Questions — CAIE Further Paper 3 (127 questions)

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CAIE Further Paper 3 2021 June Q5
5 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 attached to a fixed point \(O\). The particle completes vertical circles with centre \(O\). The points \(A\) and \(B\) are on the path of \(P\), both on the same side of the vertical through \(O\). OA makes an angle \(\theta\) with the downward vertical through \(O\) and \(O B\) makes an angle \(\theta\) with the upward vertical through \(O\). The speed of \(P\) when it is at \(A\) is \(u\) and the speed of \(P\) when it is at \(B\) is \(\sqrt { \mathrm { ag } }\). The tensions in the string at \(A\) and \(B\) are \(T _ { A }\) and \(T _ { B }\) respectively. It is given that \(T _ { A } = 7 T _ { B }\). Find the value of \(\theta\) and find an expression for \(u\) in terms of \(a\) and \(g\).
\includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-10_558_1040_258_516} Two uniform smooth spheres \(A\) and \(B\) of equal radii each have mass \(m\). The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision, A's direction of motion makes an angle \(\alpha\) with the line of centres, and \(B\) 's direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 1 } { 3 }\) and \(2 \cos \beta = \cos \alpha\).
  1. Show that the direction of motion of \(A\) after the collision is perpendicular to the line of centres.
    The total kinetic energy of the spheres after the collision is \(\frac { 3 } { 4 } \mathrm { mu } ^ { 2 }\).
  2. Find the value of \(\alpha\).
CAIE Further Paper 3 2021 June Q7
7 A particle \(P\) is projected from a point \(O\) on a horizontal plane and moves freely under gravity. The initial velocity of \(P\) is \(100 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) above the horizontal, where \(\tan \theta = \frac { 4 } { 3 }\). The two times at which \(P\) 's height above the plane is \(H \mathrm {~m}\) differ by 10 s .
  1. Find the value of \(H\).
    \includegraphics[max width=\textwidth, alt={}]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_72_1569_463_328} ........................................................................................................................................ ........................................................................................................................................
    \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_72_1573_735_324}
    \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_72_1572_826_322}
    \includegraphics[max width=\textwidth, alt={}]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_74_1572_916_322} ........................................................................................................................................ . .........................................................................................................................................
    \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_70_1570_1187_324}
    \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_67_1570_1279_324}
    \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_67_1570_1370_324}
    \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_64_1570_1462_324}
    \includegraphics[max width=\textwidth, alt={}]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_63_1570_1553_324} ......................................................................................................................................... . .......................................................................................................................................... ......................................................................................................................................... .
    \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_71_1570_1905_324}
    \includegraphics[max width=\textwidth, alt={}, center]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_74_1570_1994_324}
    \includegraphics[max width=\textwidth, alt={}]{cef5eb87-1760-4aed-8e4d-e076d5d10252-12_76_1570_2083_324} ......................................................................................................................................... .
  2. Find the magnitude and direction of the velocity of \(P\) one second before it strikes the plane.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 3 2021 June Q3
3 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(k m g\), is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(4 m\). The particle \(P\) hangs in equilibrium a distance \(x\) vertically below \(A\).
  1. Show that \(\mathrm { k } = \frac { 4 \mathrm { a } } { \mathrm { x } - \mathrm { a } }\).
    An additional particle, of mass \(2 m\), is now attached to \(P\) and the combined particle is released from rest at the original equilibrium position of \(P\). When the combined particle has descended a distance \(\frac { 1 } { 3 } a\), its speed is \(\frac { 1 } { 3 } \sqrt { \mathrm { ga } }\).
  2. Find \(x\) in terms of \(a\).
    \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-06_602_520_264_753} A uniform solid circular cone has vertical height \(k h\) and radius \(r\). A uniform solid cylinder has height \(h\) and radius \(r\). The base of the cone is joined to one of the circular faces of the cylinder so that the axes of symmetry of the two solids coincide (see diagram, which shows a cross-section). The cone and the cylinder are made of the same material.
  3. Show that the distance of the centre of mass of the combined solid from the base of the cylinder is \(\frac { \mathrm { h } \left( \mathrm { k } ^ { 2 } + 4 \mathrm { k } + 6 \right) } { 4 ( 3 + \mathrm { k } ) }\).
    The solid is placed on a plane that is inclined to the horizontal at an angle \(\theta\). The base of the cylinder is in contact with the plane. The plane is sufficiently rough to prevent sliding. It is given that \(3 h = 2 r\) and that the solid is on the point of toppling when \(\tan \theta = \frac { 4 } { 3 }\).
  4. Find the value of \(k\).
CAIE Further Paper 3 2021 June Q5
5 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 attached to a fixed point \(O\). The particle completes vertical circles with centre \(O\). The points \(A\) and \(B\) are on the path of \(P\), both on the same side of the vertical through \(O\). OA makes an angle \(\theta\) with the downward vertical through \(O\) and \(O B\) makes an angle \(\theta\) with the upward vertical through \(O\). The speed of \(P\) when it is at \(A\) is \(u\) and the speed of \(P\) when it is at \(B\) is \(\sqrt { \mathrm { ag } }\). The tensions in the string at \(A\) and \(B\) are \(T _ { A }\) and \(T _ { B }\) respectively. It is given that \(T _ { A } = 7 T _ { B }\). Find the value of \(\theta\) and find an expression for \(u\) in terms of \(a\) and \(g\).
\includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-10_558_1040_258_516} Two uniform smooth spheres \(A\) and \(B\) of equal radii each have mass \(m\). The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision, A's direction of motion makes an angle \(\alpha\) with the line of centres, and \(B\) 's direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 1 } { 3 }\) and \(2 \cos \beta = \cos \alpha\).
  1. Show that the direction of motion of \(A\) after the collision is perpendicular to the line of centres.
    The total kinetic energy of the spheres after the collision is \(\frac { 3 } { 4 } \mathrm { mu } ^ { 2 }\).
  2. Find the value of \(\alpha\).
CAIE Further Paper 3 2021 June Q7
7 A particle \(P\) is projected from a point \(O\) on a horizontal plane and moves freely under gravity. The initial velocity of \(P\) is \(100 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) above the horizontal, where \(\tan \theta = \frac { 4 } { 3 }\). The two times at which \(P\) 's height above the plane is \(H \mathrm {~m}\) differ by 10 s .
  1. Find the value of \(H\).
    \includegraphics[max width=\textwidth, alt={}]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_72_1569_463_328} ........................................................................................................................................ ........................................................................................................................................
    \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_72_1573_735_324}
    \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_72_1575_826_322}
    \includegraphics[max width=\textwidth, alt={}]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_74_1570_916_324} ........................................................................................................................................ . .........................................................................................................................................
    \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_70_1570_1187_324}
    \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_67_1570_1279_324}
    \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_67_1570_1370_324}
    \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_64_1570_1462_324}
    \includegraphics[max width=\textwidth, alt={}]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_63_1570_1553_324} ......................................................................................................................................... . .......................................................................................................................................... ......................................................................................................................................... .
    \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_71_1570_1905_324}
    \includegraphics[max width=\textwidth, alt={}, center]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_74_1570_1994_324}
    \includegraphics[max width=\textwidth, alt={}]{0671bcad-5f74-46d2-823b-48f62f5954ce-12_76_1570_2083_324} ......................................................................................................................................... .
  2. Find the magnitude and direction of the velocity of \(P\) one second before it strikes the plane.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 3 2021 June Q2
2 One end of a light elastic string of natural length 0.8 m and modulus of elasticity 36 N is attached to a fixed point \(O\) on a smooth plane. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). A particle \(P\) of mass 2 kg is attached to the other end of the string. The string lies along a line of greatest slope of the plane with the particle below the level of \(O\). The particle is projected with speed \(\sqrt { 2 } \mathrm {~ms} ^ { - 1 }\) directly down the plane from the position where \(O P\) is equal to the natural length of the string. Find the maximum extension of the string during the subsequent motion.
\includegraphics[max width=\textwidth, alt={}, center]{6dcce6fe-7a19-4c5f-9361-20e7acda458f-04_380_1173_267_447} Particles \(A\) and \(B\), of masses \(3 m\) and \(m\) respectively, are connected by a light inextensible string of length \(a\) that passes through a fixed smooth ring \(R\). Particle \(B\) hangs in equilibrium vertically below the ring. Particle \(A\) moves in horizontal circles on a smooth horizontal surface with speed \(\frac { 2 } { 5 } \sqrt { \text { ga } }\). The angle between \(A R\) and \(B R\) is \(\theta\) (see diagram). The normal reaction between \(A\) and the surface is \(\frac { 12 } { 5 } \mathrm { mg }\).
  1. Find \(\cos \theta\).
  2. Find, in terms of \(a\), the distance of \(B\) below the ring.
    \includegraphics[max width=\textwidth, alt={}, center]{6dcce6fe-7a19-4c5f-9361-20e7acda458f-06_703_481_264_785} A particle of mass \(m\) 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 initially held with the string taut at the point \(A\), where \(O A\) makes an angle \(\theta\) with the downward vertical through \(O\). The particle is then projected with speed \(u\) perpendicular to \(O A\) and begins to move upwards in part of a vertical circle. The string goes slack when the particle is at the point \(B\) where angle \(A O B\) is a right angle. The speed of the particle when it is at \(B\) is \(\frac { 1 } { 2 } u\) (see diagram). Find the tension in the string at \(A\), giving your answer in terms of \(m\) and \(g\).
CAIE Further Paper 3 2021 June Q5
5 A particle \(P\) of mass \(m \mathrm {~kg}\) is projected vertically upwards from a point \(O\), with speed \(20 \mathrm {~ms} ^ { - 1 }\), and moves under gravity. There is a resistive force of magnitude \(2 m v \mathrm {~N}\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of \(P\) at time \(t \mathrm {~s}\) after projection.
  1. Find an expression for \(v\) in terms of \(t\), while \(P\) is moving upwards.
    The displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) at time \(t \mathrm {~s}\).
  2. Find an expression for \(x\) in terms of \(t\), while \(P\) is moving upwards.
  3. Find, correct to 3 significant figures, the greatest height above \(O\) reached by \(P\).
CAIE Further Paper 3 2021 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{6dcce6fe-7a19-4c5f-9361-20e7acda458f-10_339_983_258_541} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(k m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides with sphere \(B\) which is at rest. Immediately before the collision, \(A\) 's direction of motion makes an angle \(\theta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 1 } { 3 }\).
  1. Show that the speed of \(B\) after the collision is \(\frac { 4 \mathrm { u } \cos \theta } { 3 ( 1 + \mathrm { k } ) }\).
    70\% of the total kinetic energy of the spheres is lost as a result of the collision.
  2. Given that \(\tan \theta = \frac { 1 } { 3 }\), find the value of \(k\).
CAIE Further Paper 3 2021 June Q7
7 A particle \(P\) is projected with speed \(u\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a subsequent time \(t\) are denoted by \(x\) and \(y\) respectively.
  1. Use the equation of the trajectory given in the List of formulae (MF19), together with the condition \(y = 0\), to establish an expression for the range \(R\) in terms of \(u , \theta\) and \(g\).
  2. Deduce an expression for the maximum height \(H\), in terms of \(u , \theta\) and \(g\).
    It is given that \(\mathrm { R } = \frac { 4 \mathrm { H } } { \sqrt { 3 } }\).
  3. Show that \(\theta = 60 ^ { \circ }\).
    It is given also that \(u = \sqrt { 40 } \mathrm {~ms} ^ { - 1 }\).
  4. Find, by differentiating the equation of the trajectory or otherwise, the set of values of \(x\) for which the direction of motion makes an angle of less than \(45 ^ { \circ }\) with the horizontal.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 3 2022 June Q2
2 One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). A particle of mass \(m\) is attached to the other end of the string. The particle is held at the point \(A\) with the string taut. The angle between \(O A\) and the downward vertical is equal to \(\alpha\), where \(\cos \alpha = \frac { 4 } { 5 }\). The particle is projected from \(A\), perpendicular to the string in an upwards direction, with a speed \(\sqrt { 3 \text { ga } }\). It then moves along a circular path in a vertical plane. The string first goes slack when it makes an angle \(\theta\) with the upward vertical through \(O\). Find the value of \(\cos \theta\).
CAIE Further Paper 3 2022 June Q3
4 marks
3 A particle \(P\) is moving in a horizontal straight line. Initially \(P\) is at the point \(O\) on the line and is moving with velocity \(25 \mathrm {~ms} ^ { - 1 }\). At time \(t \mathrm {~s}\) after passing through \(O\), the acceleration of \(P\) is \(\frac { 4000 } { ( 5 t + 4 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) in the direction \(P O\). The displacement of \(P\) from \(O\) at time \(t\) is \(x \mathrm {~m}\). Find an expression for \(x\) in terms of \(t\).
\includegraphics[max width=\textwidth, alt={}, center]{9067c549-00d7-4078-b47d-87b28396e2ab-06_894_809_260_628} An object is composed of a hemispherical shell of radius \(2 a\) attached to a closed hollow circular cylinder of height \(h\) and base radius \(a\). The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. \(A B\) is a diameter of the lower end of the cylinder (see diagram).
  1. Find, in terms of \(a\) and \(h\), an expression for the distance of the centre of mass of the object from \(A B\). [4]
    The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 2 } { 3 }\). The object is in equilibrium with \(A B\) in contact with the plane and lying along a line of greatest slope of the plane.
  2. Find the set of possible values of \(h\), in terms of \(a\).
    \includegraphics[max width=\textwidth, alt={}, center]{9067c549-00d7-4078-b47d-87b28396e2ab-08_629_1358_269_367} A light inextensible string \(A B\) passes through two small holes \(C\) and \(D\) in a smooth horizontal table where \(A C = 3 a\) and \(D B = a\). A particle of mass \(m\) is attached at the end \(A\) and moves in a horizontal circle with angular velocity \(\omega\). A particle of mass \(\frac { 3 } { 4 } m\) is attached to the end \(B\) and moves in a horizontal circle with angular velocity \(k \omega\). \(A C\) makes an angle \(\theta\) with the downward vertical and \(D B\) makes an angle \(\theta\) with the horizontal (see diagram). Find the value of \(k\).
CAIE Further Paper 3 2022 June Q6
6 Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(k m\) respectively. The two spheres are on a horizontal surface. Sphere \(A\) is travelling with speed \(u\) towards sphere \(B\) which is at rest. The spheres collide. Immediately before the collision, the direction of motion of \(A\) makes an angle \(\alpha\) with the line of centres. The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\).
  1. Show that the speed of \(B\) after the collision is \(\frac { 3 \mathrm { u } \cos \alpha } { 2 ( 1 + \mathrm { k } ) }\) and find also an expression for the speed of \(A\) along the line of centres after the collision, in terms of \(k , u\) and \(\alpha\).
    After the collision, the kinetic energy of \(A\) is equal to the kinetic energy of \(B\).
  2. Given that \(\tan \alpha = \frac { 2 } { 3 }\), find the possible values of \(k\).
CAIE Further Paper 3 2022 June Q7
7 Particles \(P\) and \(Q\) are projected in the same vertical plane from a point \(O\) at the top of a cliff. The height of the cliff exceeds 50 m . Both particles move freely under gravity. Particle \(P\) is projected with speed \(\frac { 35 } { 2 } \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 4 } { 3 }\). Particle \(Q\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle \(\beta\) above the horizontal, where \(\tan \beta = \frac { 1 } { 2 }\). Particle \(Q\) is projected one second after the projection of particle \(P\). The particles collide \(T\) s after the projection of particle \(Q\).
  1. Write down expressions, in terms of \(T\), for the horizontal displacements of \(P\) and \(Q\) from \(O\) when they collide and hence show that \(4 \mathrm { uT } = 21 \sqrt { 5 } ( \mathrm {~T} + 1 )\).
  2. Find the value of \(T\).
  3. Find the horizontal and vertical displacements of the particles from \(O\) when they collide.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 3 2022 June Q3
4 marks
3 A particle \(P\) is moving in a horizontal straight line. Initially \(P\) is at the point \(O\) on the line and is moving with velocity \(25 \mathrm {~ms} ^ { - 1 }\). At time \(t \mathrm {~s}\) after passing through \(O\), the acceleration of \(P\) is \(\frac { 4000 } { ( 5 t + 4 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) in the direction \(P O\). The displacement of \(P\) from \(O\) at time \(t\) is \(x \mathrm {~m}\). Find an expression for \(x\) in terms of \(t\).
\includegraphics[max width=\textwidth, alt={}, center]{c486c59a-2493-4dd3-bf1e-dde57fe744d9-06_894_809_260_628} An object is composed of a hemispherical shell of radius \(2 a\) attached to a closed hollow circular cylinder of height \(h\) and base radius \(a\). The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. \(A B\) is a diameter of the lower end of the cylinder (see diagram).
  1. Find, in terms of \(a\) and \(h\), an expression for the distance of the centre of mass of the object from \(A B\). [4]
    The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 2 } { 3 }\). The object is in equilibrium with \(A B\) in contact with the plane and lying along a line of greatest slope of the plane.
  2. Find the set of possible values of \(h\), in terms of \(a\).
    \includegraphics[max width=\textwidth, alt={}, center]{c486c59a-2493-4dd3-bf1e-dde57fe744d9-08_629_1358_269_367} A light inextensible string \(A B\) passes through two small holes \(C\) and \(D\) in a smooth horizontal table where \(A C = 3 a\) and \(D B = a\). A particle of mass \(m\) is attached at the end \(A\) and moves in a horizontal circle with angular velocity \(\omega\). A particle of mass \(\frac { 3 } { 4 } m\) is attached to the end \(B\) and moves in a horizontal circle with angular velocity \(k \omega\). \(A C\) makes an angle \(\theta\) with the downward vertical and \(D B\) makes an angle \(\theta\) with the horizontal (see diagram). Find the value of \(k\).
CAIE Further Paper 3 2022 June Q1
1 A uniform lamina \(O A B C\) is a trapezium whose vertices can be represented by coordinates in the \(x - y\) plane. The coordinates of the vertices are \(O ( 0,0 ) , A ( 15,0 ) , B ( 9,4 )\) and \(C ( 3,4 )\). Find the \(x\)-coordinate of the centre of mass of the lamina.
CAIE Further Paper 3 2022 June Q2
2 A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(\frac { 4 } { 3 } \mathrm { mg }\). The other end of the string is attached to a fixed point \(O\) on a rough horizontal surface. The particle is at rest on the surface with the string at its natural length. The coefficient of friction between \(P\) and the surface is \(\frac { 1 } { 3 }\). The particle is projected along the surface in the direction \(O P\) with a speed of \(\frac { 1 } { 2 } \sqrt { \mathrm { ga } }\). Find the greatest extension of the string during the subsequent motion.
CAIE Further Paper 3 2022 June Q3
3 A particle \(P\) is projected with speed \(25 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. After 2 s the speed of \(P\) is \(15 \mathrm {~ms} ^ { - 1 }\).
  1. Find the value of \(\sin \theta\).
  2. Find the range of the flight.
CAIE Further Paper 3 2022 June Q4
4 One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). A particle of mass \(m\) is attached to the other end of the string and is held with the string taut at the point \(A\). At \(A\) the string makes an angle \(\theta\) with the upward vertical through \(O\). The particle is projected perpendicular to the string in a downward direction from \(A\) with a speed \(u\). It moves along a circular path in the vertical plane. When the string makes an angle \(\alpha\) with the downward vertical through \(O\), the speed of the particle is \(2 u\) and the magnitude of the tension in the string is 10 times its magnitude at \(A\). It is given that \(\mathrm { u } = \sqrt { \frac { 2 } { 3 } \mathrm { ga } }\).
  1. Find, in terms of \(m\) and \(g\), the magnitude of the tension in the string at \(A\).
  2. Find the value of \(\cos \alpha\).
CAIE Further Paper 3 2022 June Q5
5 A particle \(P\) of mass 4 kg is moving in a horizontal straight line. At time \(t\) s the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) and the displacement of \(P\) from a fixed point \(O\) on the line is \(x \mathrm {~m}\). The only force acting on \(P\) is a resistive force of magnitude \(\left( 4 \mathrm { e } ^ { - x } + 12 \right) \mathrm { e } ^ { - x } \mathrm {~N}\). When \(\mathrm { t } = 0 , \mathrm { x } = 0\) and \(v = 4\).
  1. Show by integration that \(\mathrm { v } = \frac { 1 + 3 \mathrm { e } ^ { \mathrm { x } } } { \mathrm { e } ^ { \mathrm { x } } }\).
  2. Find an expression for \(x\) in terms of \(t\).
    \includegraphics[max width=\textwidth, alt={}, center]{ad8b126c-d739-4e2a-8ce3-7811a61f5876-10_510_889_269_580}
    \(A B\) and \(B C\) are two fixed smooth vertical barriers on a smooth horizontal surface, with angle \(\mathrm { ABC } = 60 ^ { \circ }\). A particle of mass \(m\) is moving with speed \(u\) on the surface. The particle strikes \(A B\) at an angle \(\theta\) with \(A B\). It then strikes \(B C\) and rebounds at an angle \(\beta\) with \(B C\) (see diagram). The coefficient of restitution between the particle and each barrier is \(e\) and \(\tan \theta = 2\). The kinetic energy of the particle after the first collision is \(40 \%\) of its kinetic energy before the first collision.
  3. Find the value of \(e\).
  4. Find the size of angle \(\beta\).
    \includegraphics[max width=\textwidth, alt={}, center]{ad8b126c-d739-4e2a-8ce3-7811a61f5876-12_965_1059_267_502} A uniform cylinder with a rough surface and of radius \(a\) is fixed with its axis horizontal. Two identical uniform rods \(A B\) and \(B C\), each of weight \(W\) and length \(2 a\), are rigidly joined at \(B\) with \(A B\) perpendicular to \(B C\). The rods rest on the cylinder in a vertical plane perpendicular to the axis of the cylinder with \(A B\) at an angle \(\theta\) to the horizontal. \(D\) and \(E\) are the midpoints of \(A B\) and \(B C\) respectively and also the points of contact of the rods with the cylinder (see diagram). The rods are about to slip in a clockwise direction. The coefficient of friction between each rod and the cylinder is \(\mu\). The normal reaction between \(A B\) and the cylinder is \(R\) and the normal reaction between \(B C\) and the cylinder is \(N\).
  5. Find the ratio \(R : N\) in terms of \(\mu\).
  6. Given that \(\mu = \frac { 1 } { 3 }\), find the value of \(\tan \theta\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 3 2023 June Q1
1 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(3 m g\), is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(m\). The string hangs with \(P\) vertically below \(O\). The particle \(P\) is pulled vertically downwards so that the extension of the string is \(2 a\). The particle \(P\) is then released from rest.
  1. Find the speed of \(P\) when it is at a distance \(\frac { 3 } { 4 } a\) below \(O\).
  2. Find the initial acceleration of \(P\) when it is released from rest.
    \includegraphics[max width=\textwidth, alt={}, center]{3b50dc98-781e-4399-8165-ad5e3065df4b-03_741_473_269_836} A particle \(P\) of mass \(m\) is moving with speed \(u\) on a fixed smooth horizontal surface. It collides at an angle \(\alpha\) with a fixed smooth vertical barrier. After the collision, \(P\) moves at an angle \(\theta\) with the barrier, where \(\tan \theta = \frac { 1 } { 2 }\) (see diagram). The coefficient of restitution between \(P\) and the barrier is \(e\). The particle \(P\) loses 20\% of its kinetic energy as a result of the collision. Find the value of \(e\).
CAIE Further Paper 3 2023 June Q3
3 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 attached to a fixed point \(O\). The particle \(P\) is held at the point \(A\), where \(O A\) makes an angle \(\theta\) with the downward vertical through \(O\), and with the string taut. The particle \(P\) is projected perpendicular to \(O A\) in an upwards direction with speed \(u\). It then starts to move along a circular path in a vertical plane. The string goes slack when \(P\) is at \(B\), where angle \(A O B\) is \(90 ^ { \circ }\) and the speed of \(P\) is \(\sqrt { \frac { 4 } { 5 } \mathrm { ag } }\).
  1. Find the value of \(\sin \theta\).
  2. Find, in terms of \(m\) and \(g\), the tension in the string when \(P\) is at \(A\).
    \includegraphics[max width=\textwidth, alt={}, center]{3b50dc98-781e-4399-8165-ad5e3065df4b-06_846_767_258_689} An object is formed from a solid hemisphere, of radius \(2 a\), and a solid cylinder, of radius \(a\) and height \(d\). The hemisphere and the cylinder are made of the same material. The cylinder is attached to the plane face of the hemisphere. The line \(O C\) forms a diameter of the base of the cylinder, where \(C\) is the centre of the plane face of the hemisphere and \(O\) is common to both circumferences (see diagram). Relative to axes through \(O\), parallel and perpendicular to \(O C\) as shown, the centre of mass of the object is ( \(\mathrm { x } , \mathrm { y }\) ).
CAIE Further Paper 3 2023 June Q5
5 A light elastic string of natural length \(a\) and modulus of elasticity \(\lambda \mathrm { mg }\) has one end attached to a fixed point \(O\) on a smooth horizontal surface. When a particle of mass \(m\) is attached to the free end of the string, it moves with speed \(v\) in a horizontal circle with centre \(O\) and radius \(x\). When, instead, a particle of mass \(2 m\) is attached to the free end of the string, this particle moves with speed \(\frac { 1 } { 2 } v\) in a horizontal circle with centre \(O\) and radius \(\frac { 3 } { 4 } x\).
  1. Find \(x\) in terms of \(a\).
  2. Given that \(\mathrm { V } = \sqrt { 12 \mathrm { ag } }\), find the value of \(\lambda\).
CAIE Further Paper 3 2023 June Q6
6 A particle \(P\) moving in a straight line has displacement \(x \mathrm {~m}\) from a fixed point \(O\) on the line and velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\). The acceleration of \(P\), in \(\mathrm { ms } ^ { - 2 }\), is given by \(6 v \sqrt { v + 9 }\). When \(t = 0 , x = 2\) and \(v = 72\).
  1. Find an expression for \(v\) in terms of \(x\).
  2. Find an expression for \(x\) in terms of \(t\).
CAIE Further Paper 3 2023 June Q7
7 At time \(t \mathrm {~s}\), a particle \(P\) is projected with speed \(40 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The greatest height achieved by \(P\) during its flight is \(H \mathrm {~m}\) and the corresponding time is \(T \mathrm {~s}\).
  1. Obtain expressions for \(H\) and \(T\) in terms of \(\theta\).
    During the time between \(t = T\) and \(t = 3 , P\) descends a distance \(\frac { 1 } { 4 } H\).
  2. Find the value of \(\theta\).
  3. Find the speed of \(P\) when \(t = 3\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 3 2023 June Q1
1 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(3 m g\), is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(m\). The string hangs with \(P\) vertically below \(O\). The particle \(P\) is pulled vertically downwards so that the extension of the string is \(2 a\). The particle \(P\) is then released from rest.
  1. Find the speed of \(P\) when it is at a distance \(\frac { 3 } { 4 } a\) below \(O\).
  2. Find the initial acceleration of \(P\) when it is released from rest.
    \includegraphics[max width=\textwidth, alt={}, center]{454be64a-204f-4fa4-a5fc-72fd88e1289f-03_741_473_269_836} A particle \(P\) of mass \(m\) is moving with speed \(u\) on a fixed smooth horizontal surface. It collides at an angle \(\alpha\) with a fixed smooth vertical barrier. After the collision, \(P\) moves at an angle \(\theta\) with the barrier, where \(\tan \theta = \frac { 1 } { 2 }\) (see diagram). The coefficient of restitution between \(P\) and the barrier is \(e\). The particle \(P\) loses 20\% of its kinetic energy as a result of the collision. Find the value of \(e\).