Questions M3 (745 questions)

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Edexcel M3 Specimen Q3
3. A particle \(P\) of mass 0.5 kg moves away from the origin \(O\) along the positive \(x\)-axis under the action of a force directed towards \(O\) of magnitude \(\frac { 2 } { x ^ { 2 } } \mathrm {~N}\), where \(O P = x\) metres. When \(x = 1\), the speed of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance of \(P\) from \(O\) when its speed has been reduced to \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(8)
Edexcel M3 Specimen Q4
4. A man of mass 75 kg is attached to one end of a light elastic rope of natural length 12 m . The other end of the rope is attached to a point on the edge of a horizontal ledge 19 m above the ground. The man steps off the ledge and falls vertically under gravity. The man is modelled as a particle falling from rest. He is brought to instantaneous rest by the rope when he is 1 m above the ground.
Find
  1. the modulus of elasticity of the rope,
    (5)
  2. the speed of the man when he is 2 m above the ground, giving your answer in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) to 3 significant figures.
    (5)
Edexcel M3 Specimen Q5
5.
\includegraphics[max width=\textwidth, alt={}, center]{e256678d-89e8-48eb-aa8a-b8e027b62ef1-3_423_357_918_847} A uniform solid, \(S\), is placed with its plane face on horizontal ground. The solid consists of a right circular cylinder, of radius \(r\) and height \(r\), joined to a right circular cone, of radius \(r\) and height \(h\). The plane face of the cone coincides with one of the plane faces of the cylinder, as shown in Fig. 3.
  1. Show that the distance of the centre of mass of \(S\) from the ground is $$\frac { 6 r ^ { 2 } + 4 r h + h ^ { 2 } } { 4 ( 3 r + h ) }$$ (8) The solid is now placed with its plane face on a rough plane which is inclined at an angle \(\alpha\) to the horizontal. The plane is rough enough to prevent \(S\) from sliding. Given that \(h = 2 r\), and that \(S\) is on the point of toppling,
  2. find, to the nearest degree, the value of \(\alpha\).
    (5)
Edexcel M3 Specimen Q6
6. A particle \(P\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is hanging in equilibrium below \(O\) when it receives a horizontal impulse giving it a speed \(u\), where \(u ^ { 2 } = 3 g a\). The string becomes slack when \(P\) is at the point \(B\). The line \(O B\) makes an angle \(\theta\) with the upward vertical.
  1. Show that \(\cos \theta = \frac { 1 } { 3 }\).
    (9)
  2. Show that the greatest height of \(P\) above \(B\) in the subsequent motion is \(\frac { 4 a } { 27 }\).
    (6)
Edexcel M3 Specimen Q7
7. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity 6 mg . The other end of the string is attached to a fixed point \(O\). When the particle hangs in equilibrium with the string vertical, the extension of the string is \(e\).
  1. Find \(e\).
    (2) The particle is now pulled down a vertical distance \(\frac { 1 } { 3 } a\) below its equilibrium position and released from rest. At time \(t\) after being released, during the time when the string remains taut, the extension of the string is \(e + x\). By forming a differential equation for the motion of \(P\) while the string remains taut,
  2. show that during this time \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { \frac { a } { 6 g } }\).
    (6)
  3. Show that, while the string remains taut, the greatest speed of \(P\) is \(\frac { 1 } { 3 } \sqrt { } ( 6 g a )\).
  4. Find \(t\) when the string becomes slack for the first time. \section*{END}
Edexcel M3 Specimen Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{689d4bd3-db24-4159-986b-40496213321a-02_438_492_328_735} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A garden game is played with a small ball \(B\) of mass \(m\) attached to one end of a light inextensible string of length 13l. The other end of the string is fixed to a point \(A\) on a vertical pole as shown in Figure 1. The ball is hit and moves with constant speed in a horizontal circle of radius \(5 l\) and centre \(C\), where \(C\) is vertically below \(A\). Modelling the ball as a particle, find
  1. the tension in the string,
  2. the speed of the ball.
Edexcel M3 Specimen Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{689d4bd3-db24-4159-986b-40496213321a-08_325_684_306_639} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle of mass 0.5 kg is attached to one end of a light elastic spring of natural length 0.9 m and modulus of elasticity \(\lambda\) newtons. The other end of the spring is attached to a fixed point \(O\) on a rough plane which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\). The coefficient of friction between the particle and the plane is 0.15 . The particle is held on the plane at a point which is 1.5 m down the line of greatest slope from \(O\), as shown in Figure 2. The particle is released from rest and first comes to rest again after moving 0.7 m up the plane. Find the value of \(\lambda\).
Edexcel M3 Specimen Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{689d4bd3-db24-4159-986b-40496213321a-12_410_579_312_689} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A container is formed by removing a right circular solid cone of height \(4 l\) from a uniform solid right circular cylinder of height \(6 l\). The centre \(O\) of the plane face of the cone coincides with the centre of a plane face of the cylinder and the axis of the cone coincides with the axis of the cylinder, as shown in Figure 3. The cylinder has radius \(2 l\) and the base of the cone has radius \(l\).
  1. Find the distance of the centre of mass of the container from \(O\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{689d4bd3-db24-4159-986b-40496213321a-12_435_560_1274_699} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} The container is placed on a plane which is inclined at an angle \(\theta ^ { \circ }\) to the horizontal. The open face is uppermost, as shown in Figure 4. The plane is sufficiently rough to prevent the container from sliding. The container is on the point of toppling.
  2. Find the value of \(\theta\).
    \includegraphics[max width=\textwidth, alt={}, center]{689d4bd3-db24-4159-986b-40496213321a-15_78_28_2588_1889}
Edexcel M3 Specimen Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{689d4bd3-db24-4159-986b-40496213321a-16_446_437_324_758} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is fixed at the point \(O\). The particle is initially held with \(O P\) horizontal and the string taut. It is then projected vertically upwards with speed \(u\), where \(u ^ { 2 } = 5 a g\). When \(O P\) has turned through an angle \(\theta\) the speed of \(P\) is \(v\) and the tension in the string is \(T\), as shown in Figure 5.
  1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(v ^ { 2 }\).
  2. Find, in terms of \(m , g\) and \(\theta\), an expression for \(T\).
  3. Prove that \(P\) moves in a complete circle.
  4. Find the maximum speed of \(P\).
Edexcel M3 Q5
5. A cyclist is travelling around a circular track which is banked at \(25 ^ { \circ }\) to the horizontal. The coefficient of friction between the cycle's tyres and the track is 0.6 . The cyclist moves with constant speed in a horizontal circle of radius 40 m , without the tyres slipping. Find the maximum speed of the cyclist.
AQA M3 Q5
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
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
57 marks
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
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
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
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
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
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
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
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
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
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
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
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 \(A\).
AQA M3 2007 June Q5
5 A ball is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\alpha\) above the horizontal so as to hit a point \(P\) on a wall. The ball travels in a vertical plane through the point of projection. During the motion, the horizontal and upward vertical displacements of the ball from the point of projection are \(x\) metres and \(y\) metres respectively.
  1. Show that, during the flight, the equation of the trajectory of the ball is given by $$y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } } \left( 1 + \tan ^ { 2 } \alpha \right)$$
  2. The ball is projected from a point 1 metre vertically below and \(R\) metres horizontally from the point \(P\).
    1. By taking \(g = 10 \mathrm {~ms} ^ { - 2 }\), show that \(R\) satisfies the equation $$5 R ^ { 2 } \tan ^ { 2 } \alpha - u ^ { 2 } R \tan \alpha + 5 R ^ { 2 } + u ^ { 2 } = 0$$
    2. Hence, given that \(u\) and \(R\) are constants, show that, for \(\tan \alpha\) to have real values, \(R\) must satisfy the inequality $$R ^ { 2 } \leqslant \frac { u ^ { 2 } \left( u ^ { 2 } - 20 \right) } { 100 }$$
    3. Given that \(R = 5\), determine the minimum possible speed of projection.