6.03j Perfectly elastic/inelastic: collisions

5 Two smooth spheres, \(A\) and \(B\), of equal radii and different masses are moving on a smooth horizontal surface when they collide. Just before the collision, \(A\) is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the line of centres of the spheres, and \(B\) is moving with speed \(3 \mathrm {~ms} ^ { - 1 }\) perpendicular to the line of centres, as shown in the diagram below. \begin{figure}[h] \end{figure} Immediately after the collision, \(A\) and \(B\) move with speeds \(u\) and \(v\) in directions which make angles of \(90 ^ { \circ }\) and \(40 ^ { \circ }\) respectively with the line of centres, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-4_392_1102_1155_392}

1. Show that \(v = 4.67 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
2. Find the coefficient of restitution between the spheres.
3. Given that the mass of \(A\) is 0.5 kg , show that the magnitude of the impulse exerted on \(A\) during the collision is 2.17 Ns , correct to three significant figures.
4. Find the mass of \(B\).

6 A smooth sphere \(A\) of mass \(m\) is moving with speed \(5 u\) in a straight line on a smooth horizontal table. The sphere \(A\) collides directly with a smooth sphere \(B\) of mass \(7 m\), having the same radius as \(A\) and moving with speed \(u\) in the same direction as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(e\). \includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-5_287_880_529_571}

1. Show that the speed of \(B\) after the collision is \(\frac { u } { 2 } ( e + 3 )\).
2. Given that the direction of motion of \(A\) is reversed by the collision, show that \(e > \frac { 3 } { 7 }\).
3. Subsequently, \(B\) hits a wall fixed at right angles to the direction of motion of \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\). Given that after \(B\) rebounds from the wall both spheres move in the same direction and collide again, show also that \(e < \frac { 9 } { 13 }\).
   (4 marks)

4 Two smooth spheres \(A\) and \(B\) of equal radius are free to move on a smooth horizontal surface. The masses of \(A\) and \(B\) are \(m\) and \(4 m\) respectively.
The coefficient of restitution between the spheres is \(e\).
The spheres are projected directly towards each other, each with speed \(u\), and subsequently collide. 4

1. Show that the speed of \(B\) immediately after the impact with \(A\) is $$\frac { u ( 3 - 2 e ) } { 5 }$$ 4
2. Find the speed of \(A\) in terms of \(u\) and \(e\).
   4
3. Comment on the direction of motion of the spheres after the collision, justifying your answer.
   4
4. The magnitude of the impulse on \(B\) due to the collision is \(I\).
   Deduce that $$\frac { 8 m u } { 5 } \leq I \leq \frac { 16 m u } { 5 }$$

7 The particles \(A\) and \(B\) are moving on a smooth horizontal surface directly towards each other. Particle \(A\) has mass 0.4 kg and particle \(B\) has mass 0.2 kg
Particle \(A\) has speed \(4 \mathrm {~ms} ^ { - 1 }\) and particle \(B\) has speed \(2 \mathrm {~ms} ^ { - 1 }\) when they collide, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{ec39a757-5867-4798-b26c-73cd5746581c-08_392_1064_625_488} The coefficient of restitution between the particles is \(e\) 7

1. Find the magnitude of the total momentum of the particles before the collision.
   [0pt] [2 marks] 7
2. (i) Show that the speed of \(B\) immediately after the collision is \(( 4 e + 2 ) \mathrm { ms } ^ { - 1 }\) [0pt] [3 marks]
   7 (b) (ii) Find an expression, in terms of \(e\), for the speed of \(A\) immediately after the collision.
   7
3. Explain what happens to particle \(A\) when the collision is perfectly elastic.

2 Two particles are moving directly towards each other when they collide.
Given that the collision is perfectly elastic, state the value of the coefficient of restitution. Circle your answer. \(e = - 1\) \(e = 0\) \(e = \frac { 1 } { 2 }\) \(e = 1\)

7 Two smooth, equally sized spheres, \(A\) and \(B\), are moving in the same direction along a straight line on a smooth horizontal surface, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{78120346-4a16-4545-925a-d6fab4b750e9-06_314_465_420_849} The spheres subsequently collide.
Immediately after the collision, \(A\) has speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) has speed \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The coefficient of restitution between the spheres is \(e\) 7

1. 1. Show that \(A\) does not change its direction of motion as a result of the collision.
      7
      1. (ii) Find the value of \(e\) 7
   2. Given that the mass of \(B\) is 0.6 kg , find the mass of \(A\)

18 J
34 J 2 Two particles are moving directly towards each other when they collide.
Given that the collision is perfectly elastic, state the value of the coefficient of restitution. Circle your answer. \(e = - 1\) \(e = 0\) \(e = \frac { 1 } { 2 }\) \(e = 1\) 3 A stone of mass 0.2 kg is thrown vertically upwards with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the initial kinetic energy of the stone.
Circle your answer.
[0pt] [1 mark]
1 J
5 J
10 J

1 A particle \(P\) of mass 4.5 kg is moving in a straight line on a smooth horizontal surface at a speed of \(2.4 \mathrm {~ms} ^ { - 1 }\) when it strikes a vertical wall directly. It rebounds at a speed of \(1.6 \mathrm {~ms} ^ { - 1 }\).

1. Find the coefficient of restitution between \(P\) and the wall.
2. Determine the impulse applied to \(P\) by the wall, stating its direction.
3. Find the loss of kinetic energy of \(P\) as a result of the collision.
4. State, with a reason, whether the collision is perfectly elastic.

1. Two smooth spheres \(A\) and \(B\) are moving on a smooth horizontal plane when they collide obliquely. When the spheres collide, the line joining their centres is parallel to the vector \(\mathbf { j }\), as shown in the diagram below.

Immediately before the collision, sphere \(A\) has velocity ( \(6 \mathbf { i } - 3 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) and sphere \(B\) has velocity \(( - 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Sphere \(A\) has mass 6 kg and sphere \(B\) has mass 2 kg . \includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-02_595_972_753_534} Immediately after the collision, sphere \(B\) has velocity \(( - 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).

1. Find the velocity of \(A\) immediately after the collision.
2. Calculate the coefficient of restitution between \(A\) and \(B\).
3. Find the angle through which the direction of motion of \(B\) is deflected as a result of the collision. Give your answer correct to the nearest degree.
4. After the collision, sphere \(B\) continues to move with velocity \(( - 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) until it collides with another sphere \(C\), which exerts an impulse of \(( - 20 \mathbf { i } + 18 \mathbf { j } )\) Ns on \(B\). Find the velocity of \(B\) after the collision with \(C\).
   \section*{PLEASE DO NOT WRITE ON THIS PAGE}

1. A particle \(P\) has mass \(5 m\) and a particle \(Q\) has mass \(2 m\).

The particles are moving in opposite directions along the same straight line on a smooth horizontal surface.
Particle \(P\) collides directly with particle \(Q\).
Immediately before the collision, the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(3 u\). Immediately after the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\).
The direction of motion of \(Q\) is reversed as a result of the collision.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the set of values of \(e\) for which the direction of motion of \(P\) is unchanged as a result of the collision. In the collision, \(Q\) receives an impulse of magnitude \(\frac { 60 } { 7 } m u\)
2. Show that \(e = \frac { 1 } { 5 }\) After the collision, \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). Particle \(Q\) rebounds and there is a second collision between \(P\) and \(Q\).
   The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\)
3. Find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(Q\) in the second collision between \(P\) and \(Q\).

10 \includegraphics[max width=\textwidth, alt={}, center]{d8ca5464-435f-45e0-8e19-1830415a7c60-5_432_949_258_598} A smooth sphere \(P\) of mass \(3 m\) is at rest on a smooth horizontal table. A second smooth sphere \(Q\) of mass \(m\) and the same radius as \(P\) is moving along the table towards \(P\) and strikes it obliquely (see diagram). After the collision, the directions of motion of the two spheres are perpendicular.

1. Find the coefficient of restitution.
2. Given that one sixth of the original kinetic energy is lost as a result of the collision, find the angle between the initial direction of motion of \(Q\) and the line of centres.

10 \includegraphics[max width=\textwidth, alt={}, center]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-4_81_949_1283_598} Three particles \(A , B\) and \(C\), having masses \(1 \mathrm {~kg} , 2 \mathrm {~kg}\) 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 \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). 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.
2. Show that \(B\) reverses direction after an impact with \(C\).
3. Find the distance between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time.

11 A smooth sphere of mass 2 kg has velocity \(( 24 \mathbf { i } - 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and is travelling on a horizontal plane, where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in the horizontal plane. The sphere strikes a vertical wall. The line of intersection of the wall and the plane is in the direction \(( 4 \mathbf { i } + 3 \mathbf { j } )\).

1. Show that the acute angle between the path of the sphere before the impact and the direction of the wall is \(\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right)\).
2. After the impact, the velocity of the sphere is \(( 7.2 \mathbf { i } + 15.4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
   1. the coefficient of restitution between the sphere and the wall,
   2. the magnitude of the impulse exerted by the sphere on the wall.

9 A particle of mass 0.5 kg moving on a smooth horizontal plane with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) collides directly with another particle of mass \(k \mathrm {~kg}\) (where \(k\) is a constant) which is at rest. After the collision the first particle comes to rest but the second particle moves off with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find \(v\) in terms of \(k\) and \(u\).
2. The coefficient of restitution between the two particles is \(e\). Find \(e\) in terms of \(k\) only.
3. Show that \(k \geqslant \frac { 1 } { 2 }\).

10 \includegraphics[max width=\textwidth, alt={}, center]{a19fab61-da1c-4803-9dbc-38d618a0c58e-5_435_951_1528_557} A smooth sphere \(P\) of mass \(3 m\) is at rest on a smooth horizontal table. A second smooth sphere \(Q\) of mass \(m\) and the same radius as \(P\) is moving along the table towards \(P\) and strikes it obliquely (see diagram). After the collision, the directions of motion of the two spheres are perpendicular.

1. Find the coefficient of restitution.
2. Given that one-sixth of the original kinetic energy is lost as a result of the collision, find the angle between the initial direction of motion of \(Q\) and the line of centres.

9 \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-5_118_851_1265_607} Three particles \(A , B\) and \(C\), having masses of \(1 \mathrm {~kg} , 2 \mathrm {~kg}\) 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 \(14 \mathrm {~ms} ^ { - 1 }\). 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.
2. Show that \(B\) reversed direction after the impact with \(C\).
3. Find the distances between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time.

9 \includegraphics[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-5_118_851_1265_607} Three particles \(A , B\) and \(C\), having masses of \(1 \mathrm {~kg} , 2 \mathrm {~kg}\) 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 \(14 \mathrm {~ms} ^ { - 1 }\). 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.
2. Show that \(B\) reversed direction after the impact with \(C\).
3. Find the distances between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time.

11 A particle of mass 0.2 kg is projected so that it hits a smooth sloping plane \(\Pi\) that makes an angle of \(\sin ^ { - 1 } 0.6\) above the horizontal. The path of the particle is in a vertical plane containing a line of greatest slope of \(\Pi\). Immediately before the first impact between the particle and \(\Pi\), the particle is moving horizontally with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between the particle and \(\Pi\) is 0.5 .

1. Find the magnitude of the impulse on the particle from \(\Pi\) at the first impact, and state the direction of this impulse.
2. Find the distance between the points on \(\Pi\) where the first and second impacts occur.
3. Find the time taken between the first and third impacts.

11 \includegraphics[max width=\textwidth, alt={}, center]{f4acd242-eb78-4124-bfa2-fdecaa188690-6_438_953_264_557} A smooth sphere \(P\) of mass \(3 m\) is at rest on a smooth horizontal table. A second smooth sphere \(Q\) of mass \(m\) and the same radius as \(P\) is moving along the table towards \(P\) and strikes it obliquely (see diagram). After the collision, the directions of motion of the two spheres are perpendicular.

1. Find the coefficient of restitution.
2. Given that one-sixth of the original kinetic energy is lost as a result of the collision, find the angle between the initial direction of motion of \(Q\) and the line of centres.

A small uniform sphere \(A\), of mass \(2m\), is moving with speed \(u\) on a smooth horizontal surface when it collides directly with a small uniform sphere \(B\), of mass \(m\), which is at rest. The spheres have equal radii and the coefficient of restitution between them is \(e\). Find expressions for the speeds of \(A\) and \(B\) immediately after the collision. [4] Subsequently \(B\) collides with a vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(0.4\). After \(B\) has collided with the wall, the speeds of \(A\) and \(B\) are equal. Find \(e\). [2] Initially \(B\) is at a distance \(d\) from the wall. Find the distance of \(B\) from the wall when it next collides with \(A\). [4]

Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(5m\) and \(2m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is moving towards it with speed \(2u\). The coefficient of restitution between the spheres is \(e\).

1. Show that the speed of \(B\) after the collision is \(\frac{1}{7}u(1 + 15e)\) and find an expression for the speed of \(A\). [4]

In the collision, the speed of \(A\) is halved and its direction of motion is reversed.

1. Find the value of \(e\). [2]
2. For this collision, find the ratio of the loss of kinetic energy of \(A\) to the loss of kinetic energy of \(B\). [3]

Two uniform small smooth spheres \(A\) and \(B\) have equal radii and masses \(2m\) and \(m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides directly with sphere \(B\) which is at rest. The coefficient of restitution between the spheres is \(\frac{2}{3}\).

1. Find, in terms of \(u\), the speeds of \(A\) and \(B\) after this collision. [4]
2. Sphere \(B\) is initially at a distance \(d\) from a fixed smooth vertical wall which is perpendicular to the direction of motion of \(A\). The coefficient of restitution between \(B\) and the wall is \(\frac{1}{2}\). Find, in terms of \(d\) and \(u\), the time that elapses between the first and second collisions between \(A\) and \(B\). [5]

A particle \(P\) of mass \(0.3\text{ kg}\), lying on a smooth plane inclined at \(30°\) to the horizontal, is released from rest. \(P\) slides down the plane for a distance of \(2.5\text{ m}\) and then reaches a horizontal plane. There is no change in speed when \(P\) reaches the horizontal plane. A particle \(Q\) of mass \(0.2\text{ kg}\) lies at rest on the horizontal plane \(1.5\text{ m}\) from the end of the inclined plane (see diagram). \(P\) collides directly with \(Q\). \includegraphics{figure_7}

1. It is given that the horizontal plane is smooth and that, after the collision, \(P\) continues moving in the same direction, with speed \(2\text{ m s}^{-1}\). Find the speed of \(Q\) after the collision. [5]
2. It is given instead that the horizontal plane is rough and that when \(P\) and \(Q\) collide, they coalesce and move with speed \(1.2\text{ m s}^{-1}\). Find the coefficient of friction between \(P\) and the horizontal plane. [5]

A small ball is projected vertically downwards with speed \(5\text{ m s}^{-1}\) from a point \(A\) at a height of \(7.2\text{ m}\) above horizontal ground. The ball hits the ground with speed \(V\text{ m s}^{-1}\) and rebounds vertically upwards with speed \(\frac{1}{2}V\text{ m s}^{-1}\). The highest point the ball reaches after rebounding is \(B\). Find \(V\) and hence find the total time taken for the ball to reach the ground from \(A\) and rebound to \(B\). [5]

\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(5m\) respectively. Sphere \(A\) is moving on a smooth horizontal surface with speed \(u\) when it collides with sphere \(B\) which is at rest on the surface. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(\theta\) with the line of centres. After the collision, the kinetic energies of \(A\) and \(B\) are equal. The coefficient of restitution between the spheres is \(\frac{1}{3}\). Find the value of \(\tan\theta\). [6]