Oblique and successive collisions

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 Two smooth billiard balls \(A\) and \(B\), of identical size and equal mass, move towards each other on a horizontal surface and collide. Just before the collision, \(A\) has velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(30 ^ { \circ }\) to the line of centres of the balls, and \(B\) has velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(60 ^ { \circ }\) to the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-4_508_1420_532_294} The coefficient of restitution between the balls is \(\frac { 1 } { 2 }\).

1. Find the speed of \(B\) immediately after the collision.
2. Find the angle between the velocity of \(B\) and the line of centres of the balls immediately after the collision.

8 Two smooth circular discs, \(A\) and \(B\), have equal radii and are free to move on a smooth horizontal plane. The masses of \(A\) and \(B\) are 1 kg and \(m \mathrm {~kg}\) respectively. \(B\) is initially placed at rest with its centre at the origin, \(O\). \(A\) is projected towards \(B\) with a velocity of \(u \mathrm {~ms} ^ { - 1 }\) at an angle of \(\theta\) to the negative \(y\)-axis where \(\tan \theta = \frac { 5 } { 2 }\). At the instant of collision the line joining their centres lies on the \(x\)-axis. There are two straight vertical walls on the plane. One is perpendicular to the \(x\)-axis and the other is perpendicular to the \(y\)-axis. The walls are an equal distance from \(O\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{857eca7f-c42d-49a9-ac39-a2fb5bcb9cd5-7_944_1241_694_242} After \(A\) and \(B\) have collided with each other, each of them goes on to collide with a wall. Each then rebounds and they collide again at the same place as their first collision, with disc \(B\) again at \(O\). The coefficient of restitution between \(A\) and \(B\) is denoted by \(e\). The coefficient of restitution between \(A\) and the wall that it collides with is also \(e\) while the coefficient of restitution between \(B\) and the wall that it collides with is \(\frac { 5 } { 9 } e\). It is assumed that any resistance to the motion of \(A\) and \(B\) may be ignored.

1. Explain why it must be the case that the collision between \(A\) and the wall that it collides with is not inelastic.
2. Show that \(\mathrm { e } = \frac { 1 } { \mathrm {~m} }\).
3. Show that \(m = \frac { 5 } { 3 }\).
4. State one limitation of the model used.

2. Two small smooth spheres \(A\) and \(B\) have equal radii. The mass of \(A\) is \(2 m \mathrm {~kg}\) and the mass of \(B\) is \(m \mathrm {~kg}\). The spheres are moving on a smooth horizontal plane and they collide. Immediately before the collision the velocity of \(A\) is \(( 2 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - 3 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(( \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the speed of \(B\) immediately after the collision.
(5)

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

A particle \(A\) has mass 2 kg and a particle \(B\) has mass 3 kg . The particles are moving on a smooth horizontal plane when they collide. Immediately before the collision, the velocity of \(A\) is \(5 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the velocity of \(B\) is \(( 3 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) Immediately after the collision, the velocity of \(A\) is \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)

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 received by \(A\) in the collision. Given that, in the collision, the impulse of \(A\) on \(B\) is equal and opposite to the impulse of \(B\) on \(A\),
3. find the velocity of \(B\) immediately after the collision.

7 Two smooth spheres, \(A\) and \(B\), have equal radii and masses \(2 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres are moving on a smooth horizontal plane. The sphere \(A\) has velocity \(( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it collides with the sphere \(B\), which has velocity \(( 2 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after the collision, the velocity of the sphere \(B\) is \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the velocity of \(A\) immediately after the collision.
2. Show that the impulse exerted on \(B\) in the collision is \(( 6 m \mathbf { j } )\) Ns.
3. Find the coefficient of restitution between the two spheres.
4. After the collision, each sphere moves in a straight line with constant speed. Given that the radius of each sphere is 0.05 m , find the time taken, from the collision, until the centres of the spheres are 1.10 m apart.

1. A smooth sphere \(S\) is moving on a smooth horizontal plane with speed \(u\) when it collides with a smooth fixed vertical wall. At the instant of collision the direction of motion of \(S\) makes an angle of \(30 ^ { \circ }\) with the wall. The coefficient of restitution between \(S\) and the wall is \(\frac { 1 } { 3 }\).

Find the speed of \(S\) immediately after the collision.

2 \includegraphics[max width=\textwidth, alt={}, center]{2c6b6722-ebba-4ade-9a9d-dd70e61cf52b-2_312_409_525_868} A small smooth ball \(P\) is moving on a smooth horizontal plane with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It strikes a smooth vertical barrier at an angle \(\alpha\) (see diagram). The coefficient of restitution between \(P\) and the barrier is 0.4 . Given that the speed of \(P\) is halved as a result of the collision, find the value of \(\alpha\).

5 Two spheres \(A\) and \(B\), of equal radius, have masses \(m _ { 1 }\) and \(m _ { 2 }\) respectively. They lie at rest on a smooth horizontal plane. Sphere \(A\) is projected directly towards sphere \(B\) with speed \(u\) and, as a result of the collision, \(A\) is brought to rest. Show that

1. the speed of \(B\) immediately after the collision cannot exceed \(u\),
2. \(m _ { 1 } \leqslant m _ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-3_273_611_1745_767} After the collision, \(B\) hits a smooth vertical wall which is at an angle of \(60 ^ { \circ }\) to the direction of motion of \(B\) (see diagram). In the impact with the wall \(B\) loses \(\frac { 2 } { 3 }\) of its kinetic energy. Find the coefficient of restitution between \(B\) and the wall and show that the direction of motion of \(B\) turns through \(90 ^ { \circ }\).

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.

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 \(P\).

6 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. 6).

\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.

2 Three uniform small smooth spheres \(A , B\) and \(C\), of equal radii and of masses \(4 m , \lambda m\) and \(m\) respectively, are at rest in a straight line on a smooth horizontal plane, with \(B\) between \(A\) and \(C\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\). The coefficient of restitution between \(A\) and \(B\), and between \(B\) and \(C\), is \(\frac { 1 } { 2 }\). Show that the speed of \(B\) after it is struck by \(A\) is \(\frac { 6 u } { \lambda + 4 }\). Given that the speed of \(C\) after it is struck by \(B\) is \(u\), find the value of \(\lambda\).

1

1. In this part-question, all the objects move along the same straight line on a smooth horizontal plane. All their collisions are direct. The masses of the objects \(\mathrm { P } , \mathrm { Q }\) and R and the initial velocities of P and Q (but not R ) are shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-2_177_1011_488_529} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} A force of 21 N acts on P for 2 seconds in the direction \(\mathrm { PQ } . \mathrm { P }\) does not reach Q in this time.
   1. Calculate the speed of P after the 2 seconds. The force of 21 N is removed after the 2 seconds. When P collides with Q they stick together (coalesce) to form an object S of mass 6 kg .
   2. Show that immediately after the collision S has a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards R . The collision between S and R is elastic with coefficient of restitution \(\frac { 1 } { 4 }\). After the collision, S has a velocity of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of its motion before the collision.
   3. Find the velocities of R before and after the collision. \section*{(b) In this part-question take \(\boldsymbol { g } = \mathbf { 1 0 }\).} A particle of mass 0.2 kg is projected vertically downwards with initial speed \(5 \mathrm {~ms} ^ { - 1 }\) and it travels 10 m before colliding with a fixed smooth plane. The plane is inclined at \(\alpha\) to the vertical where \(\tan \alpha = \frac { 3 } { 4 }\). Immediately after its collision with the plane, the particle has a speed of \(13 \mathrm {~ms} ^ { - 1 }\). This information is shown in Fig. 1.2. Air resistance is negligible. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-2_383_341_1795_854} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure}
   4. Calculate the angle between the direction of motion of the particle and the plane immediately after the collision. Calculate also the coefficient of restitution in the collision.
   5. Calculate the magnitude of the impulse of the plane on the particle.

3 Three uniform small smooth spheres \(A , B\) and \(C\) have equal radii and masses \(m , k m\) and \(m\) respectively, where \(k\) is a constant. The spheres are moving in the same direction along a straight line on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). The speeds of \(A , B\) and \(C\) are \(2 u , u\) and \(\frac { 4 } { 3 } u\) respectively. The coefficient of restitution between any pair of the spheres is \(\frac { 1 } { 2 }\). After sphere \(A\) has collided with sphere \(B\), sphere \(B\) collides with sphere \(C\).

1. Find an inequality satisfied by \(k\).
2. Given that \(k = 2\), show that after \(B\) has collided with \(C\) there are no further collisions between any of the three spheres.

\includegraphics{figure_5} \(AB\) and \(BC\) are two fixed smooth vertical barriers on a smooth horizontal surface, with angle \(ABC = 60°\). A particle of mass \(m\) is moving with speed \(u\) on the surface. The particle strikes \(AB\) at an angle \(\theta\) with \(AB\). It then strikes \(BC\) and rebounds at an angle \(\beta\) with \(BC\) (see diagram). The coefficient of restitution between the particle and each barrier is \(e\) and \(\tan \theta = 2\). The kinetic energy of the particle after the first collision is 40% of its kinetic energy before the first collision.

1. Find the value of \(e\). [4]
2. Find the size of angle \(\beta\). [4]

7. \begin{figure}[h] \end{figure} Figure 4 represents the plan view of part of a smooth horizontal floor, where \(A B\) and \(B C\) are smooth vertical walls. The angle between \(A B\) and \(B C\) is \(120 ^ { \circ }\). A ball is projected along the floor towards \(A B\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a path at an angle of \(60 ^ { \circ }\) to \(A B\). The ball hits \(A B\) and then hits \(B C\). The ball is modelled as a particle. The coefficient of restitution between the ball and each wall is \(\frac { 1 } { 2 }\)

1. Show that the speed of the ball immediately after it has hit \(A B\) is \(\frac { \sqrt { 7 } } { 4 } u\). The speed of the ball immediately after it has hit \(B C\) is \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Find \(w\) in terms of \(u\).

2 \includegraphics[max width=\textwidth, alt={}, center]{58728f93-bfdb-4f76-a9b9-3a1d1592bfc9-2_531_760_927_696} Two smooth vertical walls each with their base on a smooth horizontal surface intersect at an angle of \(60 ^ { \circ }\). A small smooth sphere \(P\) is moving on the horizontal surface with speed \(u\) when it collides with the first vertical wall at the point \(D\). The angle between the direction of motion of \(P\) and the wall is \(\alpha ^ { \circ }\) before the collision and \(75 ^ { \circ }\) after the collision. The speed of \(P\) after this collision is \(v\) and the coefficient of restitution between \(P\) and the first wall is \(e\). Sphere \(P\) then collides with the second vertical wall at the point \(E\). The speed of \(P\) after this second collision is \(\frac { 1 } { 4 } u\) (see diagram). The coefficient of restitution between \(P\) and the second wall is \(\frac { 3 } { 4 }\).

1. By considering the collision at \(E\), show that \(v = \frac { \sqrt { } 2 } { 5 } u\).
2. Find the value of \(\alpha\) and the value of \(e\).

3. A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is free to rotate about a fixed smooth axis which passes through \(A\) and is perpendicular to the rod. The rod has angular speed \(\omega\) when it strikes a particle \(P\) of mass \(m\) and adheres to it. Immediately before the rod strikes \(P , P\) is at rest and at a distance \(x\) from \(A\). Immediately after the rod strikes \(P\), the angular speed of the rod is \(\frac { 3 } { 4 } \omega\). Find \(x\) in terms of \(a\).
(5)

4 A particle \(P\), of mass \(m \mathrm {~kg}\), collides with a particle \(Q\), of mass 2 kg Immediately before the collision the velocity of \(P\) is \(\left[ \begin{array} { c } 4 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(\left[ \begin{array} { c } - 3 \\ 5 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) As a result of the collision the particles coalesce into a single particle which moves with velocity \(\left[ \begin{array} { l } k \\ 0 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant. Find the value of \(k\).

5 \includegraphics[max width=\textwidth, alt={}, center]{1b542910-a57e-4f58-a19f-92e67ee9bdf7-08_323_515_260_813} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is held with the string taut and horizontal. It is projected downwards with speed \(\sqrt { } ( 12 a g )\). At the lowest point of its motion, \(P\) collides directly with a particle \(Q\) of mass \(k m\) which is at rest (see diagram). In the collision, \(P\) and \(Q\) coalesce. The tension in the string immediately after the collision is half of its value immediately before the collision. Find the possible values of \(k\).

4 Two uniform spheres \(A\) and \(B\), of equal radius, are at rest on a smooth horizontal table. Sphere \(A\) has mass \(3 m\) and sphere \(B\) has mass \(m\). Sphere \(A\) is projected directly towards \(B\), with speed \(u\). The coefficient of restitution between the spheres is 0.6 . Find the speeds of \(A\) and \(B\) after they collide. Sphere \(B\) now strikes a wall that is perpendicular to its path, rebounds and collides with \(A\) again. The coefficient of restitution between \(B\) and the wall is \(e\). Given that the second collision between \(A\) and \(B\) brings \(A\) to rest, find \(e\).

7 Two small smooth spheres, \(A\) and \(B\), are the same size and have masses \(2 m\) and \(m\) respectively. Initially, the spheres are at rest on a smooth horizontal surface. The sphere \(A\) receives an impulse of magnitude \(J\) and moves with speed \(2 u\) directly towards \(B\).

1. \(\quad\) Find \(J\) in terms of \(m\) and \(u\).
2. The sphere \(A\) collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\). Find, in terms of \(u\), the speeds of \(A\) and \(B\) immediately after the collision.
3. At the instant of collision, the centre of \(B\) is at a distance \(s\) from a fixed smooth vertical wall which is at right angles to the direction of motion of \(A\) and \(B\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-20_280_1114_1048_497} Subsequently, \(B\) collides with the wall. The radius of each sphere is \(r\).
   Show that the distance of the centre of \(A\) from the wall at the instant that \(B\) hits the wall is \(\frac { 3 s + 12 r } { 5 }\).
4. The diagram below shows the positions of \(A\) and \(B\) when \(B\) hits the wall. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-20_330_1109_1822_493} The sphere \(B\) collides with \(A\) again after rebounding from the wall. The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 5 }\). Find the distance of the centre of \(\boldsymbol { B }\) from the wall at the instant when \(A\) and \(B\) collide again.
   [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-24_2488_1728_219_141}

3 \includegraphics[max width=\textwidth, alt={}, center]{f8dd2aee-4ed5-4588-aa03-5dd56d9e7529-2_159_707_1443_721} Two perfectly elastic small smooth spheres \(A\) and \(B\) have masses \(3 m\) and \(m\) respectively. They lie at rest on a smooth horizontal plane with \(B\) at a distance \(a\) from a smooth vertical barrier. The line of centres of the spheres is perpendicular to the barrier, and \(B\) is between \(A\) and the barrier (see diagram). Sphere \(A\) is projected towards sphere \(B\) with speed \(u\) and, after the collision between the spheres, \(B\) hits the barrier. The coefficient of restitution between \(B\) and the barrier is \(\frac { 1 } { 2 }\). Find the speeds of \(A\) and \(B\) immediately after they first collide, and the distance from the barrier of the point where they collide for the second time.

1 Two uniform small smooth spheres, \(A\) and \(B\), of equal radii and masses 2 kg and 3 kg respectively, are at rest and not in contact on a smooth horizontal plane. Sphere \(A\) receives an impulse of magnitude 8 N s in the direction \(A B\). The coefficient of restitution between the spheres is \(e\). Find, in terms of \(e\), the speeds of \(A\) and \(B\) after \(A\) collides with \(B\). Given that the spheres move in opposite directions after the collision, show that \(e > \frac { 2 } { 3 }\).

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.

3 Two identical uniform small spheres \(A\) and \(B\), each of mass \(m\), are moving towards each other in a straight line on a smooth horizontal surface. Their speeds are \(u\) and \(k u\) respectively, and they collide directly. The coefficient of restitution between the spheres is \(e\). Sphere \(B\) is brought to rest by the collision.

1. Show that \(e = \frac { k - 1 } { k + 1 }\).
2. Given that \(60 \%\) of the total initial kinetic energy is lost in the collision, find the values of \(k\) and \(e\).

2 A particle \(P\) of mass 0.2 kg is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits a horizontal smooth surface. The direction of motion of \(P\) immediately before impact makes an angle of \(27 ^ { \circ }\) with the surface. Given that the coefficient of restitution between the particle and the surface is 0.6 , find

1. the vertical component of the velocity of \(P\) immediately after impact,
2. the magnitude of the impulse exerted on \(P\).

3. A particle of mass 0.6 kg is moving with constant velocity ( \(c \mathbf { i } + 2 c \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\), where \(c\) is a positive constant. The particle receives an impulse of magnitude \(2 \sqrt { 10 } \mathrm {~N} \mathrm {~s}\). Immediately after receiving the impulse the particle has velocity ( \(2 c \mathbf { i } - c \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\). Find the value of \(c\).
(6)

2 Three particles, \(A , B\) and \(C\), of masses \(2 \mathrm {~kg} , 3 \mathrm {~kg}\) and 5 kg respectively, are at rest in a straight line on a smooth horizontal plane with \(B\) between \(A\) and \(C\). Collisions between \(A\) and \(B\) are perfectly elastic. The coefficient of restitution for collisions between \(B\) and \(C\) is \(e\). \(A\) is projected towards \(B\) with a speed of \(5 u \mathrm {~ms} ^ { - 1 }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{709f3a7a-d857-4813-98ab-de6b41a3a8dc-02_190_885_1151_260} Show that only two collisions occur.

A particle \(P\), of mass \(m\), is able to move in a vertical circle on the smooth inner surface of a sphere with centre \(O\) and radius \(a\). Points \(A\) and \(B\) are on the inner surface of the sphere and \(A O B\) is a horizontal diameter. Initially, \(P\) is projected vertically downwards with speed \(\sqrt { } \left( \frac { 21 } { 2 } a g \right)\) from \(A\) and begins to move in a vertical circle. At the lowest point of its path, vertically below \(O\), the particle \(P\) collides with a stationary particle \(Q\), of mass \(4 m\), and rebounds. The speed acquired by \(Q\), as a result of the collision, is just sufficient for it to reach the point \(B\).

1. Find the speed of \(P\) and the speed of \(Q\) immediately after their collision.
   In its subsequent motion, \(P\) loses contact with the inner surface of the sphere at the point \(D\), where the angle between \(O D\) and the upward vertical through \(O\) is \(\theta\).
2. Find \(\cos \theta\).

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 .