Momentum and Collisions 2

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.

1. A particle of mass 4 kg is moving with velocity \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) when it receives an impulse of \(( 7 \mathbf { i } - 5 \mathbf { j } )\) Ns.

Find the speed of the particle immediately after receiving the impulse.

1. A particle of mass 4 kg is moving with velocity \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) when it receives an impulse of \(( 7 \mathbf { i } - 5 \mathbf { j } )\) Ns.

Find the speed of the particle immediately after receiving the impulse.

6 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 \(\mathrm { I } = \frac { \sqrt { 3 } \mathrm { mv } } { 2 ( 1 + \mathrm { m } ) }\).
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.

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}

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 \(90 ^ { \circ }\).

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.

1 A smooth sphere of mass 0.3 kg bounces on a fixed horizontal surface. Immediately before the sphere bounces the components of its velocity horizontally and vertically downwards are \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The speed of the sphere immediately after it bounces is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the vertical component of the velocity of the sphere immediately after impact is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and hence find the coefficient of restitution between the surface and the sphere.
2. State the direction of the impulse on the sphere and find its magnitude.

7. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane] A small smooth ball of mass \(m\) kg is moving on a smooth horizontal plane and strikes a fixed smooth vertical wall. The plane and the wall intersect in a straight line which is parallel to the vector \(2 \mathbf { i } + \mathbf { j }\). The velocity of the ball immediately before the impact is \(b \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(b\) is positive. The velocity of the ball immediately after the impact is \(a ( \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(a\) is positive.

1. Show that the impulse received by the ball when it strikes the wall is parallel to \(( - \mathbf { i } + 2 \mathbf { j } )\). Find
2. the coefficient of restitution between the ball and the wall,
3. the fraction of the kinetic energy of the ball that is lost due to the impact.

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.

1. Particles \(A , B\) and \(C\) of masses \(4 m , 3 m\) and \(m\) respectively, lie at rest in a straight line on a smooth horizontal plane with \(B\) between \(A\) and \(C\). Particles \(A\) and \(B\) are projected towards each other with speeds \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, and collide directly.

As a result of the collision, \(A\) is brought to rest and \(B\) rebounds with speed \(k v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 3 } { 4 }\).

1. Show that \(u = 3 v\).
2. Find the value of \(k\). Immediately after the collision between \(A\) and \(B\), particle \(C\) is projected with speed \(2 v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(B\) so that \(B\) and \(C\) collide directly.
3. Show that there is no further collision between \(A\) and \(B\).

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.

2 A tennis ball of mass 0.057 kg has speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball receives an impulse of magnitude 0.6 N s which reduces the speed of the ball to \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Using an impulse-momentum triangle, or otherwise, find the angle the impulse makes with the original direction of motion of the ball.

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

A particle \(Q\) of mass 0.5 kg is moving on a smooth horizontal surface. Particle \(Q\) is moving with velocity \(( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse of \(( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { Ns }\).

1. Find the speed of \(Q\) immediately after receiving the impulse. As a result of receiving the impulse, the direction of motion of \(Q\) is turned through an angle \(\theta ^ { \circ }\)
2. Find the value of \(\theta\)

1 A ball of mass 0.4 kg is moving in a straight line, with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when it is struck by a bat. The bat exerts an impulse of magnitude 20 N s and the ball is deflected through an angle of \(90 ^ { \circ }\). Calculate

1. the direction of the impulse,
2. the speed of the ball immediately after it is struck.

10 A particle P of mass 2 kg is projected vertically upwards from horizontal ground with an initial speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same instant a particle \(Q\) of mass 8 kg is released from rest 5 m vertically above P . During the subsequent motion P and Q collide. The coefficient of restitution between P and Q is \(\frac { 11 } { 14 }\). Determine the time between this collision and P subsequently hitting the ground.

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.

5. \begin{figure}[h] \end{figure} Figure 3 represents the scene of a road accident. A car of mass 600 kg collided at the point \(X\) with a stationary van of mass 800 kg . After the collision the van came to rest at the point \(A\) having travelled a horizontal distance of 45 m , and the car came to rest at the point \(B\) having travelled a horizontal distance of 21 m . The angle \(A X B\) is \(90 ^ { \circ }\). The accident investigators are trying to establish the speed of the car before the collision and they model both vehicles as small spheres.

1. Find the coefficient of restitution between the car and the van.
   (5) The investigators assume that after the collision, and until the vehicles came to rest, the van was subject to a constant horizontal force of 500 N acting along \(A X\) and the car to a constant horizontal force of 300 N along \(B X\).
2. Find the speed of the car immediately before the collision.
   (9)

2
\includegraphics[max width=\textwidth, alt={}, center]{833c338f-53c1-436e-a772-0cdaf17fa72d-2_547_811_625_667} A uniform sphere \(P\) of mass \(m\) is at rest on a smooth horizontal table. The sphere is projected along the table with speed \(u\) and strikes a smooth vertical barrier \(A\) at an acute angle \(\alpha\). It then strikes another smooth vertical barrier \(B\) which is at right angles to \(A\) (see diagram). The coefficient of restitution between \(P\) and each of the barriers is \(e\). Show that the final direction of motion of \(P\) makes an angle \(\frac { 1 } { 2 } \pi - \alpha\) with the barrier \(B\) and find the total loss in kinetic energy as a result of the two impacts. [7]

2. \begin{figure}[h] \end{figure} Figure 2 represents part of the smooth rectangular floor of a sports hall. A ball is at \(B\), 4 m from one wall of the hall and 5 m from an adjacent wall. These two walls are smooth and meet at the corner \(C\). The ball is kicked so that it travels along the floor, bounces off the first wall at the point \(X\) and hits the second wall at the point \(Y\). The point \(Y\) is 7.5 m from the corner \(C\).
The coefficient of restitution between the ball and the first wall is \(\frac { 3 } { 4 }\).
Modelling the ball as a particle, find the distance \(C X\).

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 body, \(Q\), of mass 2 kg moves in a straight line under the action of a single force which acts in the direction of motion of \(Q\). Initially the speed of \(Q\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\), the magnitude \(F \mathrm {~N}\) of the force is given by $$F = t ^ { 2 } + 3 \mathrm { e } ^ { t } , \quad 0 \leq t \leq 4$$

1. Calculate the impulse of the force over the time interval.
2. Hence find the speed of \(Q\) when \(t = 4\).

1 A bullet of mass 0.01 kg is fired horizontally into a fixed vertical barrier which exerts a constant resisting force of magnitude 1000 N . The bullet enters the barrier with speed \(320 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). You may assume that the motion takes place in a horizontal straight line. Find

1. the magnitude of the impulse that acts on the bullet,
2. the thickness of the barrier,
3. the time taken for the bullet to pass through the barrier.

6 A ball of mass 0.15 kg is hit directly by a vertical cricket bat. Immediately before the impact, the ball is travelling horizontally with speed \(28 \mathrm {~ms} ^ { - 1 }\) Immediately after the impact, the ball is travelling horizontally with speed \(14 \mathrm {~ms} ^ { - 1 }\) in the opposite direction. 6

1. Find the magnitude of the impulse exerted by the bat on the ball.
   [0pt] [2 marks]
   6
2. In a simple model the force, \(F\) newtons, exerted by the bat on the ball, \(t\) seconds after the initial impact, is given by $$F = 10 k t ( 0.05 - t )$$ where \(k\) is a constant.
   Given the ball is in contact with the bat for 0.05 seconds, find the value of \(k\)
   [0pt] [3 marks]
   \(7 \quad\) Use \(g\) as \(9.81 \mathrm {~ms} ^ { - 2 }\) in this question. A light elastic string has one end attached to a fixed point \(A\) on a smooth plane inclined at \(25 ^ { \circ }\) to the horizontal. The other end of the string is attached to a wooden block of mass 2.5 kg , which rests on the plane. The elastic string has natural length 3 metres and modulus of elasticity 125 newtons.
   The block is pulled down the line of greatest slope of the plane to a point 4.5 metres from \(A\) and then released.

2 State the dimensions of impulse.
Circle your answer.
[0pt] [1 mark]
\(M L T ^ { - 2 }\)
\(M L T ^ { - 1 }\)
MLT
\(M L T { } ^ { 2 }\)

2 State the dimensions of impulse.
Circle your answer.
[0pt] [1 mark]
\(M L T ^ { - 2 }\)
\(M L T ^ { - 1 }\)
MLT
\(M L T { } ^ { 2 }\)

2
\includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-2_231_971_539_587} When a tennis ball of mass 0.057 kg bounces it receives an impulse of magnitude \(I \mathrm {~N} \mathrm {~s}\) at an angle of \(\theta\) to the horizontal. Immediately before the ball bounces it has speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction of \(30 ^ { \circ }\) to the horizontal. Immediately after the ball bounces it has speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction of \(30 ^ { \circ }\) to the horizontal (see diagram). Find \(I\) and \(\theta\).

5. A spaceship is moving in deep space with no external forces acting on it. Initially it has total mass \(M\) and is moving with speed \(V\). The spaceship reduces its speed to \(\frac { 2 } { 3 } V\) by ejecting fuel from its front end with a speed of \(c\) relative to itself and in the same direction as its own motion. Find the mass of fuel ejected.
(11 marks)

1 A rocket in deep space starts from rest and moves in a straight line. The initial mass of the rocket is \(m _ { 0 }\) and the propulsion system ejects matter at a constant mass rate \(k\) with constant speed \(u\) relative to the rocket. At time \(t\) the speed of the rocket is \(v\).

1. Show that while mass is being ejected from the rocket, \(\left( m _ { 0 } - k t \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = u k\).
2. Hence find an expression for \(v\) in terms of \(t\).
3. Find the speed of the rocket when its mass is \(\frac { 1 } { 3 } m _ { 0 }\).

5. A rocket is launched vertically upwards from rest. Initially, the total mass of the rocket and its fuel is 1000 kg . The rocket burns fuel at a rate of \(10 \mathrm {~kg} \mathrm {~s} ^ { - 1 }\). The burnt fuel is ejected vertically downwards with a speed of \(2000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the rocket, and burning stops after one minute. At time \(t\) seconds, \(t \leq 60\), after the launch, the speed of the rocket is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Air resistance is assumed to be negligible.

1. Show that $$- 9.8 ( 100 - t ) = ( 100 - t ) \frac { \mathrm { d } v } { \mathrm {~d} t } - 2000 .$$
2. Find the speed of the rocket when burning stops.

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 \(k\).

3 Two particles, \(A\) and \(B\), are moving on a smooth horizontal plane when they collide. The mass of \(A\) is 6 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Before the collision, the velocity of \(A\) is \(\left[ \begin{array} { l } 2 \\ 4 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { r } 3 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). After the collision, the velocity of \(A\) is \(\left[ \begin{array} { l } 1 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { l } 7 \\ b \end{array} \right] \mathrm { ms } ^ { - 1 }\).

1. Find \(m\).
2. \(\quad\) Find \(b\).
   (2 marks)
   .......... \(\_\_\_\_\)
   \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-07_40_118_529_159}
   \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-07_39_117_623_159}

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 \(k\).

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

2 A particle \(P\) of mass 3.5 kg is moving down a line of greatest slope of a rough inclined plane. At the instant that its speed is \(2.1 \mathrm {~ms} ^ { - 1 } P\) is at a point \(A\) on the plane. At that instant an impulse of magnitude 33.6 Ns , directed up the line of greatest slope, acts on \(P\).

1. Show that as a result of the impulse \(P\) starts moving up the plane with a speed of \(7.5 \mathrm {~ms} ^ { - 1 }\). While still moving up the plane, \(P\) has speed \(1.5 \mathrm {~ms} ^ { - 1 }\) at a point \(B\) where \(A B = 4.2 \mathrm {~m}\). The plane is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The frictional force exerted by the plane on \(P\) is modelled as constant.
2. Calculate the work done against friction as \(P\) moves from \(A\) to \(B\).
3. Hence find the magnitude of the frictional force acting on \(P\).
   \(P\) first comes to instantaneous rest at point \(C\) on the plane.
4. Calculate \(A C\).

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. A ball of mass 0.5 kg is moving with velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(( 4 \mathbf { i } + 5 \mathbf { j } ) \mathrm { Ns }\). Find the gain in kinetic energy of the ball due to the impulse.
   (6)
   

1. A particle \(P\) of mass 0.6 kg is moving with velocity ( \(4 \mathbf { i } - 2 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(\mathbf { I } \mathrm { N }\) s. Immediately after receiving the impulse, \(P\) has velocity ( \(2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

Find

1. the magnitude of \(\mathbf { I }\),
2. the kinetic energy lost by \(P\) as a result of receiving the impulse.

1. A ball of mass 0.5 kg is moving with velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(( 4 \mathbf { i } + 5 \mathbf { j } ) \mathrm { Ns }\). Find the gain in kinetic energy of the ball due to the impulse.
   (6)
   

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)

6
\includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-4_547_515_267_772} A hollow cylinder is fixed with its axis horizontal. \(O\) is the centre of a vertical cross-section of the cylinder and \(D\) is the highest point on the cross-section. \(A\) and \(C\) are points on the circumference of the cross-section such that \(A O\) and \(C O\) are both inclined at an angle of \(30 ^ { \circ }\) below the horizontal diameter through \(O\). The inner surface of the cylinder is smooth and has radius 0.8 m (see diagram). A particle \(P\), of mass \(m \mathrm {~kg}\), and a particle \(Q\), of mass \(5 m \mathrm {~kg}\), are simultaneously released from rest from \(A\) and \(C\), respectively, inside the cylinder. \(P\) and \(Q\) collide; the coefficient of restitution between them is 0.95 .

1. Show that, immediately after the collision, \(P\) moves with speed \(6.3 \mathrm {~ms} ^ { - 1 }\), and find the speed and direction of motion of \(Q\).
2. Find, in terms of \(m\), an expression for the normal reaction acting on \(P\) when it subsequently passes through \(D\).

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. 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)

1. Find the value of \(e\).
2. Find the loss in the total kinetic energy of the spheres as a result of the collision.

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.