient of restitution between \(A\) and \(B\).

3. \begin{figure}[h] \end{figure} A smooth uniform sphere \(A\), moving on a smooth horizontal table, collides with a second identical sphere \(B\) which is at rest on the table. When the spheres collide the line joining their centres makes an angle of \(30 ^ { \circ }\) with the direction of motion of \(A\), as shown in Fig. 1. The coefficient of restitution between the spheres is \(e\). The direction of motion of \(A\) is deflected through an angle \(\theta\) by the collision. Show that \(\tan \theta = \frac { ( 1 + e ) \sqrt { 3 } } { 5 - 3 e }\).
(10 marks)

4. \begin{figure}[h] \end{figure} Two smooth uniform spheres \(A\) and \(B\), of equal radius, are moving on a smooth horizontal plane. Sphere \(A\) has mass 3 kg and velocity \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), and sphere \(B\) has mass 5 kg and velocity \(( - \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When the spheres collide the line joining their centres is parallel to \(\mathbf { i }\), as shown in Fig. 2. Given that the direction of \(A\) is deflected through a right angle by the collision, find

1. the velocity of \(A\) after the collision,
2. the coefficient of restitution between the spheres.

5 Two small uniform discs, A of mass \(2 m \mathrm {~kg}\) and B of mass \(3 m \mathrm {~kg}\), slide on a smooth horizontal surface and collide obliquely with smooth contact. Immediately before the collision, A is moving towards B along the line of centres with speed \(2 \mathrm {~ms} ^ { - 1 }\) and B is moving towards A with speed \(\sqrt { 3 } \mathrm {~ms} ^ { - 1 }\) in a direction making an angle of \(30 ^ { \circ }\) with the line of centres, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{feb9a438-26b0-41d3-b044-6acd6efccde0-5_366_976_539_244}

1. Explain how you know that the motion of A will be along the line of centres after the collision.
2. - Determine the maximum possible speed of A after the collision.
   • Find the value of the coefficient of restitution in this case.
   • - Determine the minimum possible speed of B after the collision.
   • Find the value of the coefficient of restitution in this case.
   When the speed of B after the collision is a minimum, the loss of kinetic energy in the collision is 1.4625 J .
3. Determine the value of \(m\).

4 Two small smooth discs, A of mass 0.5 kg and B of mass 0.4 kg , collide while sliding on a smooth horizontal plane. Immediately before the collision A and B are moving towards each other, A with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) with speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Before the collision the direction of motion of A makes an angle \(\alpha\) with the line of centres, where \(\tan \alpha = 0.75\), and the direction of motion of B makes an angle of \(60 ^ { \circ }\) with the line of centres, as shown in Fig. 4. \begin{figure}[h] \end{figure} After the collision, one of the discs moves in a direction perpendicular to the line of centres, and the other disc moves in a direction making an angle \(\beta\) with the line of centres.

1. Explain why the disc which moves perpendicular to the line of centres must be A .
2. Determine the value of \(\beta\).
3. Determine the kinetic energy lost in the collision.
4. Determine the value of the coefficient of restitution between A and B .

4 Two uniform circular discs with the same radius, A of mass 1 kg and B of mass 5.25 kg , slide on a smooth horizontal surface and collide obliquely with smooth contact. Fig. 4 gives information about the velocities of the discs just before and just after the collision.

• The line XY passes through the centres of the discs at the moment of collision
• The components parallel and perpendicular to XY of the velocities of A are shown
• Before the collision, B is at rest and after it is moving at \(2 \mathrm {~ms} ^ { - 1 }\) in the direction XY

\begin{figure}[h] \end{figure} The coefficient of restitution between the two discs is \(\frac { 2 } { 3 }\).

1. Find the values of \(U\) and \(u\).
2. What information in the question tells you that \(v = V\) ? The speed of disc A before the collision is \(8.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the speed of disc A after the collision. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-5_398_396_397_475} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-5_399_332_399_945} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-5_305_326_493_1354} \captionsetup{labelformat=empty} \caption{Fig. 5.3}
   \end{figure} Fig. 5.1 shows a vertical light elastic spring. It is fixed to a horizontal table at one end. Fig 5.2 shows the spring with a particle of mass \(m \mathrm {~kg}\) attached to it at the other end. The system is in equilibrium when the spring is compressed by a distance \(h \mathrm {~m}\).

12 Two small uniform smooth spheres A and B are of equal radius and have masses \(m\) and \(\lambda m\) respectively. The spheres are on a smooth horizontal surface. Sphere A is moving on the surface with velocity \(u _ { 1 } \mathbf { i } + u _ { 2 } \mathbf { j }\) towards B , which is at rest.
The spheres collide obliquely. When the spheres collide, the line joining their centres is parallel to \(\mathbf { i }\). The coefficient of restitution between A and B is \(e\).

1. 1. Explain why, when the spheres collide, the impulse of A on B is in the direction of \(\mathbf { i }\).
   2. Determine this impulse in terms of \(\lambda , m , e\) and \(u _ { 1 }\). The loss in kinetic energy due to the collision between A and B is \(\frac { 1 } { 8 } \mathrm { mu } _ { 1 } { } ^ { 2 }\).
2. Determine the range of possible values of \(\lambda\).

11 Two uniform small smooth spheres A and B have equal radii and equal masses. The spheres are on a smooth horizontal surface. Sphere A is moving at an acute angle \(\alpha\) to the line of centres, when it collides with B, which is stationary. After the impact A is moving at an acute angle \(\beta\) to the line of centres. The coefficient of restitution between A and B is \(\frac { 1 } { 3 }\).

1. Show that \(\tan \beta = 3 \tan \alpha\).
2. Explain why the assumption that the contact between the spheres is smooth is needed in answering part (a). It is given that A is deflected through an angle \(\gamma\).
3. Determine, in terms of \(\alpha\), an expression for \(\tan \gamma\).
4. Determine the maximum value of \(\gamma\). You do not need to justify that this value is a maximum. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cce64530-6284-409d-867a-e26c27d3e50a-09_488_903_264_258} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} Fig. 12 shows a hemispherical bowl. The rim of this bowl is a circle with centre O and radius \(r\). The bowl is fixed with its rim horizontal and uppermost. A particle P , of mass \(m\), is connected by a light inextensible string of length \(l\) to the lowest point A on the bowl and describes a horizontal circle with constant angular speed \(\omega\) on the smooth inner surface of the bowl. The string is taut, and AP makes an angle \(\alpha\) with the vertical.
5. Show that the normal contact force between P and the bowl is of magnitude \(m g + 2 m r \omega ^ { 2 } \cos ^ { 2 } \alpha\).
6. Deduce that \(g < r \omega ^ { 2 } \left( k _ { 1 } + k _ { 2 } \cos ^ { 2 } \alpha \right)\), stating the value of the constants \(k _ { 1 }\) and \(k _ { 2 }\).

5 Two small uniform smooth spheres A and B , of equal radius, have masses 2 kg and 4 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, A has speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along the line of centres, and B has speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along a line which is perpendicular to the line of centres (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-03_389_764_1592_244} The direction of motion of B after the collision makes an angle of \(45 ^ { \circ }\) with the line of centres. Determine the coefficient of restitution between A and B .

5. In this question, \(\mathbf { i }\) and \(\mathbf { j }\) represent unit vectors due east and due north respectively. Two smooth spheres \(P\) and \(Q\), of equal radii, are moving on a smooth horizontal surface. The mass of \(P\) is 2 kg and the mass of \(Q\) is 6 kg . The velocity of \(P\) is \(( 8 \mathbf { i } - 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the velocity of \(Q\) is \(( 4 \mathbf { i } + 10 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). At a particular instant, \(Q\) is positioned 12 m east and 48 m south of \(P\).

1. Prove that \(P\) and \(Q\) will collide. At the instant the spheres collide, the line joining their centres is parallel to the vector \(\mathbf { j }\). Immediately after the collision, sphere \(Q\) has speed \(5 \mathrm {~ms} ^ { - 1 }\).
2. Determine the coefficient of restitution between the spheres and hence calculate the velocity of \(P\) immediately after the collision.
3. Find the magnitude of the impulse required to stop sphere \(P\) after 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 \(A\) 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.

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 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 \(B\).

5. \section*{Figure 1} A smooth uniform sphere \(S\) of mass \(m\) is moving on a smooth horizontal table. The sphere \(S\) collides with another smooth uniform sphere \(T\), of the same radius as \(S\) but of mass \(k m , k > 1\), which is at rest on the table. The coefficient of restitution between the spheres is \(e\). Immediately before the spheres collide the direction of motion of \(S\) makes an angle \(\theta\) with the line joing their centres, as shown in Fig. 1. Immediately after the collision the directions of motion of \(S\) and \(T\) are perpendicular.

1. Show that \(e = \frac { 1 } { k }\).
   (6) Given that \(k = 2\) and that the kinetic energy lost in the collision is one quarter of the initial kinetic energy,
2. find the value of \(\theta\).
   (6)

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.

12 Two small uniform discs A and B , of equal radius, have masses 3 kg and 5 kg respectively. The discs are sliding on a smooth horizontal surface and collide obliquely. The contact between the discs is smooth and A is stationary after the collision.
Immediately before the collision B is moving with speed \(2 \mathrm {~ms} ^ { - 1 }\) in a direction making an angle of \(60 ^ { \circ }\) with the line of centres, XY (see diagram below).
\includegraphics[max width=\textwidth, alt={}, center]{dc8515fb-7104-4d3d-a8ea-ada2b75c70a2-7_458_985_632_242}

1. Explain how you can tell that A must have been moving along XY before the collision. The coefficient of restitution between A and B is 0.8 .
2. - Determine the speed of A immediately before the collision.
   • Determine the speed of B immediately after the collision.
   • Determine the angle turned through by the direction of B in the collision.
   Disc B subsequently collides with a smooth wall, which is parallel to XY. The kinetic energy of B after the collision with the wall is \(95 \%\) of the kinetic energy of B before the collision with the wall.
3. Determine the coefficient of restitution between B and the wall.

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.