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CAIE M2 2017 November Q1
4 marks Challenging +1.2
1 A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) on a smooth horizontal surface. A horizontal force of magnitude \(t \mathrm { e } ^ { - v } \mathrm {~N}\) directed away from \(O\) acts on \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after release. Find the velocity of \(P\) when \(t = 2\).
CAIE M2 2017 November Q2
3 marks Standard +0.3
2 A uniform solid cone has height 0.6 m and base radius 0.2 m . A uniform hollow cylinder, open at both ends, has the same dimensions. An object is made by putting the cone inside the cylinder so that the base of the cone coincides with one end of the cylinder (see diagram, which shows a cross-section). The total weight of the object is 60 N and its centre of mass is 0.25 m from the base of the cone. Calculate the weight of the cone.
CAIE M2 2017 November Q3
6 marks Challenging +1.2
3 A particle \(P\) of mass 0.4 kg is released from rest at a point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) moves down the line of greatest slope through \(O\). The velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its displacement from \(O\) is \(x \mathrm {~m}\). A retarding force of magnitude \(0.2 v ^ { 2 } \mathrm {~N}\) acts on \(P\) in the direction \(P O\).
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 - 0.5 v ^ { 2 }\).
  2. Express \(v\) in terms of \(x\).
CAIE M2 2017 November Q4
9 marks Challenging +1.2
4 \includegraphics[max width=\textwidth, alt={}, center]{6b220343-1d64-4dbc-a42d-77967eef9c6d-06_264_839_260_653} A light elastic string has natural length 2 m and modulus of elasticity 39 N . The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and 2.4 m apart. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of the string and hangs in equilibrium at a point 0.5 m below \(A B\) (see diagram).
  1. Show that \(m = 0.9\). \(P\) is projected vertically downwards from the equilibrium position, and comes to instantaneous rest at a point 1.6 m below \(A B\).
  2. Calculate the speed of projection of \(P\).
CAIE M2 2017 November Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{6b220343-1d64-4dbc-a42d-77967eef9c6d-08_449_890_262_630} \(O A B\) is a uniform lamina in the shape of a quadrant of a circle with centre \(O\) and radius 0.8 m which has its centre of mass at \(G\). The lamina is smoothly hinged at \(A\) to a fixed point and is free to rotate in a vertical plane. A horizontal force of magnitude 12 N acting in the plane of the lamina is applied to the lamina at \(B\). The lamina is in equilibrium with \(A G\) horizontal (see diagram).
  1. Calculate the length \(A G\).
  2. Find the weight of the lamina.
CAIE M2 2017 November Q6
9 marks Standard +0.3
6 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 8 N is attached to a fixed point \(O\) on a smooth horizontal plane. The other end of the string is attached to a particle \(P\) of mass 0.2 kg which moves on the plane in a circular path with centre \(O\). The speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the extension of the string is \(x \mathrm {~m}\).
  1. Given that \(v = 2.5\), find \(x\).
    It is given instead that the kinetic energy of \(P\) is twice the elastic potential energy stored in the string.
  2. Form two simultaneous equations and hence find \(x\) and \(v\).
CAIE M2 2017 November Q7
11 marks Standard +0.8
7 A small ball \(B\) is projected from a point \(O\) which is \(h \mathrm {~m}\) above a horizontal plane. At time 2 s after projection \(B\) has speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving in the direction \(30 ^ { \circ }\) above the horizontal.
  1. Find the initial speed and the angle of projection of \(B\). \(B\) has speed \(38 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) immediately before it strikes the plane.
  2. Calculate \(h\). \(B\) bounces when it strikes the plane, and leaves the plane with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) but with its horizontal component of velocity unchanged.
  3. Find the total time which elapses between the initial projection of \(B\) and the instant when it strikes the plane for the second time.
CAIE M2 2017 November Q3
7 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{2e4e6e32-eafc-4196-aaa8-42909cc2078e-04_305_510_264_813} One end of a light inextensible string of length 0.4 m is attached to a fixed point \(A\) which is above a smooth horizontal surface. A particle \(P\) of mass 0.6 kg is attached to the other end of the string. \(P\) moves in a circle on the surface with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), with the string taut and making an angle of \(60 ^ { \circ }\) with the horizontal (see diagram).
  1. Given that \(v = 0.5\), calculate the magnitude of the force that the surface exerts on \(P\).
  2. Find the greatest possible value of \(v\) for which \(P\) remains in contact with the surface.
CAIE M2 2018 November Q1
4 marks Standard +0.3
1 A small ball \(B\) is projected with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the horizontal from a point on horizontal ground. Find the time after projection when the speed of \(B\) is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the second time.
[0pt] [4] \includegraphics[max width=\textwidth, alt={}, center]{ef38fda2-230b-431a-8064-82e4a3bff393-04_620_668_255_742} A uniform object is made by attaching the base of a solid hemisphere to the base of a solid cone so that the object has an axis of symmetry. The base of the cone has radius 0.3 m , and the hemisphere has radius 0.2 m . The object is placed on a horizontal plane with a point \(A\) on the curved surface of the hemisphere and a point \(B\) on the circumference of the cone in contact with the plane (see diagram).
  1. Given that the object is on the point of toppling about \(B\), find the distance of the centre of mass of the object from the base of the cone.
  2. Given instead that the object is on the point of toppling about \(A\), calculate the height of the cone.
    [0pt] [The volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\). The volume of a hemisphere is \(\frac { 2 } { 3 } \pi r ^ { 3 }\).]
CAIE M2 2018 November Q3
7 marks Standard +0.3
3 A particle \(P\) of mass 0.4 kg is attached to a fixed point \(O\) by a light elastic string of natural length 0.5 m and modulus of elasticity 20 N . The particle \(P\) is released from rest at \(O\).
  1. Find the greatest speed of \(P\) in the subsequent motion.
  2. Find the distance below \(O\) of the point at which \(P\) comes to instantaneous rest.
CAIE M2 2018 November Q4
8 marks Challenging +1.8
4 \includegraphics[max width=\textwidth, alt={}, center]{ef38fda2-230b-431a-8064-82e4a3bff393-08_152_885_262_630} A particle \(P\) of mass 0.5 kg is projected along a smooth horizontal surface towards a fixed point \(A\). Initially \(P\) is at a point \(O\) on the surface, and after projection, \(P\) has a displacement from \(O\) of \(x \mathrm {~m}\) and velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle \(P\) is connected to \(A\) by a light elastic string of natural length 0.8 m and modulus of elasticity 16 N . The distance \(O A\) is 1.6 m (see diagram). The motion of \(P\) is resisted by a force of magnitude \(24 x ^ { 2 } \mathrm {~N}\).
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 32 - 40 x - 48 x ^ { 2 }\) while \(P\) is in motion and the string is stretched.
    The maximum value of \(v\) is 4.5 .
  2. Find the initial value of \(v\).
CAIE M2 2018 November Q5
8 marks Moderate -0.3
5 A particle \(P\) of mass 0.1 kg is attached to one end of a light inextensible string of length 0.5 m . The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a circle which has its centre \(O\) on a smooth horizontal surface 0.3 m below \(A\). The tension in the string has magnitude \(T \mathrm {~N}\) and the magnitude of the force exerted on \(P\) by the surface is \(R \mathrm {~N}\).
  1. Given that the speed of \(P\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), calculate \(T\) and \(R\).
  2. Given instead that \(T = R\), calculate the angular speed of \(P\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ef38fda2-230b-431a-8064-82e4a3bff393-12_449_621_260_762} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows the cross-section \(A B C D E\) through the centre of mass \(G\) of a uniform prism. The crosssection consists of a rectangle \(A B C F\) from which a triangle \(D E F\) has been removed; \(A B = 0.6 \mathrm {~m}\), \(B C = 0.7 \mathrm {~m}\) and \(D F = E F = 0.3 \mathrm {~m}\).
  3. Show that the distance of \(G\) from \(B C\) is 0.276 m , and find the distance of \(G\) from \(A B\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ef38fda2-230b-431a-8064-82e4a3bff393-13_494_583_258_781} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The prism is placed with \(C D\) on a rough horizontal surface. A force of magnitude 2 N acting in the plane of the cross-section is applied to the prism. The line of action of the force passes through \(G\) and is perpendicular to \(D E\) (see Fig. 2). The prism is on the point of toppling about the edge through \(D\).
  4. Calculate the weight of the prism. \includegraphics[max width=\textwidth, alt={}, center]{ef38fda2-230b-431a-8064-82e4a3bff393-14_512_520_258_817} A small object is projected with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) at the foot of a plane inclined at \(45 ^ { \circ }\) to the horizontal. The angle of projection of the object is \(15 ^ { \circ }\) above a line of greatest slope of the plane (see diagram). At time \(t \mathrm {~s}\) after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  5. Express \(x\) and \(y\) in terms of \(t\), and hence find the value of \(t\) for the instant when the object strikes the plane.
  6. Express the vertical height of the object above the plane in terms of \(t\) and hence find the greatest vertical height of the object above 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. \(\overline { \text { Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable } }\) effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
    To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series.
    Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
CAIE M2 2018 November Q1
3 marks Standard +0.3
1 A small ball \(B\) is projected with speed \(38 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the horizontal from a point on horizontal ground. Find the speed of \(B\) when the path of \(B\) makes an angle of \(20 ^ { \circ }\) above the horizontal.
CAIE M2 2018 November Q2
7 marks Standard +0.3
2 A uniform solid object is made by attaching a cone to a cylinder so that the circumferences of the base of the cone and a plane face of the cylinder coincide. The cone and the cylinder each have radius 0.3 m and height 0.4 m .
  1. Calculate the distance of the centre of mass of the object from the vertex of the cone.
    [0pt] [The volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
    The object has weight \(W \mathrm {~N}\) and is placed with its plane circular face on a rough horizontal surface. A force of magnitude \(k W \mathrm {~N}\) acting at \(30 ^ { \circ }\) to the upward vertical is applied to the vertex of the cone. The object does not slip.
  2. Find the greatest possible value of \(k\) for which the object does not topple.
CAIE M2 2018 November Q3
7 marks Challenging +1.8
3 A particle \(P\) of mass 0.4 kg is projected horizontally along a smooth horizontal plane from a point \(O\). After projection the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its displacement from \(O\) is \(x \mathrm {~m}\). A force of magnitude \(8 x \mathrm {~N}\) directed away from \(O\) acts on \(P\) and a force of magnitude ( \(2 \mathrm { e } ^ { - x } + 4\) ) N opposes the motion of \(P\). One end of a light elastic string of natural length 0.5 m is attached to \(O\) and the other end of the string is attached to \(P\).
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 20 x - 10 - 5 \mathrm { e } ^ { - x }\) before the elastic string becomes taut.
  2. Given that the initial velocity of \(P\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find \(v\) when the string first becomes taut.
    When the string is taut, the acceleration of \(P\) is proportional to \(\mathrm { e } ^ { - x }\).
  3. Find the modulus of elasticity of the string.
CAIE M2 2018 November Q4
8 marks Moderate -0.3
4 A small object is projected horizontally with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) above horizontal ground. At time \(t \mathrm {~s}\) after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(t\) and hence show that the equation of the path of the object is \(y = - \frac { 5 x ^ { 2 } } { V ^ { 2 } }\).
    The object passes through points with coordinates \(( a , - a )\) and \(\left( a ^ { 2 } , - 16 a \right)\), where \(a\) is a positive constant.
  2. Find the value of \(a\).
  3. Given that the object strikes the ground at the point where \(x = 5 a\), find the height of \(O\) above the ground .
CAIE M2 2018 November Q5
8 marks Standard +0.3
5 A particle \(P\) of mass 0.7 kg is attached to a fixed point \(O\) by a light elastic string of natural length 0.6 m and modulus of elasticity 15 N . The particle \(P\) is projected vertically downwards from the point \(A , 0.8 \mathrm {~m}\) vertically below \(O\). The initial speed of \(P\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the distance below \(A\) of the point at which \(P\) comes to instantaneous rest.
  2. Find the greatest speed of \(P\) in the motion. \includegraphics[max width=\textwidth, alt={}, center]{f922bf53-94a0-4ccc-8c38-959d2f795629-10_478_652_260_751} The diagram shows a uniform lamina \(A B C D E F G H\). The lamina consists of a quarter-circle \(O A B\) of radius \(r \mathrm {~m}\), a rectangle \(D E F G\) and two isosceles right-angled triangles \(C O D\) and \(G O H\). The rectangle has \(D G = E F = r \mathrm {~m}\) and \(D E = F G = x \mathrm {~m}\).
  3. Given that the centre of mass of the lamina is at \(O\), express \(x\) in terms of \(r\).
  4. Given instead that the rectangle \(D E F G\) is a square with edges of length \(r \mathrm {~m}\), state with a reason whether the centre of mass of the lamina lies within the square or the quarter-circle. \includegraphics[max width=\textwidth, alt={}, center]{f922bf53-94a0-4ccc-8c38-959d2f795629-12_384_693_258_726} A rough horizontal rod \(A B\) of length 0.45 m rotates with constant angular velocity \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a vertical axis through \(A\). A small ring \(R\) of mass 0.2 kg can slide on the rod. A particle \(P\) of mass 0.1 kg is attached to the mid-point of a light inextensible string of length 0.6 m . One end of the string is attached to \(R\) and the other end of the string is attached to \(B\), with angle \(R P B = 60 ^ { \circ }\) (see diagram). \(R\) and \(P\) move in horizontal circles as the system rotates. \(R\) is in limiting equilibrium.
  5. Show that the tension in the portion \(P R\) of the string is 1.66 N , correct to 3 significant figures.
  6. Find the coefficient of friction between the ring and the rod.
    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 M2 2018 November Q1
4 marks Standard +0.3
1 A small ball \(B\) is projected with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the horizontal from a point on horizontal ground. Find the time after projection when the speed of \(B\) is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the second time.
[0pt] [4] \includegraphics[max width=\textwidth, alt={}, center]{5ab17f54-2408-4cf5-b852-611dd5aa112e-04_620_668_255_742} A uniform object is made by attaching the base of a solid hemisphere to the base of a solid cone so that the object has an axis of symmetry. The base of the cone has radius 0.3 m , and the hemisphere has radius 0.2 m . The object is placed on a horizontal plane with a point \(A\) on the curved surface of the hemisphere and a point \(B\) on the circumference of the cone in contact with the plane (see diagram).
  1. Given that the object is on the point of toppling about \(B\), find the distance of the centre of mass of the object from the base of the cone.
  2. Given instead that the object is on the point of toppling about \(A\), calculate the height of the cone.
    [0pt] [The volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\). The volume of a hemisphere is \(\frac { 2 } { 3 } \pi r ^ { 3 }\).]
CAIE M2 2018 November Q4
8 marks Challenging +1.2
4 \includegraphics[max width=\textwidth, alt={}, center]{5ab17f54-2408-4cf5-b852-611dd5aa112e-08_152_885_262_630} A particle \(P\) of mass 0.5 kg is projected along a smooth horizontal surface towards a fixed point \(A\). Initially \(P\) is at a point \(O\) on the surface, and after projection, \(P\) has a displacement from \(O\) of \(x \mathrm {~m}\) and velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle \(P\) is connected to \(A\) by a light elastic string of natural length 0.8 m and modulus of elasticity 16 N . The distance \(O A\) is 1.6 m (see diagram). The motion of \(P\) is resisted by a force of magnitude \(24 x ^ { 2 } \mathrm {~N}\).
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 32 - 40 x - 48 x ^ { 2 }\) while \(P\) is in motion and the string is stretched.
    The maximum value of \(v\) is 4.5 .
  2. Find the initial value of \(v\).
CAIE M2 2018 November Q5
8 marks Moderate -0.3
5 A particle \(P\) of mass 0.1 kg is attached to one end of a light inextensible string of length 0.5 m . The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a circle which has its centre \(O\) on a smooth horizontal surface 0.3 m below \(A\). The tension in the string has magnitude \(T \mathrm {~N}\) and the magnitude of the force exerted on \(P\) by the surface is \(R \mathrm {~N}\).
  1. Given that the speed of \(P\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), calculate \(T\) and \(R\).
  2. Given instead that \(T = R\), calculate the angular speed of \(P\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5ab17f54-2408-4cf5-b852-611dd5aa112e-12_449_621_260_762} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows the cross-section \(A B C D E\) through the centre of mass \(G\) of a uniform prism. The crosssection consists of a rectangle \(A B C F\) from which a triangle \(D E F\) has been removed; \(A B = 0.6 \mathrm {~m}\), \(B C = 0.7 \mathrm {~m}\) and \(D F = E F = 0.3 \mathrm {~m}\).
  3. Show that the distance of \(G\) from \(B C\) is 0.276 m , and find the distance of \(G\) from \(A B\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5ab17f54-2408-4cf5-b852-611dd5aa112e-13_494_583_258_781} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The prism is placed with \(C D\) on a rough horizontal surface. A force of magnitude 2 N acting in the plane of the cross-section is applied to the prism. The line of action of the force passes through \(G\) and is perpendicular to \(D E\) (see Fig. 2). The prism is on the point of toppling about the edge through \(D\).
  4. Calculate the weight of the prism. \includegraphics[max width=\textwidth, alt={}, center]{5ab17f54-2408-4cf5-b852-611dd5aa112e-14_512_520_258_817} A small object is projected with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) at the foot of a plane inclined at \(45 ^ { \circ }\) to the horizontal. The angle of projection of the object is \(15 ^ { \circ }\) above a line of greatest slope of the plane (see diagram). At time \(t \mathrm {~s}\) after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  5. Express \(x\) and \(y\) in terms of \(t\), and hence find the value of \(t\) for the instant when the object strikes the plane.
  6. Express the vertical height of the object above the plane in terms of \(t\) and hence find the greatest vertical height of the object above 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 M2 2019 November Q2
6 marks Moderate -0.3
2 A particle is projected from a point on horizontal ground with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. The particle strikes the ground 2 s after projection.
  1. Find \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{4cd525d5-d59b-4ab9-85a3-fc3d97fd09fc-03_67_1571_438_328}
  2. Calculate the time after projection at which the direction of motion of the particle is \(20 ^ { \circ }\) below the horizontal.
CAIE M2 2019 November Q3
6 marks Challenging +1.2
3 A smooth horizontal surface has two fixed points \(O\) and \(A\) which are 0.8 m apart. A particle \(P\) of mass 0.25 kg is projected with velocity \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) horizontally from \(A\) in the direction away from \(O\). The velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\). A force of magnitude \(k v ^ { 2 } x ^ { - 2 } \mathrm {~N}\) opposes the motion of \(P\).
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 4 k v ^ { 2 } x ^ { - 2 }\).
  2. Express \(v\) in terms of \(k\) and \(x\).
CAIE M2 2019 November Q4
7 marks Moderate -0.3
4 A small ball \(B\) is projected with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal from a point \(O\). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of \(B\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(t\) and hence find the equation of the trajectory of the ball.
  2. Find the value of \(x\) for which \(O B\) makes an angle of \(45 ^ { \circ }\) above the horizontal.
CAIE M2 2019 November Q5
9 marks Standard +0.3
5 A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 9 N . The other end of the string is attached to a fixed point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. \(O A\) is a line of greatest slope of the plane with \(A\) below the level of \(O\) and \(O A = 0.8 \mathrm {~m}\). The particle \(P\) is released from rest at \(A\).
  1. Find the initial acceleration of \(P\).
  2. Find the greatest speed of \(P\). \(6 \quad A\) and \(B\) are two fixed points on a vertical axis with \(A 0.6 \mathrm {~m}\) above \(B\). A particle \(P\) of mass 0.3 kg is attached to \(A\) by a light inextensible string of length 0.5 m . The particle \(P\) is attached to \(B\) by a light elastic string with modulus of elasticity 46 N . The particle \(P\) moves with constant angular speed \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle with centre at the mid-point of \(A B\).
  3. Find the speed of \(P\).
  4. Calculate the tension in the string \(B P\) and hence find the natural length of this string. \includegraphics[max width=\textwidth, alt={}, center]{4cd525d5-d59b-4ab9-85a3-fc3d97fd09fc-10_540_574_260_781} \(A B C\) is the cross-section through the centre of mass of a uniform prism which rests with \(A B\) on a rough horizontal surface. \(A B = 0.4 \mathrm {~m}\) and \(C\) is 0.9 m above the surface (see diagram). The prism is on the point of toppling about its edge through \(B\).
  5. Show that angle \(B A C = 48.4 ^ { \circ }\), correct to 3 significant figures.
    A force of magnitude 18 N acting in the plane of the cross-section and perpendicular to \(A C\) is now applied to the prism at \(C\). The prism is on the point of rotating about its edge through \(A\).
  6. Calculate the weight of the prism.
  7. Given also that the prism is on the point of slipping, calculate the coefficient of friction between the prism and the surface.
    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 M2 2019 November Q1
3 marks Standard +0.3
1 A particle of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 9 N . The other end of the string is attached to a fixed point \(O\) on a smooth horizontal surface. The particle is projected horizontally from \(O\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the greatest distance of the particle from \(O\).