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Edexcel M2 Q4
12 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-3_222_350_242_788} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows an earring consisting of a uniform wire \(A B C D\) of length \(6 a\) bent to form right angles at \(B\) and \(C\) such that \(A B\) and \(C D\) are of length \(2 a\) and \(a\) respectively.
  1. Find, in terms of \(a\), the distance of the centre of mass from
    1. \(\quad A B\),
    2. \(B C\). The earring is to be worn such that it hangs in equilibrium suspended from the point \(A\).
  2. Find, to the nearest degree, the angle made by \(A B\) with the downward vertical.
    (4 marks)
Edexcel M2 Q5
13 marks Moderate -0.3
5. A lorry of mass 40 tonnes moves up a straight road inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 1 } { 20 }\). The lorry moves at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In a model of the motion of the lorry, the non-gravitational resistance to motion is assumed to be constant and of magnitude 4400 N .
  1. Show that the engine of the lorry is working at a rate of 480 kW . The road becomes horizontal. The lorry's engine continues to work at the same rate and the resistance to motion is assumed to remain unchanged.
  2. Find the initial acceleration of the lorry.
  3. Find, correct to 3 significant figures, the maximum speed of the lorry.
  4. Using your answer to part (c), comment on the suitability of the modelling assumption.
Edexcel M2 Q6
13 marks Moderate -0.3
6. Particle \(S\) of mass \(2 M\) is moving with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane when it collides directly with a particle \(T\) of mass \(5 M\) which is lying at rest on the plane. The coefficient of restitution between \(S\) and \(T\) is \(\frac { 3 } { 4 }\). Given that the speed of \(T\) after the collision is \(4 \mathrm {~ms} ^ { - 1 }\),
  1. find \(U\). As a result of the collision, \(T\) is projected horizontally from the top of a building of height 19.6 m and falls freely under gravity. \(T\) strikes the ground at the point \(X\) as shown in Figure 3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-4_663_928_740_523} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure}
  2. Find the time taken for \(T\) to reach \(X\).
  3. Show that the angle between the horizontal and the direction of motion of \(T\), just before it strikes the ground at \(X\), is \(78.5 ^ { \circ }\) correct to 3 significant figures.
    (4 marks)
Edexcel M2 Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ef2dd10c-5a3c-4868-af00-aaf7f2937d7e-5_495_604_214_580} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Figure 4 shows a particle \(P\) projected from the point \(A\) up the line of greatest slope of a rough plane which is inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 4 } { 5 } . P\) is projected with speed \(5.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the coefficient of friction between \(P\) and the plane is \(\frac { 4 } { 7 }\). Given that \(P\) first comes to rest at point \(B\),
  1. use the Work-Energy principle to show that the distance \(A B\) is 1.4 m . The particle then slides back down the plane.
  2. Find, correct to 2 significant figures, the speed of \(P\) when it returns to \(A\).
Edexcel M2 Q1
5 marks Moderate -0.3
  1. An ice hockey puck of mass 0.5 kg is moving with velocity \(( 5 \mathbf { i } - 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors, when it is struck by a stick. After the impact, the puck travels with velocity \(( 13 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Find the magnitude of the impulse exerted by the stick on the puck.
(5 marks)
Edexcel M2 Q2
8 marks Moderate -0.3
2. A car of mass 1 tonne is climbing a hill inclined at an angle \(\theta\) to the horizontal where \(\sin \theta = \frac { 1 } { 7 }\). When the car passes a point \(X\) on the hill, it is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the car passes the point \(Y , 200 \mathrm {~m}\) further up the hill, it has speed \(10 \mathrm {~ms} ^ { - 1 }\). In a preliminary model of the situation, the car engine is assumed only to be doing work against gravity. Using this model,
  1. find the change in the total mechanical energy of the car as it moves from \(X\) to \(Y\).
    (6 marks)
    In a more sophisticated model, the car engine is also assumed to work against other forces.
  2. Write down two other forces which this model might include.
    (2 marks)
Edexcel M2 Q3
8 marks Moderate -0.8
3. A particle moves along a straight horizontal track such that its displacement, \(s\) metres, from a fixed point \(O\) on the line after \(t\) seconds is given by $$s = 2 t ^ { 3 } - 13 t ^ { 2 } + 20 t$$
  1. Find the values of \(t\) for which the particle is at \(O\).
  2. Find the values of \(t\) at which the particle comes instantaneously to rest.
Edexcel M2 Q4
9 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0ea2267e-6c46-4a4f-9a38-c242de57901d-3_378_730_196_609} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a uniform rod \(A B\) of length 2 m and mass 6 kg inclined at an angle of \(30 ^ { \circ }\) to the horizontal with \(A\) on smooth horizontal ground and \(B\) supported by a rough peg. The rod is in limiting equilibrium and the coefficient of friction between \(B\) and the peg is \(\mu\).
  1. Find, in terms of \(g\), the magnitude of the reactions at \(A\) and \(B\).
  2. Show that \(\mu = \frac { 1 } { \sqrt { 3 } }\).
Edexcel M2 Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0ea2267e-6c46-4a4f-9a38-c242de57901d-3_405_718_1169_555} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} During a cricket match, a batsman hits the ball giving it an initial velocity of \(22 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 7 } { 8 }\). When the batsman strikes the ball it is 1.6 metres above the ground, as shown in Figure 2, and it subsequently moves freely under gravity.
  1. Find, correct to 3 significant figures, the maximum height above the ground reached by the ball. The ball is caught by a fielder when it is 0.2 metres above the ground.
  2. Find the length of time for which the ball is in the air. Assuming that the fielder who caught the ball ran at a constant speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  3. find, correct to 3 significant figures, the maximum distance that the fielder could have been from the ball when it was struck.
Edexcel M2 Q6
16 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0ea2267e-6c46-4a4f-9a38-c242de57901d-4_433_282_196_726} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows a uniform rectangular lamina \(A B C D\) of mass \(8 m\) in which the sides \(A B\) and \(B C\) are of length \(a\) and \(2 a\) respectively. Particles of mass \(2 m , 6 m\) and \(4 m\) are fixed to the lamina at the points \(A , B\) and \(D\) respectively.
  1. Write down the distance of the centre of mass from \(A D\).
  2. Show that the distance of the centre of mass from \(A B\) is \(\frac { 4 } { 5 } a\). Another particle of mass \(k m\) is attached to the lamina at the point \(B\).
  3. Show that the distance of the centre of mass from \(A D\) is now given by \(\frac { ( 10 + k ) a } { 20 + k }\).
    (4 marks)
    Given that when the lamina is suspended freely from the point \(A\) the side \(A B\) makes an angle of \(45 ^ { \circ }\) with the vertical,
  4. find the value of \(k\).
Edexcel M2 Q7
17 marks Standard +0.3
7. Particle \(A\) of mass 7 kg is moving with speed \(u _ { 1 }\) on a smooth horizontal surface when it collides directly with particle \(B\) of mass 4 kg moving in the same direction as \(A\) with speed \(u _ { 2 }\). After the impact, \(A\) continues to move in the same direction but its speed has been halved. Given that the coefficient of restitution between the particles is \(e\),
  1. show that \(8 u _ { 2 } ( e + 1 ) = u _ { 1 } ( 8 e - 3 )\). Given also that \(u _ { 1 } = 14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(u _ { 2 } = 3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  2. find \(e\),
  3. show that the percentage of the kinetic energy of the particles lost as a result of the impact is \(9.6 \%\), correct to 2 significant figures.
AQA M3 Q5
Moderate -0.3
5 A football is kicked from a point \(O\) on a horizontal football ground with a velocity of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(30 ^ { \circ }\). During the motion, the horizontal and upward vertical displacements of the football from \(O\) are \(x\) metres and \(y\) metres respectively.
  1. Show that \(x\) and \(y\) satisfy the equation $$y = x \tan 30 ^ { \circ } - \frac { g x ^ { 2 } } { 800 \cos ^ { 2 } 30 ^ { \circ } }$$
  2. On its downward flight the ball hits the horizontal crossbar of the goal at a point which is 2.5 m above the ground. Using the equation given in part (a), find the horizontal distance from \(O\) to the goal.
    (4 marks) \includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-04_330_1411_1902_303}
  3. State two modelling assumptions that you have made.
AQA M3 Q6
Standard +0.8
6 Two smooth billiard balls \(A\) and \(B\), of identical size and equal mass, move towards each other on a horizontal surface and collide. Just before the collision, \(A\) has velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(30 ^ { \circ }\) to the line of centres of the balls, and \(B\) has velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(60 ^ { \circ }\) to the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-05_508_1420_532_294} The coefficient of restitution between the balls is \(\frac { 1 } { 2 }\).
  1. Find the speed of \(B\) immediately after the collision.
  2. Find the angle between the velocity of \(B\) and the line of centres of the balls immediately after the collision.
AQA M3 Q7
Challenging +1.8
7 A projectile is fired from a point \(O\) on the slope of a hill which is inclined at an angle \(\alpha\) to the horizontal. The projectile is fired up the hill with velocity \(U\) at an angle \(\theta\) above the hill and first strikes it at a point \(A\). The projectile is modelled as a particle and the hill is modelled as a plane with \(O A\) as a line of greatest slope.
    1. Find, in terms of \(U , g , \alpha\) and \(\theta\), the time taken by the projectile to travel from \(O\) to \(A\).
    2. Hence, or otherwise, show that the magnitude of the component of the velocity of the projectile perpendicular to the hill, when it strikes the hill at the point \(A\), is the same as it was initially at \(O\).
  2. The projectile rebounds and strikes the hill again at a point \(B\). The hill is smooth and the coefficient of restitution between the projectile and the hill is \(e\). \includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-06_428_1332_1023_338} Find the ratio of the time of flight from \(O\) to \(A\) to the time of flight from \(A\) to \(B\). Give your answer in its simplest form.
AQA M3 2006 June Q1
7 marks Moderate -0.5
1 The time \(T\) taken for a simple pendulum to make a single small oscillation is thought to depend only on its length \(l\), its mass \(m\) and the acceleration due to gravity \(g\). By using dimensional analysis:
  1. show that \(T\) does not depend on \(m\);
  2. express \(T\) in terms of \(l , g\) and \(k\), where \(k\) is a dimensionless constant.
AQA M3 2006 June Q2
12 marks Standard +0.3
2 Three smooth spheres \(A , B\) and \(C\) of equal radii and masses \(m , m\) and \(2 m\) respectively lie at rest on a smooth horizontal table. The centres of the spheres lie in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between any two spheres is \(e\). The sphere \(A\) is projected directly towards \(B\) with speed \(u\) and collides with \(B\).
  1. Find, in terms of \(u\) and \(e\), the speed of \(B\) immediately after the impact between \(A\) and \(B\).
  2. The sphere \(B\) subsequently collides with \(C\). The speed of \(C\) immediately after this collision is \(\frac { 3 } { 8 } u\). Find the value of \(e\).
AQA M3 2006 June Q3
9 marks Standard +0.3
3 A ball of mass 0.45 kg is travelling horizontally with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it strikes a fixed vertical bat directly and rebounds from it. The ball stays in contact with the bat for 0.1 seconds. At time \(t\) seconds after first coming into contact with the bat, the force exerted on the ball by the bat is \(1.4 \times 10 ^ { 5 } \left( t ^ { 2 } - 10 t ^ { 3 } \right)\) newtons, where \(0 \leqslant t \leqslant 0.1\). In this simple model, ignore the weight of the ball and model the ball as a particle.
  1. Show that the magnitude of the impulse exerted by the bat on the ball is 11.7 Ns , correct to three significant figures.
  2. Find, to two significant figures, the speed of the ball immediately after the impact.
  3. Give a reason why the speed of the ball immediately after the impact is different from the speed of the ball immediately before the impact.
AQA M3 2006 June Q4
10 marks Moderate -0.3
4 The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and due north respectively.
Two cyclists, Aazar and Ben, are cycling on straight horizontal roads with constant velocities of \(( 6 \mathbf { i } + 12 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and \(( 12 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) respectively. Initially, Aazar and Ben have position vectors \(( 5 \mathbf { i } - \mathbf { j } ) \mathrm { km }\) and \(( 18 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\) respectively, relative to a fixed origin.
  1. Find, as a vector in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of Ben relative to Aazar.
  2. The position vector of Ben relative to Aazar at time \(t\) hours after they start is \(\mathbf { r } \mathrm { km }\). Show that $$\mathbf { r } = ( 13 + 6 t ) \mathbf { i } + ( 6 - 20 t ) \mathbf { j }$$
  3. Find the value of \(t\) when Aazar and Ben are closest together.
  4. Find the closest distance between Aazar and Ben.
AQA M3 2006 June Q5
13 marks Moderate -0.3
5 A football is kicked from a point \(O\) on a horizontal football ground with a velocity of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(30 ^ { \circ }\). During the motion, the horizontal and upward vertical displacements of the football from \(O\) are \(x\) metres and \(y\) metres respectively.
  1. Show that \(x\) and \(y\) satisfy the equation $$y = x \tan 30 ^ { \circ } - \frac { g x ^ { 2 } } { 800 \cos ^ { 2 } 30 ^ { \circ } }$$
  2. On its downward flight the ball hits the horizontal crossbar of the goal at a point which is 2.5 m above the ground. Using the equation given in part (a), find the horizontal distance from \(O\) to the goal.
    (4 marks) \includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-3_330_1411_1902_303}
  3. State two modelling assumptions that you have made.
AQA M3 2006 June Q6
11 marks Standard +0.3
6 Two smooth billiard balls \(A\) and \(B\), of identical size and equal mass, move towards each other on a horizontal surface and collide. Just before the collision, \(A\) has velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(30 ^ { \circ }\) to the line of centres of the balls, and \(B\) has velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(60 ^ { \circ }\) to the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-4_508_1420_532_294} The coefficient of restitution between the balls is \(\frac { 1 } { 2 }\).
  1. Find the speed of \(B\) immediately after the collision.
  2. Find the angle between the velocity of \(B\) and the line of centres of the balls immediately after the collision.
AQA M3 2006 June Q7
13 marks Challenging +1.2
7 A projectile is fired from a point \(O\) on the slope of a hill which is inclined at an angle \(\alpha\) to the horizontal. The projectile is fired up the hill with velocity \(U\) at an angle \(\theta\) above the hill and first strikes it at a point \(A\). The projectile is modelled as a particle and the hill is modelled as a plane with \(O A\) as a line of greatest slope.
    1. Find, in terms of \(U , g , \alpha\) and \(\theta\), the time taken by the projectile to travel from \(O\) to \(A\).
    2. Hence, or otherwise, show that the magnitude of the component of the velocity of the projectile perpendicular to the hill, when it strikes the hill at the point \(A\), is the same as it was initially at \(O\).
  2. The projectile rebounds and strikes the hill again at a point \(B\). The hill is smooth and the coefficient of restitution between the projectile and the hill is \(e\). \includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-5_428_1332_1023_338} Find the ratio of the time of flight from \(O\) to \(A\) to the time of flight from \(A\) to \(B\). Give your answer in its simplest form.
AQA M3 2007 June Q1
8 marks Moderate -0.5
1 The magnitude of the gravitational force, \(F\), between two planets of masses \(m _ { 1 }\) and \(m _ { 2 }\) with centres at a distance \(x\) apart is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { x ^ { 2 } }$$ where \(G\) is a constant.
  1. By using dimensional analysis, find the dimensions of \(G\).
  2. The lifetime, \(t\), of a planet is thought to depend on its mass, \(m\), its initial radius, \(R\), the constant \(G\) and a dimensionless constant, \(k\), so that $$t = k m ^ { \alpha } R ^ { \beta } G ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
    Find the values of \(\alpha , \beta\) and \(\gamma\).
AQA M3 2007 June Q2
10 marks Standard +0.3
2 The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are directed due east, due north and vertically upwards respectively. Two helicopters, \(A\) and \(B\), are flying with constant velocities of \(( 20 \mathbf { i } - 10 \mathbf { j } + 20 \mathbf { k } ) \mathrm { ms } ^ { - 1 }\) and \(( 30 \mathbf { i } + 10 \mathbf { j } + 10 \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) respectively. At noon, the position vectors of \(A\) and \(B\) relative to a fixed origin, \(O\), are \(( 8000 \mathbf { i } + 1500 \mathbf { j } + 3000 \mathbf { k } ) \mathrm { m }\) and \(( 2000 \mathbf { i } + 500 \mathbf { j } + 1000 \mathbf { k } ) \mathrm { m }\) respectively.
  1. Write down the velocity of \(A\) relative to \(B\).
  2. Find the position vector of \(A\) relative to \(B\) at time \(t\) seconds after noon.
  3. Find the value of \(t\) when \(A\) and \(B\) are closest together.
AQA M3 2007 June Q3
9 marks Moderate -0.3
3 A particle \(P\), of mass 2 kg , is initially at rest at a point \(O\) on a smooth horizontal surface. The particle moves along a straight line, \(O A\), under the action of a horizontal force. When the force has been acting for \(t\) seconds, it has magnitude \(( 4 t + 5 ) \mathrm { N }\).
  1. Find the magnitude of the impulse exerted by the force on \(P\) between the times \(t = 0\) and \(t = 3\).
  2. Find the speed of \(P\) when \(t = 3\).
  3. The speed of \(P\) at \(A\) is \(37.5 \mathrm {~ms} ^ { - 1 }\). Find the time taken for the particle to reach \(A\).
AQA M3 2007 June Q4
9 marks Standard +0.3
4 Two small smooth spheres, \(A\) and \(B\), of equal radii have masses 0.3 kg and 0.2 kg respectively. They are moving on a smooth horizontal surface directly towards each other with speeds \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively when they collide. The coefficient of restitution between \(A\) and \(B\) is 0.8 .
  1. Find the speeds of \(A\) and \(B\) immediately after the collision.
  2. Subsequently, \(B\) collides with a fixed smooth vertical wall which is at right angles to the path of the sphere. The coefficient of restitution between \(B\) and the wall is 0.7 . Show that \(B\) will collide again with \(A\).