6.03b Conservation of momentum: 1D two particles

Two spheres A and B are free to move on a smooth horizontal surface. The masses of A and B are 2 kg and 3 kg respectively. Both A and B are initially at rest. Sphere A is set in motion directly towards sphere B with speed 4 m s\(^{-1}\) and subsequently collides with sphere B The coefficient of restitution between the spheres is \(e\)

1. 1. Show that the speed of B immediately after the collision is $$\frac{8(1 + e)}{5}$$ [4 marks]
   2. Find an expression, in terms of \(e\), for the velocity of A immediately after the collision. [2 marks]
2. It is given that the spheres both move in the same direction after the collision. Find the range of possible values of \(e\) [2 marks]
3. 1. The impulse of sphere A on sphere B is \(I\) The impulse of sphere B on sphere A is \(J\) Given that the collision is perfectly inelastic, find the value of \(I + J\) [1 mark]
   2. State, giving a reason for your answer, whether the value found in part (c)(i) would change if the collision was not perfectly inelastic. [2 marks]

Two spheres, \(A\) and \(B\), of equal size are moving in the same direction along a straight line on a smooth horizontal surface. Sphere \(A\) has mass \(m\) and is moving with speed \(4u\) Sphere \(B\) has mass \(6m\) and is moving with speed \(u\) The diagram shows the spheres and their velocities. \includegraphics{figure_8} Subsequently \(A\) collides directly with \(B\) The coefficient of restitution between \(A\) and \(B\) is \(e\)

1. Find, in terms of \(m\) and \(u\), the total momentum of the spheres before the collision. [1 mark]
2. Show that the speed of \(B\) immediately after the collision is \(\frac{u(3e + 10)}{7}\) [4 marks]
3. After the collision sphere \(A\) moves in the opposite direction. Find the range of possible values for \(e\) [5 marks]

\includegraphics{figure_5} The masses of two spheres \(A\) and \(B\) are \(3m\) kg and \(m\) kg respectively. The spheres are moving towards each other with constant speeds \(2u \, \text{m s}^{-1}\) and \(u \, \text{m s}^{-1}\) respectively along the same straight line towards each other on a smooth horizontal surface (see diagram). The two spheres collide and the coefficient of restitution between the spheres is \(e\). After colliding, \(A\) and \(B\) both move in the same direction with speeds \(v \, \text{m s}^{-1}\) and \(w \, \text{m s}^{-1}\), respectively.

1. Find an expression for \(v\) in terms of \(e\) and \(u\). [6]
2. Write down unsimplified expressions in terms of \(e\) and \(u\) for
   1. the total kinetic energy of the spheres before the collision, [1]
   2. the total kinetic energy of the spheres after the collision. [2]
3. Given that the total kinetic energy of the spheres after the collision is \(\lambda\) times the total kinetic energy before the collision, show that $$\lambda = \frac{27e^2 + 25}{52}.$$ [3]
4. Comment on the cases when
   1. \(\lambda = 1\),
   2. \(\lambda = \frac{25}{52}\). [3]

Two smooth circular discs \(A\) and \(B\) are moving on a smooth horizontal plane when they collide. The mass of \(A\) is \(5\) kg and the mass of \(B\) is \(3\) kg. At the instant before they collide, • the velocity of \(A\) is \(4\) m s\(^{-1}\) at an angle of \(60°\) to the line of centres, • the velocity of \(B\) is \(6\) m s\(^{-1}\) along the line of centres (see diagram). \includegraphics{figure_3} The coefficient of restitution for collisions between the two discs is \(\frac{3}{4}\). Determine the angle that the velocity of \(A\) makes with the line of centres after the collision. [7]

Two uniform smooth spheres A and B have equal radii and are moving on a smooth horizontal surface. The mass of A is 0.2kg and the mass of B is 0.6kg. The spheres collide obliquely. When the spheres collide the line joining their centres is parallel to \(\mathbf{i}\). Immediately before the collision the velocity of A is \(\mathbf{u}_A\)ms\(^{-1}\) and the velocity of B is \(\mathbf{u}_B\)ms\(^{-1}\). The coefficient of restitution between A and B is 0.5. Immediately after the collision the velocity of A is \((-4\mathbf{i} + 2\mathbf{j})\)ms\(^{-1}\) and the velocity of B is \((2\mathbf{i} + 3\mathbf{j})\)ms\(^{-1}\).

1. Find \(\mathbf{u}_A\) and \(\mathbf{u}_B\). [7]

After the collision B collides with a smooth vertical wall which is parallel to \(\mathbf{j}\). The loss in kinetic energy of B caused by the collision with the wall is 1.152J.

1. Find the coefficient of restitution between B and the wall. [3]
2. Find the angle through which the direction of motion of B is deflected as a result of the collision with the wall. [4]

Two small uniform smooth spheres A and B, of masses 2 kg and 3 kg respectively, are moving in opposite directions along the same straight line towards each other on a smooth horizontal surface. Sphere A has speed 2 m s\(^{-1}\) and B has speed 1 m s\(^{-1}\) before they collide. The coefficient of restitution between A and B is \(e\).

1. Show that the velocity of B after the collision, in the original direction of motion of A, is \(\frac{1}{5}(1 + 6e)\) m s\(^{-1}\) and find a similar expression for the velocity of A after the collision. [5]
2. The following three parts are independent of each other, and each considers a different scenario regarding the collision between A and B.
   1. In the collision between A and B the spheres coalesce to form a combined body C. State the speed of C after the collision. [1]
   2. In the collision between A and B the direction of motion of A is reversed. Find the range of possible values of \(e\). [2]
   3. The total loss in kinetic energy due to the collision is 3 J. Determine the value of \(e\). [4]

\includegraphics{figure_2} Two small uniform smooth spheres A and B have masses 0.5 kg and 2 kg respectively. The two spheres are travelling in the same direction in the same straight line on a smooth horizontal surface. Sphere A is moving towards B with speed \(6 \text{ m s}^{-1}\) and B is moving away from A with speed \(2 \text{ m s}^{-1}\) (see diagram). Spheres A and B collide. After this collision A moves with speed \(0.2 \text{ m s}^{-1}\). Determine the possible speeds with which B moves after the collision. [4]

Two small uniform smooth spheres A and B are of equal radius and have masses \(m\) and \(\lambda m\) respectively. The spheres are on a smooth horizontal surface. Sphere A is moving on the surface with velocity \(u_1 \mathbf{i} + u_2 \mathbf{j}\) towards B, which is at rest. The spheres collide obliquely. When the spheres collide, the line joining their centres is parallel to \(\mathbf{i}\). The coefficient of restitution between A and B is \(e\).

1. 1. Explain why, when the spheres collide, the impulse of A on B is in the direction of \(\mathbf{i}\). [1]
   2. Determine this impulse in terms of \(\lambda\), \(m\), \(e\) and \(u_1\). [6]

The loss in kinetic energy due to the collision between A and B is \(\frac{1}{8}mu_1^2\).

1. Determine the range of possible values of \(\lambda\). [6]

A car A of mass 1200 kg is about to tow another car B of mass 800 kg in a straight line along a horizontal road by means of a tow-rope attached between A and B. The tow-rope is modelled as being light and inextensible. Just before the tow-rope tightens, A is travelling at a speed of \(1.5 \text{ m s}^{-1}\) and B is at rest. Just after the tow-rope tightens, both cars have a speed of \(v \text{ m s}^{-1}\).

1. Find the value of \(v\). [2]
2. Calculate the magnitude of the impulse on A when the tow-rope tightens. [2]

Two small uniform discs A and B, of equal radius, have masses 3 kg and 5 kg respectively. The discs are sliding on a smooth horizontal surface and collide obliquely. The contact between the discs is smooth and A is stationary after the collision. Immediately before the collision B is moving with speed \(2 \text{ m s}^{-1}\) in a direction making an angle of \(60°\) with the line of centres, XY (see diagram below). \includegraphics{figure_12}

1. Explain how you can tell that A must have been moving along XY before the collision. [1]

The coefficient of restitution between A and B is 0.8.

1. • Determine the speed of A immediately before the collision. • Determine the speed of B immediately after the collision. [7]
2. Determine the angle turned through by the direction of B in the collision. [3]

Disc B subsequently collides with a smooth wall, which is parallel to XY. The kinetic energy of B after the collision with the wall is 95% of the kinetic energy of B before the collision with the wall.

1. Determine the coefficient of restitution between B and the wall. [4]

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

1. Show that \(\tan\beta = 3\tan\alpha\). [5]
2. Explain why the assumption that the contact between the spheres is smooth is needed in answering part (a). [1] It is given that A is deflected through an angle \(\gamma\).
3. Determine, in terms of \(\alpha\), an expression for \(\tan\gamma\). [2]
4. Determine the maximum value of \(\gamma\). You do not need to justify that this value is a maximum. [5]

\includegraphics{figure_9} Fig. 9 shows the instant of impact of two identical uniform smooth spheres, A and B, each with mass \(m\). Immediately before they collide, the spheres are sliding towards each other on a smooth horizontal table in the directions shown in the diagram, each with speed \(v\). The coefficient of restitution between the spheres is \(\frac{1}{2}\).

1. Show that, immediately after the collision, the speed of A is \(\frac{1}{8}v\). Find its direction of motion. [6]
2. Find the percentage of the original kinetic energy that is lost in the collision. [7]
3. State where in your answer to part (i) you have used the assumption that the contact between the spheres is smooth. [1]

Two objects, \(A\) of mass 18 kg and \(B\) of mass 7 kg, are moving in the same straight line on a smooth horizontal surface. Initially, they are moving with the same speed of \(4\text{ ms}^{-1}\) and in the same direction. Object \(B\) collides with a vertical wall which is perpendicular to its direction of motion and rebounds with a speed of \(3\text{ ms}^{-1}\). Subsequently, the two objects \(A\) and \(B\) collide directly. The coefficient of restitution between the two objects is \(\frac{5}{7}\).

1. Find the coefficient of restitution between \(B\) and the wall. [1]
2. Determine the speed of \(A\) and the speed of \(B\) immediately after the two objects collide. [7]
3. Calculate the impulse exerted by \(A\) on \(B\) due to the collision and clearly state its units. [2]
4. Find the loss in energy due to the collision between \(A\) and \(B\). [2]
5. State the direction of motion of \(A\) relative to the wall after the collision with \(B\). [1]

Two spheres \(A\) and \(B\), of equal radii, are moving towards each other on a smooth horizontal surface and collide directly. Sphere \(A\) has mass \(4m\) kg and sphere \(B\) has mass \(3m\) kg. Just before the collision, \(A\) has speed \(9\text{ ms}^{-1}\) and \(B\) has speed \(3.5\text{ ms}^{-1}\). Immediately after the collision, \(A\) has speed \(1.5\text{ ms}^{-1}\) in the direction of its original motion.

1. Show that the speed of \(B\) immediately after the collision is \(6.5\text{ ms}^{-1}\). [3]
2. Calculate the coefficient of restitution between \(A\) and \(B\). [3]
3. Given that the magnitude of the impulse exerted by \(B\) on \(A\) is 36 Ns, find the value of \(m\). [3]
4. Give a reason why it is not necessary to model the spheres as particles in this question. [1]

The diagram below shows two spheres \(A\) and \(B\), of equal radii, moving in the same direction on a smooth horizontal surface. Sphere \(A\), of mass \(3\) kg, is moving with speed \(4\) ms\(^{-1}\) and sphere \(B\), of mass \(2\) kg, is moving with speed \(10\) ms\(^{-1}\). \includegraphics{figure_5} Sphere \(B\) is then given an impulse after which it moves in the opposite direction with speed \(6\) ms\(^{-1}\).

1. Calculate the magnitude and direction of the impulse exerted on \(B\). [3]

Sphere \(B\) continues to move with speed \(6\) ms\(^{-1}\) so that it collides directly with sphere \(A\). The kinetic energy lost due to the collision is \(45\) J.

1. Calculate the speed of \(A\) and the speed of \(B\) immediately after the two spheres collide. State the direction in which each sphere is moving relative to its motion immediately before the collision. [8]

By burning a charge, a cannon fires a cannon ball of mass 12 kg horizontally. As the cannon ball leaves the cannon, its speed is 600 ms\(^{-1}\). The recoiling part of the cannon has a mass of 1600 kg.

1. Determine the speed of the recoiling part immediately after the cannon ball leaves the cannon. [3]
2. Find the energy created by the burning of the charge. State any assumption you have made in your solution and briefly explain how the assumption affects your answer. [5]
3. Calculate the constant force needed to bring the recoiling part to rest in 1.2 m. State, with a reason, whether your answer is an overestimate or an underestimate of the actual force required. [4]

Three particles, \(P\), \(Q\) and \(R\), are at rest on a smooth horizontal plane. The particles lie along a straight line with \(Q\) between \(P\) and \(R\). The particles \(Q\) and \(R\) have masses \(m\) and \(km\) respectively, where \(k\) is a constant. Particle \(Q\) is projected towards \(R\) with speed \(u\) and the particles collide directly. The coefficient of restitution between each pair of particles is \(e\).

1. Find, in terms of \(e\), the range of values of \(k\) for which there is a second collision. [9] Given that the mass of \(P\) is \(km\) and that there is a second collision,
2. write down, in terms of \(u\), \(k\) and \(e\), the speed of \(Q\) after this second collision. [1]

Three particles \(A\), \(B\) and \(C\) are free to move in the same straight line on a large smooth horizontal surface. Their masses are 3.3 kg, 2.2 kg and 1 kg respectively. The coefficient of restitution in collisions between any two of them is \(e\). Initially, \(B\) and \(C\) are at rest and \(A\) is moving towards \(B\) with speed \(u \text{ ms}^{-1}\) (see diagram). \(A\) collides directly with \(B\) and \(B\) then goes on to collide directly with \(C\). \includegraphics{figure_4}

1. The velocities of \(A\) and \(B\) immediately after the first collision are denoted by \(v_A \text{ ms}^{-1}\) and \(v_B \text{ ms}^{-1}\) respectively. \(\bullet\) Show that \(v_A = \frac{u(3-2e)}{5}\). \(\bullet\) Find an expression for \(v_B\) in terms of \(u\) and \(e\). [4]
2. Find an expression in terms of \(u\) and \(e\) for the velocity of \(B\) immediately after its collision with \(C\). [4]

After the collision between \(B\) and \(C\) there is a further collision between \(A\) and \(B\).

1. Determine the range of possible values of \(e\). [4]

A point \(O\) is situated a distance \(h\) above a smooth horizontal plane, and a particle \(A\) of mass \(m\) is attached to \(O\) by a light inextensible string of length \(h\). A particle \(B\) of mass \(2m\) is at rest on the plane, directly below \(O\), and is attached to a point \(C\) on the plane, where \(BC = l\), by a light inextensible string of length \(l\). \(A\) is released from rest with the string \(OA\) taut and making an acute angle \(\theta\) with the downward vertical (see diagram). \includegraphics{figure_8} \(A\) moves in a vertical plane perpendicular to \(CB\) and collides directly with \(B\). As a result of this collision, \(A\) is brought to rest and \(B\) moves on the plane in a horizontal circle with centre \(C\). After \(B\) has made one complete revolution the particles collide again.

1. Show that, on the next occasion that \(A\) comes to rest, the string \(OA\) makes an angle \(\phi\) with the downward vertical through \(O\), where \(\cos \phi = \frac{3 + \cos \theta}{4}\). [9]

\(A\) and \(B\) collide again when \(AO\) is next vertical.

1. Find the percentage of the original energy of the system that remains immediately after this collision. [5]
2. Explain why the total momentum of the particles immediately before the first collision is the same as the total momentum of the particles immediately after the second collision. [1]
3. Explain why the total momentum of the particles immediately before the first collision is different from the total momentum of the particles immediately after the third collision. [1]

\includegraphics{figure_12} A particle \(P\) of mass 2 kg can move along a line of greatest slope on a smooth plane, inclined at \(30°\) to the horizontal. \(P\) is initially at rest at a point on the plane, and a force of constant magnitude 20 N is applied to \(P\) parallel to and up the slope (see diagram).

1. Copy and complete the diagram, showing all forces acting on \(P\). [1]
2. Find the velocity of \(P\) in terms of time \(t\) seconds, whilst the force of 20 N is applied. [4]

After 3 seconds the force is removed at the instant that \(P\) collides with a particle of mass 1 kg moving down the slope with speed 5 m s\(^{-1}\). The coefficient of restitution between the particles is 0.2.

1. Express the velocity of \(P\) as a function of time after the collision. [6]

\includegraphics{figure_8} A smooth sphere with centre \(A\) and of mass 2 kg, moving at 13 m s\(^{-1}\) on a smooth horizontal plane, strikes a smooth sphere with centre \(B\) and of mass 3 kg moving at 5 m s\(^{-1}\) on the same smooth horizontal plane. The spheres have equal radii. The directions of motion immediately before impact are at angles \(\tan^{-1}\left(\frac{2}{13}\right)\) to \(\overrightarrow{AB}\) and \(\tan^{-1}\left(\frac{4}{3}\right)\) to \(\overrightarrow{BA}\) respectively (see diagram). Given that the coefficient of restitution is \(\frac{2}{3}\), find the speeds of the spheres after impact. [9]

\includegraphics{figure_9} Three particles \(A\), \(B\) and \(C\), having masses of 1 kg, 2 kg and 5 kg respectively, are placed 1 metre apart in a straight line on a smooth horizontal plane (see diagram). The particles \(B\) and \(C\) are initially at rest and \(A\) is moving towards \(B\) with speed 14 m s\(^{-1}\). The coefficient of restitution between each pair of particles is 0.5.

1. Find the velocity of \(B\) immediately after the first impact and show that \(A\) comes to rest. [4]
2. Show that \(B\) reversed direction after the impact with \(C\). [3]
3. Find the distances between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time. [3]

\includegraphics{figure_9} Three particles \(A\), \(B\) and \(C\), having masses of 1 kg, 2 kg and 5 kg respectively, are placed 1 metre apart in a straight line on a smooth horizontal plane (see diagram). The particles \(B\) and \(C\) are initially at rest and \(A\) is moving towards \(B\) with speed 14 ms\(^{-1}\). The coefficient of restitution between each pair of particles is 0.5.

1. Find the velocity of \(B\) immediately after the first impact and show that \(A\) comes to rest. [4]
2. Show that \(B\) reversed direction after the impact with \(C\). [3]
3. Find the distances between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time. [3]

1. Determine the impulse of a force of magnitude \(6\) N that acts on a particle of mass \(3\) kg for \(1.5\) seconds. [1]

Particles \(A\) and \(B\), of masses \(0.1\) kg and \(0.2\) kg respectively, can move on a smooth horizontal table. Initially \(A\) is moving with speed \(3\) m s\(^{-1}\) towards \(B\), which is moving with speed \(1\) m s\(^{-1}\) in the same direction as the motion of \(A\). During a collision \(B\) experiences an impulse from \(A\) of magnitude \(0.2\) kg m s\(^{-1}\).

1. Find the speeds of the particles immediately after the collision. [4]
2. Determine the coefficient of restitution between the particles. [2]