6.03b Conservation of momentum: 1D two particles

1. Two particles, \(A\) and \(B\), are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.

Particle \(A\) has mass \(5 m\) and particle \(B\) has mass \(3 m\).
The coefficient of restitution between \(A\) and \(B\) is \(e\), where \(e > 0\) Immediately after the collision the speed of \(A\) is \(v\) and the speed of \(B\) is \(2 v\).
Given that \(A\) and \(B\) are moving in the same direction after the collision,

1. find the set of possible values of \(e\). Given also that the kinetic energy of \(A\) immediately after the collision is \(16 \%\) of the kinetic energy of \(A\) immediately before the collision,
2. find
   1. the value of \(e\),
   2. the magnitude of the impulse received by \(A\) in the collision, giving your answer in terms of \(m\) and \(v\).

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

Immediately before the collision, the speed of \(A\) is \(k u\) and the speed of \(B\) is \(u\). Immediately after the collision, the speed of \(A\) is \(v\) and the speed of \(B\) is \(2 v\). The magnitude of the impulse received by \(B\) in the collision is \(\frac { 3 } { 2 } m u\).

1. Find \(v\) in terms of \(u\) only.
2. Find the two possible values of \(k\).

A particle \(P\) of mass \(2 m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal plane. It collides directly with a particle \(Q\) of mass \(m\) that is moving on the plane with speed \(2 u\) in the opposite direction to \(P\).
The coefficient of restitution between \(P\) and \(Q\) is \(e\), where \(e > \frac { 4 } { 5 }\)

1. Show that the speed of \(Q\) immediately after the collision is \(\frac { ( 4 + 10 e ) u } { 3 }\) After the collision \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). The coefficient of restitution between \(Q\) and the wall is \(f\).
2. Find, in terms of \(\boldsymbol { e }\), the set of values of \(f\) for which there will be a second collision between \(P\) and \(Q\).

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

A particle \(A\) has mass 3 kg and a particle \(B\) has mass 2 kg .
The particles are moving on a smooth horizontal plane when they collide directly.
Immediately before the collision, the velocity of \(A\) is \(( 3 \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the velocity of \(B\) is \(( - 6 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) Immediately after the collision the velocity of \(A\) is \(\left( - 2 \mathbf { i } + \frac { 2 } { 3 } \mathbf { j } \right) \mathrm { ms } ^ { - 1 }\)

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 exerted on \(A\) by \(B\) in the collision.
3. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) immediately after the collision.

1. A particle \(A\) of mass \(2 m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal plane. Particle \(A\) collides directly with a particle \(B\) of mass \(m\) which is at rest on the plane.

The coefficient of restitution between \(A\) and \(B\) is \(e\), where \(e > 0\)

1. Show that the speed of \(B\) immediately after the collision is \(2 u ( 1 + e )\). After the collision, \(B\) hits a smooth fixed vertical wall which is perpendicular to the direction of motion of \(B\).
2. Show that there will be a second collision between \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\) Find, in simplified form, in terms of \(m\), \(u\) and \(e\),
3. the magnitude of the impulse received by \(B\) in its collision with the wall,
4. the loss in kinetic energy of \(B\) due to its collision with the wall.

1. A particle \(P\) of mass \(2 m\) and a particle \(Q\) of mass \(5 m\) are moving along the same straight line on a smooth horizontal plane.

They are moving in opposite directions towards each other and collide directly.
Immediately before the collision the speed of \(P\) is \(2 u\) and the speed of \(Q\) is \(u\).
The direction of motion of \(Q\) is reversed by the collision.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the range of possible values of \(e\). Given that \(e = \frac { 1 } { 3 }\)
2. show that the kinetic energy lost in the collision is \(\frac { 40 m u ^ { 2 } } { 7 }\).
3. Without doing any further calculation, state how the amount of kinetic energy lost in the collision would change if \(e > \frac { 1 } { 3 }\)

5 Two discs, \(A\) and \(B\), have masses 1.4 kg and 2.1 kg respectively. They are sliding towards each other in the same straight line across a large sheet of horizontal ice. Immediately before the collision \(A\) has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) has speed \(3 \mathrm {~ms} ^ { - 1 }\). Immediately after the collision \(A\) 's speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Explain why it is impossible for \(A\) to be travelling in the same direction after the collision as it was before the collision.
2. Find the velocity of \(B\) immediately after the collision.
3. Calculate the coefficient of restitution between \(A\) and \(B\).
4. State what your answer to part (iii) means about the kinetic energy of the system. The discs are made from the same material. The discs will be damaged if subjected to an impulse of magnitude greater than 6.5 Ns .
5. Determine whether \(B\) will be damaged as a result of the collision.
6. Explain why \(A\) will be damaged if, and only if, \(B\) is damaged.

7 The masses of two particles \(A\) and \(B\) are \(m\) and \(2 m\) respectively. They are moving towards each other on a smooth horizontal table. Just before they collide their speeds are \(u\) and \(2 u\) respectively. After the collision the kinetic energy of \(A\) is 8 times the kinetic energy of \(B\). Find the coefficient of restitution between \(A\) and \(B\). \section*{END OF QUESTION PAPER}

5 Two particles \(A\) and \(B\) are on a smooth horizontal floor with \(B\) between \(A\) and a vertical wall. The masses of \(A\) and \(B\) are 4 kg and 11 kg respectively. Initially, \(B\) is at rest and \(A\) is moving towards \(B\) with a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). \(A\) collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(e\). \includegraphics[max width=\textwidth, alt={}, center]{bf86ac88-0fd1-4d49-a705-9b8d06fbac2a-3_209_803_1658_630}

1. Show that immediately after the collision the speed of \(B\) is \(\frac { 4 } { 15 } u ( 1 + e )\). After the collision between \(A\) and \(B\) the direction of motion of \(A\) is reversed. \(B\) subsequently collides directly with the vertical wall. The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 } e\).
2. Given that there is a second collision between \(A\) and \(B\), find the range of possible values of \(e\).

3 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]{493f11f4-e25c-4eeb-a0ab-20ec6d7a7a7d-2_186_903_2330_251} Show that only two collisions occur.

5 \includegraphics[max width=\textwidth, alt={}, center]{c397fca5-e7e8-4f3d-b519-cd92a983ebcc-04_221_1233_367_328} The masses of two spheres \(A\) and \(B\) are \(3 m \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively. The spheres are moving towards each other with constant speeds \(2 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 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 \(v \mathrm {~ms} ^ { - 1 }\) and \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\), respectively.

1. Find an expression for \(v\) in terms of \(e\) and \(u\).
2. Write down unsimplified expressions in terms of \(e\) and \(u\) for
   1. the total kinetic energy of the spheres before the collision,
   2. the total kinetic energy of the spheres after the collision.
   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 $$\lambda = \frac { 27 e ^ { 2 } + 25 } { 52 }$$
   4. Comment on the cases when
      (a) \(\lambda = 1\),
      (b) \(\lambda = \frac { 25 } { 52 }\). \includegraphics[max width=\textwidth, alt={}, center]{c397fca5-e7e8-4f3d-b519-cd92a983ebcc-05_789_981_324_543} The fixed points \(A\), \(B\) and \(C\) are in a vertical line with \(A\) above \(B\) and \(B\) above \(C\). A particle \(P\) of mass 2.5 kg is joined to \(A\), to \(B\) and to a particle \(Q\) of mass 2 kg , by three light rods where the length of rod \(A P\) is 1.5 m and the length of rod \(P Q\) is 0.75 m . Particle \(P\) moves in a horizontal circle with centre \(B\). Particle \(Q\) moves in a horizontal circle with centre \(C\) at the same constant angular speed \(\omega\) as \(P\), in such a way that \(A , B , P\) and \(Q\) are coplanar. The rod \(A P\) makes an angle of \(60 ^ { \circ }\) with the downward vertical, rod \(P Q\) makes an angle of \(30 ^ { \circ }\) with the downward vertical and rod \(B P\) is horizontal (see diagram).
      1. Find the tension in the \(\operatorname { rod } P Q\).
      2. Find \(\omega\).
      3. Find the speed of \(P\).
      4. Find the tension in the \(\operatorname { rod } A P\).
      5. Hence find the magnitude of the force in rod \(B P\). Decide whether this rod is under tension or compression.

1 A particle \(A\) moves across a smooth horizontal surface in a straight line. The particle \(A\) has mass 2 kg and speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A particle \(B\), which has mass 3 kg , is at rest on the surface. The particle \(A\) collides with the particle \(B\). \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-003_147_506_644_733}

1. If, after the collision, \(A\) is at rest and \(B\) moves away from \(A\), find the speed of \(B\).
2. If, after the collision, \(A\) and \(B\) move away from each other with speeds \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, as shown in the diagram below, find the value of \(v\). \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-003_138_506_1144_730}

1 A particle \(A\) moves across a smooth horizontal surface in a straight line. The particle \(A\) has mass 2 kg and speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A particle \(B\), which has mass 3 kg , is at rest on the surface. The particle \(A\) collides with the particle \(B\). \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-2_147_506_644_733}

1. If, after the collision, \(A\) is at rest and \(B\) moves away from \(A\), find the speed of \(B\).
2. If, after the collision, \(A\) and \(B\) move away from each other with speeds \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, as shown in the diagram below, find the value of \(v\). \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-2_138_506_1144_730}

1 Two particles, \(A\) and \(B\), are travelling in the same direction along a straight line on a smooth horizontal surface. Particle \(A\) has mass 3 kg and particle \(B\) has mass 7 kg . Particle \(A\) has a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and particle \(B\) has a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-2_186_835_653_593} Particle \(A\) and particle \(B\) collide and coalesce to form a single particle. Find the speed of this single particle after the collision.

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)

Two small smooth spheres \(A\) and \(B\), of mass 2 kg and 1 kg respectively, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of \(A\) is \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(- 2 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(\mathbf { j } \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Show that the velocity of \(B\) immediately after the collision is \(2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the impulse of \(B\) on \(A\) in the collision, giving your answer as a vector, and hence show that the line of centres is parallel to \(\mathbf { i } + \mathbf { j }\).
3. Find the coefficient of restitution between \(A\) and \(B\).
   

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] $$\left[ \begin{array} { l } 0.5 \\ 1.5 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 } \quad \left[ \begin{array} { l } 2 \\ 6 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 } \quad \left[ \begin{array} { l } 1 \\ 3 \end{array} \right] \mathrm { ms } ^ { - 1 } \quad \left[ \begin{array} { l } 3 \\ 9 \end{array} \right] \mathrm { m } \mathrm {~s} ^ { - 1 }$$

2 Two particles of equal mass are moving on a horizontal surface when they collide.
Immediately before the collision, their velocities are \(\left[ \begin{array} { l } 2 \\ 4 \end{array} \right] \mathrm { ms } ^ { - 1 }\) and \(\left[ \begin{array} { c } 6 \\ - 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\) As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle, immediately after the collision.
Circle your answer.
[0pt] [1 mark] \(\left[ \begin{array} { l } 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) \(\left[ \begin{array} { l } 4 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) \(\left[ \begin{array} { l } 8 \\ 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\) \(\left[ \begin{array} { l } 8 \\ 6 \end{array} \right] \mathrm { ms } ^ { - 1 }\)

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.

20 J
25 J
50 J
100 J 2 Two particles of equal mass are moving on a horizontal surface when they collide.
Immediately before the collision, their velocities are \(\left[ \begin{array} { l } 2 \\ 4 \end{array} \right] \mathrm { ms } ^ { - 1 }\) and \(\left[ \begin{array} { c } 6 \\ - 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\) As a result of the collision the particles coalesce to become a single particle.
Find the velocity of the single particle, immediately after the collision.
Circle your answer.
[0pt] [1 mark] \(\left[ \begin{array} { l } 4 \\ 1 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) \(\left[ \begin{array} { l } 4 \\ 3 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) \(\left[ \begin{array} { l } 8 \\ 2 \end{array} \right] \mathrm { ms } ^ { - 1 }\) \(\left[ \begin{array} { l } 8 \\ 6 \end{array} \right] \mathrm { ms } ^ { - 1 }\) 3 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A ball of mass of 0.75 kg is thrown vertically upwards with an initial speed of \(12 \mathrm {~ms} ^ { - 1 }\) The ball is thrown from ground level. 3

1. Calculate the initial kinetic energy of the ball. 3
2. The maximum height of the ball above the ground is \(h\) metres.
   Jeff and Gurjas use an energy method to find \(h\) Jeff concludes that \(h = 7.3\) Gurjas concludes that \(h < 7.3\) Explain the reasoning that they have used, showing any calculations that you make.
   4 Wavelength is defined as the distance from the highest point on one wave to the highest point on the next wave. Surfers classify waves into one of several types related to their wavelengths.
   Two of these classifications are deep water waves and shallow water waves.
   4
   1. The wavelength \(w\) of a deep water wave is given by $$w = \frac { g t ^ { 2 } } { k }$$ where \(g\) is the acceleration due to gravity and \(t\) is the time period between consecutive waves. Given that the formula for a deep water wave is dimensionally consistent, show that \(k\) is a dimensionless constant. 4
   2. The wavelength \(w\) of a shallow water wave is given by $$w = ( g d ) ^ { \alpha } t ^ { \beta }$$ where \(g\) is the acceleration due to gravity, \(d\) is the depth of water and \(t\) is the time period between consecutive waves. Use dimensional analysis to find the values of \(\alpha\) and \(\beta\) 5 A car, of mass 1000 kg , is travelling on a straight horizontal road. When the car travels at a speed of \(v \mathrm {~ms} ^ { - 1 }\), it experiences a resistance force of magnitude \(25 v\) newtons. The car has a maximum speed of \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on the straight road.
      Find the maximum power output of the car.
      Fully justify your answer.

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 \(A\)

2 \includegraphics[max width=\textwidth, alt={}, center]{60f72141-4a99-4907-93b1-adb0cd66948e-2_211_1276_1427_365} Three particles \(A , B\) and \(C\) are free to move in the same straight line on a large smooth horizontal surface. Their masses are \(1.2 \mathrm {~kg} , 1.8 \mathrm {~kg}\) and \(m \mathrm {~kg}\) respectively (see diagram). The coefficient of restitution in collisions between any two of them is \(\frac { 3 } { 4 }\). Initially, \(B\) and \(C\) are at rest and \(A\) is moving with a velocity of \(4.0 \mathrm {~ms} ^ { - 1 }\) towards \(B\).
a) Show that immediately after the collision between \(A\) and \(B\) the speed of \(B\) is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
b) Find the velocity of \(A\) immediately after this collision. \(B\) subsequently collides with \(C\).
c) Find, in terms of \(m\), the velocity of \(B\) after its collision with \(C\).
d) Given that the direction of motion of \(B\) is reversed by the collision with \(C\), find the range of possible values of \(m\). The car is attached to a trailer of mass 200 kg by a light rigid horizontal tow bar. The greatest steady speed of the car and trailer on the road is now \(30 \mathrm {~ms} ^ { - 1 }\). The resistance to motion of the trailer may also be assumed constant.
(b) Find the magnitude of the resistance force on the trailer. The car and trailer again travel along the road. At one instant their speed is \(15 \mathrm {~ms} ^ { - 1 }\) and their acceleration is \(0.57 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
(c) (i) Find the power of the engine of the car at this instant.
(ii) Find the magnitude of the tension in the tow bar at this instant. In a refined model of the motion of the car and trailer the resistance to the motion of each is assumed to be zero until they reach a speed of \(10 \mathrm {~ms} ^ { - 1 }\). When the speed is \(10 \mathrm {~ms} ^ { - 1 }\) or above the same constant resistance forces as in the first model are assumed to apply to each. The car and trailer start at rest on the road and accelerate, using maximum power.
(d) Without carrying out any further calculations,

1. explain whether the time taken to attain a speed of \(20 \mathrm {~m} ^ { - 1 }\) would be predicted to be lower, the same or higher using the refined model compared with the original model,
2. explain whether the greatest steady speed of the system would be predicted to be lower, the same or higher using the refined model compared with the original model.

4 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 \(I = \frac { \sqrt { 3 } m v } { 2 ( 1 + m ) }\).
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.

3 Two smooth circular discs \(A\) and \(B\) are moving on a horizontal plane. The masses of \(A\) and \(B\) are 3 kg and 4 kg respectively. At the instant before they collide

• the velocity of \(A\) is \(2 \mathrm {~ms} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the line joining their centres,
• the velocity of \(B\) is \(5 \mathrm {~ms} ^ { - 1 }\) towards \(A\) along the line joining their centres (see Fig. 3).

\begin{figure}[h] \end{figure} Given that the velocity of \(A\) after the collision is perpendicular to the velocity of \(A\) before the collision, find

1. the coefficient of restitution between \(A\) and \(B\),
2. the total loss of kinetic energy as a result of the collision.