Momentum and Collisions

3. A cannon of mass 600 kg lies on a rough horizontal surface and is used to fire a 3 kg shell horizontally at \(200 \mathrm {~ms} ^ { - 1 }\).

1. Find the impulse which the shell exerts on the cannon.
2. Find the speed with which the cannon recoils. Given that the coefficient of friction between the cannon and the surface is 0.75 ,
3. calculate, to the nearest centimetre, the distance that the cannon travels before coming to rest.

1. A particle \(A\) of mass 2 kg is moving along a straight horizontal line with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Another particle \(B\) of mass \(m \mathrm {~kg}\) is moving along the same straight line, in the opposite direction to \(A\), with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particles collide. The direction of motion of \(A\) is unchanged by the collision. Immediately after the collision, \(A\) is moving with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) is moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
   1. the magnitude of the impulse exerted by \(B\) on \(A\) in the collision,
   2. the value of \(m\).
      
   3. An athlete runs along a straight road. She starts from rest and moves with constant acceleration for 5 seconds, reaching a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). This speed is then maintained for \(T\) seconds. She then decelerates at a constant rate until she stops. She has run a total of 500 m in 75 s .
   4. In the space below, sketch a speed-time graph to illustrate the motion of the athlete.
   5. Calculate the value of \(T\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{330c2068-fe0a-4c6d-b892-79ab173c6a11-04_271_750_214_598} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} A particle of mass \(m \mathrm {~kg}\) is attached at \(C\) to two light inextensible strings \(A C\) and \(B C\). The other ends of the strings are attached to fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with \(A C\) and \(B C\) inclined to the horizontal at \(30 ^ { \circ }\) and \(60 ^ { \circ }\) respectively, as shown in Figure 1. Given that the tension in \(A C\) is 20 N , find
2. the tension in \(B C\),
3. the value of \(m\).

1. Two railway trucks \(A\) and \(B\), of masses 10000 kg and 7000 kg respectively, are moving towards each other along a horizontal straight track. The trucks collide, and in the collision \(A\) exerts an impulse on \(B\) of magnitude 84000 Ns. Immediately after the collision, the trucks move together with speed \(10 \mathrm {~ms} ^ { - 1 }\). Modelling the trucks as particles,
   1. find the speed of each truck immediately before the collision.
   When the trucks are moving together along the track, the coefficient of friction between them and the track is 0.15 . Assuming that no other resisting forces act on the trucks, calculate
2. the magnitude of the resisting force on the trucks,
3. the time taken after the collision for the trucks to come to rest.

1 A trolley, of mass 5 kg , is moving in a straight line on a smooth horizontal surface. It has a velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides with a stationary trolley, of mass \(m \mathrm {~kg}\). Immediately after the collision, the trolleys move together with velocity \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find \(m\).
(3 marks)

1 A particle \(A\), of mass 0.2 kg , collides with a particle \(B\), of mass 0.3 kg Immediately before the collision, the velocity of \(A\) is \(\left[ \begin{array} { c } 4 \\ 12 \end{array} \right] \mathrm { ms } ^ { - 1 }\)
and the velocity of \(B\) is \(\left[ \begin{array} { l } - 1 \\ - 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\)
As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle.
Circle your answer.
[0pt] [1 mark] $$\left[ \begin{array} { l } 0.5 \\ 1.5 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 } \quad \left[ \begin{array} { l } 2 \\ 6 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 } \quad \left[ \begin{array} { l } 1 \\ 3 \end{array} \right] \mathrm { ms } ^ { - 1 } \quad \left[ \begin{array} { l } 3 \\ 9 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 }$$

7
\includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-4_227_901_1352_623} Two particles \(P\) and \(Q\) have masses 0.7 kg and 0.3 kg respectively. \(P\) and \(Q\) are simultaneously projected towards each other in the same straight line on a horizontal surface with initial speeds of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram). Before \(P\) and \(Q\) collide the only horizontal force acting on each particle is friction and each particle decelerates at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The particles coalesce when they collide.

1. Given that \(P\) and \(Q\) collide 2 s after projection, calculate the speed of each particle immediately before the collision, and the speed of the combined particle immediately after the collision.
2. Given instead that \(P\) and \(Q\) collide 3 s after projection,
   (a) sketch on a single diagram the \(( t , v )\) graphs for the two particles in the interval \(0 \leqslant t < 3\),
   (b) calculate the distance between the two particles at the instant when they are projected.

1 Two particles \(P\) and \(Q\), of masses 0.1 kg and 0.4 kg respectively, are free to move on a smooth horizontal plane. Particle \(P\) is projected with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(Q\) which is stationary. After \(P\) and \(Q\) collide, the speeds of \(P\) and \(Q\) are equal. Find the two possible values of the speed of \(P\) after the collision.

1. A particle \(P\) has mass \(3 m\) and a particle \(Q\) has mass \(5 m\). The particles are moving towards each other in opposite directions along the same straight line on a smooth horizontal surface. The particles collide directly.

Immediately before the collision the speed of \(P\) is \(k u\), where \(k\) is a constant, and the speed of \(Q\) is \(2 u\). Immediately after the collision the speed of \(P\) is \(u\) and the speed of \(Q\) is \(3 u\).
The direction of motion of \(Q\) is reversed by the collision.

1. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(Q\) by \(P\) in the collision.
2. Find the two possible values of \(k\).
   \includegraphics[max width=\textwidth, alt={}, center]{5a2cf693-d966-4787-8778-ecc8a79a6265-03_2647_1837_118_114}

1 Particles \(P\) of mass 0.4 kg and \(Q\) of mass 0.5 kg are free to move on a smooth horizontal plane. \(P\) and \(Q\) are moving directly towards each other with speeds \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. After \(P\) and \(Q\) collide, the speed of \(Q\) is twice the speed of \(P\). Find the two possible values of the speed of \(P\) after the collision.

1 Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected at a speed of \(4 \mathrm {~ms} ^ { - 1 }\) directly towards \(Q\). After \(P\) and \(Q\) collide, \(Q\) begins to move with a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the speed of \(P\) after the collision.
   After the collision, \(Q\) moves directly towards a third particle \(R\), of mass \(m \mathrm {~kg}\), which is at rest on the plane. The two particles \(Q\) and \(R\) coalesce on impact and move with a speed of \(2 \mathrm {~ms} ^ { - 1 }\).
2. Find \(m\).

1 Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected at a speed of \(4 \mathrm {~ms} ^ { - 1 }\) directly towards \(Q\). After \(P\) and \(Q\) collide, \(Q\) begins to move with a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the speed of \(P\) after the collision.
   After the collision, \(Q\) moves directly towards a third particle \(R\), of mass \(m \mathrm {~kg}\), which is at rest on the plane. The two particles \(Q\) and \(R\) coalesce on impact and move with a speed of \(2 \mathrm {~ms} ^ { - 1 }\).
2. Find \(m\).

7
\includegraphics[max width=\textwidth, alt={}, center]{b2cd1b68-523f-40c3-8a51-acb2b55ae8c0-10_289_1191_269_475} The diagram shows a smooth track which lies in a vertical plane. The section \(A B\) is a quarter circle of radius 1.8 m with centre \(O\). The section \(B C\) is a horizontal straight line of length 7.0 m and \(O B\) is perpendicular to \(B C\). The section \(C F E\) is a straight line inclined at an angle of \(\theta ^ { \circ }\) above the horizontal. A particle \(P\) of mass 0.5 kg is released from rest at \(A\). Particle \(P\) collides with a particle \(Q\) of mass 0.1 kg which is at rest at \(B\). Immediately after the collision, the speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(B C\). You should assume that \(P\) is moving horizontally when it collides with \(Q\).

1. Show that the speed of \(Q\) immediately after the collision is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   When \(Q\) reaches \(C\), it collides with a particle \(R\) of mass 0.4 kg which is at rest at \(C\). The two particles coalesce. The combined particle comes instantaneously to rest at \(F\). You should assume that there is no instantaneous change in speed as the combined particle leaves \(C\), nor when it passes through \(C\) again as it returns down the slope.
2. Given that the distance \(C F\) is 0.4 m , find the value of \(\theta\).
3. Find the distance from \(B\) at which \(P\) collides with the combined particle.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 Two particles \(P\) and \(Q\), of masses 0.2 kg and 0.5 kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected towards \(Q\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Write down the momentum of \(P\).
2. After the collision \(P\) continues to move in the same direction with speed \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(Q\) after the collision.

1 Two particles \(P\) and \(Q\), of masses 0.2 kg and 0.5 kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected towards \(Q\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Write down the momentum of \(P\).
2. After the collision \(P\) continues to move in the same direction with speed \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(Q\) after the collision.

1. A constant force, \(\mathbf { F }\), acts on a particle, \(P\), of mass 5 kg causing its velocity to change from \(\left( { } ^ { - } 2 \mathbf { i } + \mathbf { j } \right) \mathrm { ms } ^ { - 1 }\) to \(( 4 \mathbf { i } - 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) in 2 seconds.
   1. Find, in the form \(a \mathbf { i } + b \mathbf { j }\), the acceleration of \(P\).
   2. Show that the magnitude of \(\mathbf { F }\) is 25 N and find, to the nearest degree, the acute angle between the line of action of \(\mathbf { F }\) and the vector \(\mathbf { j }\).
      (5 marks)
   3. A particle \(A\) of mass \(3 m\) is moving along a straight line with constant speed \(u \mathrm {~ms} ^ { - 1 }\). It collides with a particle \(B\) of mass \(2 m\) moving at the same speed but in the opposite direction. As a result of the collision, \(A\) is brought to rest.
   4. Show that, after the collision, \(B\) has changed its direction of motion and that its speed has been halved.
   Given that the magnitude of the impulse exerted by \(A\) on \(B\) is \(9 m \mathrm { Ns }\),
2. find the value of \(u\).

1. The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane. A ball of mass 0.5 kg is moving with velocity \(- 20 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is struck by a bat. The bat gives the ball an impulse of \(( 15 \mathbf { i } + 10 \mathbf { j } )\) Ns.

Find, to 3 significant figures, the speed of the ball immediately after it has been struck.
(5)

2 A particle, \(A\), of mass 12 kg is moving on a smooth horizontal surface with velocity \(\left[ \begin{array} { l } 4 \\ 7 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\). It then collides and coalesces with a second particle, \(B\), of mass 4 kg .

1. If before the collision the velocity of \(B\) was \(\left[ \begin{array} { l } 2 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), find the velocity of the combined particle after the collision.
2. If after the collision the velocity of the combined particle is \(\left[ \begin{array} { l } 1 \\ 4 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), find the velocity of \(B\) before the collision.

7. \begin{figure}[h] \end{figure} A ball \(B\) of mass 0.4 kg is struck by a bat at a point \(O\) which is 1.2 m above horizontal ground. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are respectively horizontal and vertical. Immediately before being struck, \(B\) has velocity \(( - 20 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after being struck it has velocity \(( 15 \mathbf { i } + 16 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). After \(B\) has been struck, it moves freely under gravity and strikes the ground at the point \(A\), as shown in Fig. 3. The ball is modelled as a particle.

1. Calculate the magnitude of the impulse exerted by the bat on \(B\).
2. By using the principle of conservation of energy, or otherwise, find the speed of \(B\) when it reaches \(A\).
3. Calculate the angle which the velocity of \(B\) makes with the ground when \(B\) reaches \(A\).
4. State two additional physical factors which could be taken into account in a refinement of the model of the situation which would make it more realistic.

1 A ball of mass 0.2 kg is travelling horizontally at \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits a vertical wall.
It rebounds horizontally at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
Find the magnitude of the impulse exerted on the ball by the wall.
Circle your answer.
[0pt] [1 mark]
0.4 N s
1.4 N s

1 A ball of mass 0.2 kg is travelling horizontally at \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits a vertical wall.
It rebounds horizontally at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
Find the magnitude of the impulse exerted on the ball by the wall.
Circle your answer.
[0pt] [1 mark]
0.4 N s
1.4 N s

3. A particle \(P\) of mass 0.4 kg is moving on rough horizontal ground when it hits a fixed vertical plane wall. Immediately before hitting the wall, \(P\) is moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the wall. The particle rebounds from the wall and comes to rest at a distance of 5 m from the wall. The coefficient of friction between \(P\) and the ground is \(\frac { 1 } { 8 }\). Find the magnitude of the impulse exerted on \(P\) by the wall.

2. A particle \(P\) of mass \(2 m\) is moving on a rough horizontal plane when it collides directly with a particle \(Q\) of mass \(4 m\) which is at rest on the plane. The speed of \(P\) immediately before the collision is \(3 u\). The speed of \(Q\) immediately after the collision is \(2 u\).

1. Find, in terms of \(u\), the speed of \(P\) immediately after the collision.
2. State clearly the direction of motion of \(P\) immediately after the collision. Following the collision, \(Q\) comes to rest after travelling a distance \(\frac { 6 u ^ { 2 } } { g }\) along the plane. The coefficient of friction between \(Q\) and the plane is \(\mu\).
3. Find the value of \(\mu\).

1. A particle \(P\) of mass 0.5 kg is moving with velocity \(( 4 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(( 2 \mathbf { i } - \mathbf { j } )\) Ns.

Show that the kinetic energy gained by \(P\) as a result of the impulse is 12 J .

1
\includegraphics[max width=\textwidth, alt={}, center]{cb2cec83-6f8d-4c13-90a1-03bbf4e4452f-03_471_613_254_766} A metal post is driven vertically into the ground by dropping a heavy object onto it from above. The mass of the object is 120 kg and the mass of the post is 40 kg (see diagram). The object hits the post with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and remains in contact with it after the impact.

1. Calculate the speed with which the combined post and object moves immediately after the impact.
2. There is a constant force resisting the motion of magnitude 4800 N . Calculate the distance the post is driven into the ground.

2 A particle \(A\) of mass 3.6 kg is attached by a light inextensible string to a particle \(B\) of mass 2.4 kg .
\(A\) and \(B\) are initially at rest, with the string slack, on a smooth horizontal surface. \(A\) is projected directly away from \(B\) with a speed of \(7.2 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the speed of \(A\) after the string becomes taut.
2. Find the impulse exerted on \(A\) at the instant that the string becomes taut.
3. Find the loss in kinetic energy as a result of the string becoming taut.

1 Particles \(P\) of mass \(m \mathrm {~kg}\) and \(Q\) of mass 0.2 kg are free to move on a smooth horizontal plane. \(P\) is projected at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(Q\) which is stationary. After the collision \(P\) and \(Q\) move in opposite directions with speeds of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Find \(m\).

1. Two particles \(P\) and \(Q\) have mass 0.4 kg and 0.6 kg respectively. The particles are initially at rest on a smooth horizontal table. Particle \(P\) is given an impulse of magnitude 3 N s in the direction \(P Q\).
   1. Find the speed of \(P\) immediately before it collides with \(Q\).
   Immediately after the collision between \(P\) and \(Q\), the speed of \(Q\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Show that immediately after the collision \(P\) is at rest.

1 A bullet of mass \(m \mathrm {~kg}\) is fired into a fixed vertical barrier. It enters the barrier horizontally with speed \(280 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges horizontally after 0.01 s with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a constant horizontal resisting force of magnitude 1500 N . Find \(m\).

6 Two smooth spheres \(A\) and \(B\) have equal radii and masses \(m\) and \(2 m\) respectively. Sphere \(B\) is at rest on a smooth horizontal floor. Sphere \(A\) is moving on the floor with velocity \(u\) and collides directly with \(B\). The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(u\) and \(e\), the velocities of \(A\) and \(B\) after the collision.
   Subsequently, \(B\) collides with a fixed vertical wall which makes an angle \(\theta\) with the direction of motion of \(B\), where \(\tan \theta = \frac { 3 } { 4 }\). The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 3 }\). Immediately after \(B\) collides with the wall, the kinetic energy of \(A\) is \(\frac { 5 } { 32 }\) of the kinetic energy of \(B\).
2. Find the possible values of \(e\).

6 A particle of mass 0.5 kg is held at rest at a point \(P\), which is at the bottom of an inclined plane. The particle is given an impulse of 1.8 Ns directed up a line of greatest slope of the plane.

1. Find the speed at which the particle starts to move. The particle subsequently moves up the plane to a point \(Q\), which is 0.3 m above the level of \(P\).
2. Given that the plane is smooth, find the speed of the particle at \(Q\). It is given instead that the plane is rough. The particle is now projected up the plane from \(P\) with initial speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and comes to rest at a point \(R\) which is 0.2 m above the level of \(P\).
3. Given that the plane is inclined at \(30 ^ { \circ }\) to the horizontal, find the magnitude of the frictional force on the particle.

1 The graph shows how a force, \(F\) newtons, varies during a 5 second period of time.
\includegraphics[max width=\textwidth, alt={}, center]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-02_575_1182_680_429} Calculate the magnitude of the impulse of the force.
Circle your answer.
[0pt] [1 mark]
17.5 N s
25 Ns
35 Ns
70 Ns

6
\includegraphics[max width=\textwidth, alt={}, center]{7febbd80-4cbb-4b2e-b022-d6a20e7e13aa-10_426_1191_267_438} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(k m\) respectively. The two spheres are moving on a horizontal surface with speeds \(u\) and \(\frac { 5 } { 8 } u\) respectively. Immediately before the spheres collide, \(A\) is travelling along the line of centres, and \(B\) 's direction of motion makes an angle \(\alpha\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 2 } { 3 }\) and \(\tan \alpha = \frac { 3 } { 4 }\). After the collision, the direction of motion of \(B\) is perpendicular to the line of centres.

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

5 A particle \(A\) of mass 0.5 kg is projected vertically upwards from horizontal ground with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the speed of \(A\) when it reaches a height of 20 m above the ground.
   When \(A\) reaches a height of 20 m , it collides with a particle \(B\) of mass 0.3 kg which is moving downwards in the same vertical line as \(A\) with speed \(32.5 \mathrm {~ms} ^ { - 1 }\). In the collision between the two particles, \(B\) is brought to instantaneous rest.
2. Show that the velocity of \(A\) immediately after the collision is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards.
3. Find the time interval between \(A\) and \(B\) reaching the ground. You should assume that \(A\) does not bounce when it reaches the ground.

2 A light spring \(A B\) has natural length \(a\) and modulus of elasticity 5 mg . The end \(A\) of the spring is attached to a fixed point on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the end \(B\) of the spring. The spring and particle \(P\) are at rest on the surface. Another particle \(Q\) of mass \(k m\) is moving with speed \(\sqrt { 4 \mathrm { ga } }\) along the horizontal surface towards \(P\) in the direction \(B A\). The particles \(P\) and \(Q\) collide directly and coalesce. In the subsequent motion the greatest amount by which the spring is compressed is \(\frac { 1 } { 5 } a\). Find the value of \(k\).
\includegraphics[max width=\textwidth, alt={}, center]{e34abb4b-1c6c-4f39-836d-467ed18345eb-04_307_1088_274_470} Particles \(A\) and \(B\), of masses \(m\) and \(3 m\) respectively, are connected by a light inextensible string of length \(a\) that passes through a fixed smooth ring \(R\). Particle \(B\) hangs in equilibrium vertically below the ring. Particle \(A\) moves in horizontal circles with speed \(v\). Particles \(A\) and \(B\) are at the same horizontal level. The angle between \(A R\) and \(B R\) is \(\theta\) (see diagram).

1. Show that \(\cos \theta = \frac { 1 } { 3 }\).
2. Find an expression for \(v\) in terms of \(a\) and \(g\).
   \includegraphics[max width=\textwidth, alt={}, center]{e34abb4b-1c6c-4f39-836d-467ed18345eb-06_597_803_258_625} An object is formed by removing a solid cylinder, of height \(k a\) and radius \(\frac { 1 } { 2 } a\), from a uniform solid hemisphere of radius \(a\). The axes of symmetry of the hemisphere and the cylinder coincide and one circular face of the cylinder coincides with the plane face of the hemisphere. \(A B\) is a diameter of the circular face of the hemisphere (see diagram).

2 Small smooth spheres \(A\) and \(B\), of equal radii and of masses 6 kg and 2 kg respectively, lie on a smooth horizontal plane. Initially \(A\) is moving towards \(B\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) is moving towards \(A\) with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the spheres collide, both \(A\) and \(B\) move in the same direction and the difference in the speeds of the spheres is \(2 \mathrm {~ms} ^ { - 1 }\). Find the loss of kinetic energy of the system due to the collision.