6.03l Newton's law: oblique impacts

3 A disc of mass 0.5 kg is moving with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal surface when it receives a horizontal impulse in a direction perpendicular to its direction of motion. Immediately after the impulse, the disc has speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the magnitude of the impulse received by the disc.
2. Before the impulse, the disc is moving parallel to a smooth vertical wall, as shown in the diagram. \section*{11/1/1/1/1/1/1/1/1/1/1/1/ Wall} $$\overbrace { 3 \mathrm {~ms} ^ { - 1 } } ^ { \underset { < } { \bigcirc } } \text { Disc }$$ After the impulse, the disc hits the wall and rebounds with speed \(3 \sqrt { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   Find the coefficient of restitution between the disc and the wall.
   [0pt] [4 marks]

4 Three uniform smooth spheres, \(A , B\) and \(C\), have equal radii and masses \(m , 2 m\) and \(6 m\) respectively. The spheres lie at rest in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). The sphere \(A\) is projected with speed \(u\) directly towards \(B\) and collides with it. \includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-10_218_1164_500_438} The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).

1. 1. Show that the speed of \(B\) immediately after the collision is \(\frac { 5 } { 9 } u\).
   2. Find, in terms of \(u\), the speed of \(A\) immediately after the collision.
2. Subsequently, \(B\) collides with \(C\). The coefficient of restitution between \(B\) and \(C\) is \(e\). Show that \(B\) will collide with \(A\) again if \(e > k\), where \(k\) is a constant to be determined.
3. Explain why it is not necessary to model the spheres as particles in this question.
   [0pt] [2 marks]

4 \includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-02_283_711_1754_722} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 5 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving perpendicular to the line of centres, and \(B\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution is 0.75 . Find the speed and direction of motion of each sphere immediately after the collision.

4 \includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-2_283_711_1754_722} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 5 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving perpendicular to the line of centres, and \(B\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution is 0.75 . Find the speed and direction of motion of each sphere immediately after the collision.

6 \includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-3_446_821_1007_664} Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.4 kg , and \(B\) has mass \(m \mathrm {~kg}\). Immediately before the collision, \(A\) is moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an acute angle \(\theta\) to the line of centres, and \(B\) is moving with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(30 ^ { \circ }\) to the line of centres. Immediately after the collision \(A\) is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(45 ^ { \circ }\) to the line of centres, and \(B\) is moving with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the line of centres (see diagram).

1. Find \(u\).
2. Given that \(\theta = 88.1 ^ { \circ }\) correct to 1 decimal place, calculate the approximate values of \(v\) and \(m\).
3. The coefficient of restitution is 0.75 . Show that the exact value of \(\theta\) is a root of the equation \(8 \sin \theta - 6 \cos \theta = 9 \cos 30 ^ { \circ }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{7e0f600a-18f1-458b-8549-27fca592b19c-3_419_921_267_612} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 6 kg and 3 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision the velocity of \(A\) has components \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres towards \(B\), and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the line of centres. \(B\) is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres towards \(A\) (see diagram). The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(e\), the component of the velocity of \(A\) along the line of centres immediately after the collision.
2. Given that the speeds of \(A\) and \(B\) are the same immediately after the collision, and that \(3 e ^ { 2 } = 1\), find \(v\).

5 \includegraphics[max width=\textwidth, alt={}, center]{14403602-94a6-4441-a673-65f9b98180e5-4_369_953_269_596} Two smooth uniform spheres \(A\) and \(B\), of equal radius, have masses 3 kg and 4 kg respectively. They are moving on a horizontal surface, each with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide. The directions of motion of \(A\) and \(B\) make angles \(\alpha\) and \(\beta\) respectively with the line of centres of the spheres, where \(\sin \alpha = \cos \beta = 0.6\) (see diagram). The coefficient of restitution between the spheres is 0.75 . Find the angle that the velocity of \(A\) makes, immediately after impact, with the line of centres of the spheres.
[0pt] [10]

2 \includegraphics[max width=\textwidth, alt={}, center]{08760a55-da6c-41f2-a88a-289ecc227f69-2_421_759_936_694} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 2 kg and 3 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along the line of centres, and \(B\) has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving perpendicular to the line of centres (see diagram). The coefficient of restitution is 0.6 . The direction of motion of \(B\) after the collision makes an angle of \(45 ^ { \circ }\) with the line of centres. Find the value of \(v\).

4 \includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-3_497_1157_255_493} Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.4 kg and \(B\) has mass 0.3 kg . Immediately before the collision \(A\) is moving with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an acute angle \(\theta\) to the line of centres, where \(\cos \theta = 0.6\), and \(B\) is moving with speed \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution between the spheres is 0.7. Find

1. the speed of \(B\) immediately after the collision,
2. the angle turned through by the direction of motion of \(A\) as a result of the collision.

2 \includegraphics[max width=\textwidth, alt={}, center]{43ed8ec7-67f1-418a-8d4e-ee96448647fd-2_544_816_781_603} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. They are moving in opposite directions on a horizontal surface and they collide. Immediately before the collision, each sphere has speed \(u \mathrm {~ms} ^ { - 1 }\) in a direction making an angle \(\alpha\) with the line of centres (see diagram). The coefficient of restitution between \(A\) and \(B\) is 0.5 .

1. Show that the speed of \(B\) is unchanged as a result of the collision.
2. Find the direction of motion of each of the spheres after the collision.

2 Two uniform smooth spheres \(A\) and \(B\), of equal radius and equal mass, are moving towards each other on a horizontal surface. Immediately before they collide, \(A\) has speed \(0.3 \mathrm {~ms} ^ { - 1 }\) along the line of centres and \(B\) has speed \(0.6 \mathrm {~ms} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the line of centres (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{dfe477d4-eae6-40e1-b704-1a97485f4c7e-2_302_1013_1247_502} After the collision, the direction of motion of \(B\) is at right angles to its original direction of motion. Find

1. the speed of \(B\) after the collision,
2. the speed and direction of motion of \(A\) after the collision,
3. the coefficient of restitution between \(A\) and \(B\).

5 \includegraphics[max width=\textwidth, alt={}, center]{09d3e8ca-0062-4f62-8453-7acaff591db5-3_362_841_264_651} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 2 kg and 3 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision \(A\) is moving with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(60 ^ { \circ }\) to the line of centres, and \(B\) is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution between the spheres is 0.5 . Find the speed and direction of motion of each sphere after the collision.

4 \includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-2_332_995_1375_575} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 4 kg and 3 kg respectively. They are moving on a horizontal surface, and they collide. Immediately before the collision, \(A\) is moving with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to the line of centres, where \(\sin \alpha = 0.8\), and \(B\) is moving along the line of centres with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). The coefficient of restitution between the spheres is 0.5 . Find the speed and direction of motion of each sphere after the collision.
[0pt] [10]

3 \includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-2_387_561_1055_794} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 0.8 kg and 2.0 kg respectively. The spheres are on a horizontal surface. \(A\) is moving with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(60 ^ { \circ }\) to the line of centres when it collides with \(B\), which is stationary (see diagram). The coefficient of restitution between the spheres is 0.75 . Find the speed and direction of motion of \(A\) immediately after the collision.

6 \includegraphics[max width=\textwidth, alt={}, center]{cc74a925-f1c8-4f59-a421-b46444cae5ec-5_387_867_258_575} Two smooth uniform spheres \(A\) and \(B\), of equal radius, have masses 2 kg and \(m \mathrm {~kg}\) respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving towards \(B\) at an angle of \(\alpha\) to the line of centres, where \(\cos \alpha = 0.6\). \(B\) has speed \(2 \mathrm {~ms} ^ { - 1 }\) and is moving towards \(A\) along the line of centres (see diagram). As a result of the collision, \(A\) 's loss of kinetic energy is \(7.56 \mathrm {~J} , B\) 's direction of motion is reversed and \(B\) 's speed after the collision is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the speed of \(A\) after the collision,
2. the component of \(A\) 's velocity after the collision, parallel to the line of centres, stating with a reason whether its direction is to the left or to the right,
3. the value of \(m\),
4. the coefficient of restitution between \(A\) and \(B\). \(7 S _ { A }\) and \(S _ { B }\) are light elastic strings. \(S _ { A }\) has natural length 2 m and modulus of elasticity \(120 \mathrm {~N} ; S _ { B }\) has natural length 3 m and modulus of elasticity 180 N . A particle \(P\) of mass 0.8 kg is attached to one end of each of the strings. The other ends of \(S _ { A }\) and \(S _ { B }\) are attached to fixed points \(A\) and \(B\) respectively, on a smooth horizontal table. The distance \(A B\) is \(6 \mathrm {~m} . P\) is released from rest at the point of the line segment \(A B\) which is 2.9 m from \(A\).
5. For the subsequent motion, show that the total elastic potential energy of the strings is the same when \(A P = 2.1 \mathrm {~m}\) and when \(A P = 2.9 \mathrm {~m}\). Deduce that neither string becomes slack.
6. Find, in terms of \(x\), an expression for the acceleration of \(P\) in the direction of \(A B\) when \(A P = ( 2.5 + x ) \mathrm { m }\).
7. State, giving a reason, the type of motion of \(P\) and find the time taken between successive occasions when \(P\) is instantaneously at rest. For the instant 0.6 seconds after \(P\) is released, find
8. the distance travelled by \(P\),
9. the speed of \(P\).

4 Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.1 kg and \(B\) has mass 0.2 kg . Immediately before the collision \(A\) is moving with speed \(3 \mathrm {~ms} ^ { - 1 }\) along the line of centres, and \(B\) is moving away from \(A\) with speed \(1 \mathrm {~ms} ^ { - 1 }\) at an acute angle \(\theta\) to the line of centres, where \(\cos \theta = 0.6\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-3_422_844_431_612} The coefficient of restitution between the spheres is 0.8 . Find

1. the velocity of \(A\) immediately after the collision,
2. the angle turned through by the direction of motion of \(B\) as a result of the collision.

3. A smooth uniform sphere \(P\) of mass \(m\) is falling vertically and strikes a fixed smooth inclined plane with speed \(u\). The plane is inclined at an angle \(\theta , \theta < 45 ^ { \circ }\), to the horizontal. The coefficient of restitution between \(P\) and the inclined plane is \(e\). Immediately after \(P\) strikes the plane, \(P\) moves horizontally.

1. Show that \(e = \tan ^ { 2 } \theta\).

1. \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 2 kg and sphere \(B\) has mass 3 kg . The spheres collide and at the instant of collision the line joining their centres is parallel to \(\mathbf { i }\). Before the collision \(A\) has velocity ( \(3 \mathbf { i } - \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) and after the collision it has velocity \(( - 2 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Before the collision the velocity of \(B\) makes an angle \(\alpha\) with the line of centres, as shown in Fig. 1, where \(\tan \alpha = 2\). The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\). Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) before the collision.
(9)

6. \begin{figure}[h] \end{figure} A small smooth uniform sphere \(S\) is at rest on a smooth horizontal floor at a distance \(d\) from a straight vertical wall. An identical sphere \(T\) is projected along the floor with speed \(U\) towards \(S\) and in a direction which is perpendicular to the wall. At the instant when \(T\) strikes \(S\) the line joining their centres makes an angle \(\alpha\) with the wall, as shown in Fig. 3. Each sphere is modelled as having negligible diameter in comparison with \(d\). The coefficient of restitution between the spheres is \(e\).

1. Show that the components of the velocity of \(T\) after the impact, parallel and perpendicular to the line of centres, are \(\frac { 1 } { 2 } U ( 1 - e ) \sin \alpha\) and \(U \cos \alpha\) respectively.
2. Show that the components of the velocity of \(T\) after the impact, parallel and perpendicular to the wall, are \(\frac { 1 } { 2 } U ( 1 + e ) \cos \alpha \sin \alpha\) and \(\frac { 1 } { 2 } U \left[ 2 - ( 1 + e ) \sin ^ { 2 } \alpha \right]\) respectively. The spheres \(S\) and \(T\) strike the wall at the points \(A\) and \(B\) respectively.
   Given that \(e = \frac { 2 } { 3 }\) and \(\tan \alpha = \frac { 3 } { 4 }\),
3. find, in terms of \(d\), the distance \(A B\). \section*{END}

2. \begin{figure}[h] \end{figure} A smooth uniform sphere \(P\) is at rest on a smooth horizontal plane, when it is struck by an identical sphere \(Q\) moving on the plane. Immediately before the impact, the line of motion of the centre of \(Q\) is tangential to the sphere \(P\), as shown in Fig. 1. The direction of motion of \(Q\) is turned through \(30 ^ { \circ }\) by the impact. Find the coefficient of restitution between the spheres.

1. \begin{figure}[h] \end{figure} Two smooth uniform spheres \(A\) and \(B\) have masses \(2 m \mathrm {~kg}\) and \(3 m \mathrm {~kg}\) respectively and equal radii. The spheres are moving on a smooth horizontal surface. Initially, sphere \(A\) has velocity \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and sphere \(B\) has velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). 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 spheres is \(\frac { 3 } { 7 }\). Find, in terms of \(m\), the total kinetic energy lost in the collision.

1. A smooth uniform sphere \(A\), of mass \(5 m\) and radius \(r\), is at rest on a smooth horizontal plane. A second smooth uniform sphere \(B\), of mass \(3 m\) and radius \(r\), is moving in a straight line on the plane with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and strikes \(A\). Immediately before the impact the direction of motion of \(B\) makes an angle of \(60 ^ { \circ }\) with the line of centres of the spheres. The direction of motion of \(B\) is turned through an angle of \(30 ^ { \circ }\) by the impact.

Find

1. the speed of \(B\) immediately after the impact,
2. the coefficient of restitution between the spheres.

3. \begin{figure}[h] \end{figure} Two smooth uniform spheres \(A\) and \(B\) with equal radii have masses \(m\) and \(2 m\) respectively. The spheres are moving in opposite directions on a smooth horizontal surface and collide obliquely. Immediately before the collision, \(A\) has speed \(3 u\) with its direction of motion at an angle \(\theta\) to the line of centres, and \(B\) has speed \(u\) with its direction of motion at an angle \(\theta\) to the line of centres, as shown in Figure 1. The coefficient of restitution between the spheres is \(\frac { 1 } { 8 }\) Immediately after the collision, the speed of \(A\) is twice the speed of \(B\).
Find the size of the angle \(\theta\).

1. \begin{figure}[h] \end{figure} A smooth uniform sphere \(A\) of mass \(m\) is moving on a smooth horizontal plane when it collides with a second smooth uniform sphere \(B\), which is at rest on the plane. The sphere \(B\) has mass \(4 m\) and the same radius as \(A\). Immediately before the collision the direction of motion of \(A\) makes an angle \(\alpha\) with the line of centres of the spheres, as shown in Figure 1. The direction of motion of \(A\) is turned through an angle of \(90 ^ { \circ }\) by the collision and the coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\) Find the value of \(\tan \alpha\).
1.