Direct collision with direction reversal

7. A particle \(P\) of mass \(3 m\) is moving in a straight line with speed \(u\) on a smooth horizontal table. A second particle \(Q\) of mass \(2 m\) is moving with speed \(2 u\) in the opposite direction to \(P\) along the same straight line. Particle \(P\) collides directly with \(Q\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the direction of motion of \(P\) is reversed as a result of the collision with \(Q\).
2. Find the range of values of \(e\) for which the direction of motion of \(Q\) is also reversed as a result of the collision. Given that \(e = \frac { 1 } { 2 }\)
3. find, in terms of \(m\) and \(u\), the kinetic energy lost in the collision between \(P\) and \(Q\).

6. A particle \(P\) of mass \(3 m\) is moving with speed \(2 u\) in a straight line on a smooth horizontal table. The particle \(P\) collides with a particle \(Q\) of mass \(2 m\) moving with speed \(u\) in the opposite direction to \(P\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the speed of \(Q\) after the collision is \(\frac { 1 } { 5 } u ( 9 e + 4 )\). As a result of the collision, the direction of motion of \(P\) is reversed.
2. Find the range of possible values of \(e\). Given that the magnitude of the impulse of \(P\) on \(Q\) is \(\frac { 32 } { 5 } m u\),
3. find the value of \(e\).
   (4)

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

1. Find the speed of \(Q\) immediately after the collision. Given that the direction of motion of \(P\) is reversed by the collision,
2. find the range of possible values of \(e\).

7. A particle \(P\) of mass \(2 m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal table. A second particle \(Q\) of mass \(3 m\) is moving in the opposite direction to \(P\) along the same straight line with speed \(u\). The particle \(P\) collides directly with \(Q\). The direction of motion of \(P\) is reversed by the collision. The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the speed of \(Q\) immediately after the collision is \(\frac { u } { 5 } ( 8 e + 3 )\)
2. Find the range of possible values of \(e\). The total kinetic energy of the particles before the collision is \(T\). The total kinetic energy of the particles after the collision is \(k T\). Given that \(e = \frac { 1 } { 2 }\)
3. find the value of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{82cadc37-4cb0-455e-9531-e09ec0c19533-14_104_61_2407_1836}

5 A particle \(P\) of mass \(2 m\) is moving on a smooth horizontal surface with speed \(u\) when it collides directly with a particle \(Q\) of mass \(k m\) whose speed is \(3 u\) in the opposite direction. As a result of the collision, the directions of motion of both particles are reversed and the speed of \(P\) is halved.

1. Find, in terms of \(u\) and \(k\), the speed of \(Q\) after the collision. Hence write down the range of possible values of \(k\).
2. Calculate the magnitude of the impulse which \(Q\) exerts on \(P\).
3. Given that \(k = \frac { 1 } { 2 }\), calculate the coefficient of restitution between \(P\) and \(Q\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-3_472_1143_221_242} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} One end of a light inextensible string is attached to a point \(P\). The other end is attached to a point \(Q , 1.96 \mathrm {~m}\) vertically below \(P\). A small smooth bead \(B\), of mass 0.3 kg , is threaded on the string and moves in a horizontal circle with centre \(Q\) and radius \(1.96 \mathrm {~m} . B\) rotates about \(Q\) with constant angular speed \(\omega\) rad s \(^ { - 1 }\) (see Fig. 1).

8. Two ships \(A\) and \(B\), of masses \(m\) and km respectively, are moving towards each other in heavy fog along the same straight line, both with speed \(u\). The ships collide and immediately after the collision they drift away from each other, both their directions of motion having been reversed. The speed of \(A\) after the impact is \(\frac { 1 } { 5 } u\) and the speed of \(B\) after the impact is \(v\).

1. Show that \(v = u \left( \frac { 6 } { 5 k } - 1 \right)\). The coefficient of restitution between \(A\) and \(B\) is \(e\).
2. Show that \(v = u \left( 2 e - \frac { 1 } { 5 } \right)\).
3. Use your answers to parts (a) and (b) to find the rational numbers \(p\) and \(q\) such that \(p \leq k < q\).
   (9 marks)

1

1. Two small objects, P of mass \(m \mathrm {~kg}\) and Q of mass \(k m \mathrm {~kg}\), slide on a smooth horizontal plane. Initially, P and Q are moving in the same straight line towards one another, each with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After a direct collision with P , the direction of motion of Q is reversed and it now has a speed of \(\frac { 1 } { 3 } u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of P is now \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where the positive direction is the original direction of motion of P .
   1. Draw a diagram showing the velocities of P and Q before and after the impact.
   2. By considering the linear momentum of the objects before and after the collision, show that \(v = \left( 1 - \frac { 4 } { 3 } k \right) u\).
   3. Hence find the condition on \(k\) for the direction of motion of P to be reversed. The coefficient of restitution in the collision is 0.5 .
   4. Show that \(v = - \frac { 2 } { 3 } u\) and calculate the value of \(k\).
2. Particle \(A\) has a mass of 5 kg and velocity \(\binom { 3 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Particle \(B\) has mass 3 kg and is initially at rest. A force \(\binom { 1 } { - 2 } \mathrm {~N}\) acts for 9 seconds on B and subsequently (in the absence of the force), \(A\) and \(B\) collide and stick together to form an object \(C\) that moves off with a velocity \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\).
   1. Show that \(\mathbf { V } = \binom { 3 } { - 1 }\). The object C now collides with a smooth barrier which lies in the direction \(\binom { 0 } { 1 }\). The coefficient of restitution in the collision is 0.5 .
   2. Calculate the velocity of C after the impact.

1

1. A particle, P , of mass 5 kg moving with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) collides with another particle, Q , of mass 30 kg travelling with a speed of \(\frac { u } { 3 } \mathrm {~ms} ^ { - 1 }\) towards P . The particles P and Q are moving in the same horizontal straight line with negligible resistance to their motion. As a result of the collision, the speed of P is halved and its direction of travel reversed; the speed of Q is now \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   1. Draw a diagram showing this information. Find the velocity of Q immediately after the collision in terms of \(u\). Find also the coefficient of restitution between P and Q .
   2. Find, in terms of \(u\), the impulse of P on Q in the collision.
2. Fig. 1 shows a small object R of mass 5 kg travelling on a smooth horizontal plane at \(6 \mathrm {~ms} ^ { - 1 }\). It explodes into two parts of masses 2 kg and 3 kg . The velocities of these parts are in the plane in which R was travelling with the speeds and directions indicated. The angles \(\alpha\) and \(\beta\) are given by \(\cos \alpha = \frac { 4 } { 5 }\) and \(\cos \beta = \frac { 3 } { 5 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{334b2170-3708-46db-bff7-bcad7d5fab00-2_460_1450_1050_312} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure}
   1. Calculate \(u\) and \(v\).
   2. Calculate the increase in kinetic energy resulting from the explosion.

1. Two identical particles are approaching each other along a straight horizontal track. Just before they collide, they are moving with speeds \(5 \mathrm {~ms} ^ { - 1 }\) and \(3 \mathrm {~ms} ^ { - 1 }\) respectively. The coefficient of restitution between the particles is \(\frac { 1 } { 2 }\).

Find the speeds of the particles immediately after the impact.

2. Two smooth spheres \(P\) and \(Q\) of equal radius and of mass \(2 m\) and \(5 m\) respectively, are moving towards each other along a horizontal straight line when they collide. After the collision, \(P\) and \(Q\) travel in opposite directions with speeds of \(3 \mathrm {~ms} ^ { - 1 }\) and \(4 \mathrm {~ms} ^ { - 1 }\) respectively. Given that the coefficient of restitution between the two particles is \(\frac { 1 } { 2 }\), find the speeds of \(P\) and \(Q\) before the collision.
(6 marks)

4 Two discs, \(A\) and \(B\), have equal radii and masses 0.8 kg and 0.4 kg respectively. The discs are placed on a horizontal surface. The discs are set in motion when they are 3 metres apart, so that they move directly towards each other, each travelling at a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The discs collide directly with each other. After the collision \(A\) moves in the opposite direction with a speed of \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The coefficient of restitution between the two discs is \(e\). 4

1. Assuming that the surface is smooth, show that \(e = 0.8\) 4
2. Describe one way in which the model you have used could be refined. Turn over for the next question

1. A particle \(A\) has mass \(2 m\) and a particle \(B\) has mass \(3 m\). The particles are moving in opposite directions along the same straight line and collide directly.

Immediately before the collision, the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\). Immediately after the collision, the speed of \(A\) is \(0.5 u\) and the speed of \(B\) is \(w\). Given that the direction of motion of each particle is reversed by the collision,

1. find \(w\) in terms of \(u\)
2. find the coefficient of restitution between the particles,
3. find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(A\) in the collision.

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 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. \(e = - 1\) \(e = 0\) \(e = \frac { 1 } { 2 }\) \(e = 1\)

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

9 A particle \(P\) of mass \(m \mathrm {~kg}\) is moving with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on a smooth horizontal table. \(P\) collides directly with a stationary particle \(Q\) of mass 0.5 kg . This collision reverses the direction of motion of \(P\). Immediately after the collision the speed of \(P\) is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(Q\) is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the value of \(m\),
2. the coefficient of restitution between the two particles.

A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\), and the amplitude of the motion is \(2.5\) m. The points \(L\) and \(M\) are on the line, on opposite sides of \(O\), with \(OL = 1.5\) m. The magnitudes of the accelerations of \(P\) at \(L\) and at \(M\) are in the ratio \(3 : 4\).

1. Find the distance \(OM\). [2]

The time taken by \(P\) to travel directly from \(L\) to \(M\) is \(2\) s.

1. Find the period of the motion. [5]
2. Find the speed of \(P\) when it passes through \(L\). [2]

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 particles, \(P\), of mass \(2m\), and \(Q\), of mass \(m\), are moving along the same straight line on a smooth horizontal plane. They are moving in opposite directions towards each other and collide. Immediately before the collision the speed of \(P\) is \(2u\) and the speed of \(Q\) is \(u\). The coefficient of restitution between the particles is \(e\), where \(e < 1\). Find, in terms of \(u\) and \(e\),

1. the speed of \(P\) immediately after the collision,
2. the speed of \(Q\) immediately after the collision.

[7]

Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are moving towards each other along a straight line. \(P\) has speed 4 ms\(^{-1}\). They collide directly. After the collision the direction of motion of both particles has been reversed, and \(Q\) has speed 2 ms\(^{-1}\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac{1}{3}\). Find

1. the speed of \(Q\) before the collision, [4 marks]
2. the speed of \(P\) after the collision, [4 marks]
3. the kinetic energy, in J, lost in the impact. [4 marks]

A particle \(A\) is released from rest from the top of a smooth plane, which makes an angle of 30° with the horizontal. The particle \(A\) collides 2 s later with a particle \(B\), which is moving up a line of greatest slope of the plane. The coefficient of restitution between the particles is 0.4 and the speed of \(B\) immediately before the collision is 2 ms\(^{-1}\). \(B\) has velocity 1 ms\(^{-1}\) down the plane immediately after the collision. Find

1. the speed of \(A\) immediately after the collision, [4]
2. the distance \(A\) moves up the plane after the collision. [2]

The masses of \(A\) and \(B\) are 0.5 kg and \(m\) kg, respectively.

1. Find the value of \(m\). [3]

A sledge and a child sitting on it have a combined mass of 29.5 kg. The sledge slides on horizontal ice with negligible resistance to its movement.

1. While at rest, the sledge is hit directly from behind by a ball of mass 0.5 kg travelling horizontally at \(10 \text{ m s}^{-1}\). The coefficient of restitution in the collision is 0.8. After the impact the speeds of the sledge and the ball are \(V_1 \text{ m s}^{-1}\) and \(V_2 \text{ m s}^{-1}\) respectively. Calculate \(V_1\) and \(V_2\) and state the direction in which the ball is travelling after the impact. [7]
2. While at rest, the sledge is hit directly from behind by a snowball of mass 0.5 kg travelling horizontally at \(10 \text{ m s}^{-1}\). The snowball sticks to the sledge.
   1. Calculate the velocity with which the combined sledge and snowball start to move. [3]
   2. The child scoops up the 0.5 kg of snow and drops it over the back of the sledge. What happens to the velocity of the sledge? Give a reason for your answer. [2]
3. In another situation, the sledge is travelling over the ice at \(2 \text{ m s}^{-1}\) with 10.5 kg of snow on it (giving a total mass of 40 kg). The child throws a snowball of mass 0.5 kg from the sledge, parallel to the ground and in the positive direction of the motion of the sledge. Immediately after the snowball is thrown, the sledge has a speed of \(V \text{ m s}^{-1}\) and the snowball and sledge are separating at a speed of \(10 \text{ m s}^{-1}\). Draw a diagram showing the velocities of the sledge and snowball before and after the snowball is thrown. Calculate \(V\). [5]