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CAIE M2 2015 November Q7
11 marks Challenging +1.2
A particle \(P\) is projected with speed \(V\,\text{m s}^{-1}\) at an angle of \(60°\) above the horizontal from a point \(O\). At the instant \(1\,\text{s}\) later a particle \(Q\) is projected from \(O\) with the same initial speed at an angle of \(45°\) above the horizontal. The two particles collide when \(Q\) has been in motion for \(t\,\text{s}\).
  1. Show that \(t = 2.414\), correct to \(3\) decimal places. [3]
  2. Find the value of \(V\). [4]
The collision occurs after \(P\) has passed through the highest point of its trajectory.
  1. Calculate the vertical distance of \(P\) below its greatest height when \(P\) and \(Q\) collide. [4]
CAIE M2 2016 November Q1
4 marks Standard +0.3
A particle \(P\) of mass 0.3 kg moves in a circle with centre \(O\) on a smooth horizontal surface. \(P\) is attached to \(O\) by a light elastic string of modulus of elasticity 12 N and natural length \(l\) m. The speed of \(P\) is 4 m s\(^{-1}\), and the radius of the circle in which it moves is 2l m. Calculate \(l\). [4]
CAIE M2 2016 November Q2
7 marks Standard +0.8
\includegraphics{figure_2} A uniform wire is bent to form an object which has a semicircular arc with diameter \(AB\) of length 1.2 m, with a smaller semicircular arc with diameter \(BC\) of length 0.6 m. The end \(C\) of the smaller arc is at the centre of the larger arc (see diagram). The two semicircular arcs of the wire are in the same plane.
  1. Show that the distance of the centre of mass of the object from the line \(ACB\) is 0.191 m, correct to 3 significant figures. [3]
The object is freely suspended at \(A\) and hangs in equilibrium.
  1. Find the angle between \(ACB\) and the vertical. [4]
CAIE M2 2016 November Q3
7 marks Standard +0.3
A small block \(B\) of mass 0.25 kg is released from rest at a point \(O\) on a smooth horizontal surface. After its release the velocity of \(B\) is \(v\) m s\(^{-1}\) when its displacement is \(x\) m from \(O\). The force acting on \(B\) has magnitude \((2 + 0.3x^2)\) N and is directed horizontally away from \(O\).
  1. Show that \(v\frac{dv}{dx} = 1.2x^2 + 8\). [2]
  2. Find the velocity of \(B\) when \(x = 1.5\). [3]
An extra force acts on \(B\) after \(x = 1.5\). It is given that, when \(x > 1.5\), $$v\frac{dv}{dx} = 1.2x^2 + 6 - 3x.$$
  1. Find the magnitude of this extra force and state the direction in which it acts. [2]
CAIE M2 2016 November Q4
7 marks Standard +0.3
\includegraphics{figure_4} The diagram shows the cross-section \(ABCD\) through the centre of mass of a uniform solid prism. \(AB = 0.9\) m, \(BC = 2a\) m, \(AD = a\) m and angle \(ABC =\) angle \(BAD = 90°\).
  1. Calculate the distance of the centre of mass of the prism from \(AD\). [2]
  2. Express the distance of the centre of mass of the prism from \(AB\) in terms of \(a\). [2]
The prism has weight 18 N and rests in equilibrium on a rough horizontal surface, with \(AD\) in contact with the surface. A horizontal force of magnitude 6 N is applied to the prism. This force acts through the centre of mass in the direction \(BC\).
  1. Given that the prism is on the point of toppling, calculate \(a\). [3]
CAIE M2 2016 November Q5
7 marks Standard +0.3
A small ball \(B\) of mass 0.4 kg moves in a horizontal circle with centre \(O\) and radius 0.6 m on a smooth horizontal surface. One end of a light inextensible string is attached to \(B\); the other end of the string is attached to a fixed point 0.45 m vertically above \(O\).
  1. Given that the tension in the string is 5 N, calculate the speed of \(B\). [3]
  2. Find the greatest possible tension in the string for the motion, and the corresponding angular speed of \(B\). [4]
CAIE M2 2016 November Q6
7 marks Challenging +1.2
\includegraphics{figure_6} The diagram shows a smooth narrow tube formed into a fixed vertical circle with centre \(O\) and radius 0.9 m. A light elastic string with modulus of elasticity 8 N and natural length 1.2 m has one end attached to the highest point \(A\) on the inside of the tube. The other end of the string is attached to a particle \(P\) of mass 0.2 kg. The particle is released from rest at the lowest point on the inside of the tube. By considering energy, calculate
  1. the speed of \(P\) when it is at the same horizontal level as \(O\), [4]
  2. the speed of \(P\) at the instant when the string becomes slack. [3]
CAIE M2 2016 November Q7
11 marks Standard +0.8
A particle \(P\) is projected with speed 35 m s\(^{-1}\) from a point \(O\) on a horizontal plane. In the subsequent motion, the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x\) m and \(y\) m respectively. The equation of the trajectory of \(P\) is $$y = kx - \frac{(1 + k^2)x^2}{245},$$ where \(k\) is a constant. \(P\) passes through the points \(A(14, a)\) and \(B(42, 2a)\), where \(a\) is a constant.
  1. Calculate the two possible values of \(k\) and hence show that the larger of the two possible angles of projection is 63.435°, correct to 3 decimal places. [5]
For the larger angle of projection, calculate
  1. the time after projection when \(P\) passes through \(A\), [2]
  2. the speed and direction of motion of \(P\) when it passes through \(B\). [4]
CAIE M2 2018 November Q1
4 marks Standard +0.3
A small ball \(B\) is projected with speed \(30\text{ m s}^{-1}\) at an angle of \(60°\) to the horizontal from a point on horizontal ground. Find the time after projection when the speed of \(B\) is \(25\text{ m s}^{-1}\) for the second time. [4]
CAIE M2 2018 November Q2
6 marks Standard +0.8
\includegraphics{figure_2} 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 \(0.3\text{ m}\), and the hemisphere has radius \(0.2\text{ m}\). 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. [3]
  2. Given instead that the object is on the point of toppling about \(A\), calculate the height of the cone. [3]
[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
A particle \(P\) of mass \(0.4\text{ kg}\) is attached to a fixed point \(O\) by a light elastic string of natural length \(0.5\text{ m}\) and modulus of elasticity \(20\text{ N}\). The particle \(P\) is released from rest at \(O\).
  1. Find the greatest speed of \(P\) in the subsequent motion. [4]
  2. Find the distance below \(O\) of the point at which \(P\) comes to instantaneous rest. [3]
CAIE M2 2018 November Q4
8 marks Challenging +1.2
\includegraphics{figure_4} A particle \(P\) of mass \(0.5\text{ kg}\) 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 \(x\text{ m}\) and velocity \(v\text{ m s}^{-1}\). The particle \(P\) is connected to \(A\) by a light elastic string of natural length \(0.8\text{ m}\) and modulus of elasticity \(16\text{ N}\). The distance \(OA\) is \(1.6\text{ m}\) (see diagram). The motion of \(P\) is resisted by a force of magnitude \(24x^2\text{ N}\).
  1. Show that \(v\frac{\text{d}v}{\text{d}x} = 32 - 40x - 48x^2\) while \(P\) is in motion and the string is stretched. [3]
  2. The maximum value of \(v\) is \(4.5\). Find the initial value of \(v\). [5]
CAIE M2 2018 November Q5
8 marks Standard +0.3
A particle \(P\) of mass \(0.1\text{ kg}\) is attached to one end of a light inextensible string of length \(0.5\text{ m}\). 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 \(0.3\text{ m}\) below \(A\). The tension in the string has magnitude \(T\text{ N}\) and the magnitude of the force exerted on \(P\) by the surface is \(R\text{ N}\).
  1. Given that the speed of \(P\) is \(1.5\text{ m s}^{-1}\), calculate \(T\) and \(R\). [4]
  2. Given instead that \(T = R\), calculate the angular speed of \(P\). [4]
CAIE M2 2018 November Q6
8 marks Standard +0.3
\includegraphics{figure_6} Fig. 1 shows the cross-section \(ABCDE\) through the centre of mass \(G\) of a uniform prism. The cross-section consists of a rectangle \(ABCF\) from which a triangle \(DEF\) has been removed; \(AB = 0.6\text{ m}\), \(BC = 0.7\text{ m}\) and \(DF = EF = 0.3\text{ m}\).
  1. Show that the distance of \(G\) from \(BC\) is \(0.276\text{ m}\), and find the distance of \(G\) from \(AB\). [5]
  2. The prism is placed with \(CD\) on a rough horizontal surface. A force of magnitude \(2\text{ N}\) 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 \(DE\) (see Fig. 2). The prism is on the point of toppling about the edge through \(D\). Calculate the weight of the prism. [3]
CAIE M2 2018 November Q7
9 marks Challenging +1.2
\includegraphics{figure_7} A small object is projected with speed \(24\text{ m s}^{-1}\) from a point \(O\) at the foot of a plane inclined at \(45°\) to the horizontal. The angle of projection of the object is \(15°\) above a line of greatest slope of the plane (see diagram). At time \(t\) s after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x\) m and \(y\) m respectively.
  1. Express \(x\) and \(y\) in terms of \(t\), and hence find the value of \(t\) for the instant when the object strikes the plane. [4]
  2. 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. [5]
CAIE M2 2018 November Q4
8 marks Challenging +1.2
\includegraphics{figure_4} A particle \(P\) of mass \(0.5\text{ kg}\) 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 \(x\text{ m}\) and velocity \(v\text{ m s}^{-1}\). The particle \(P\) is connected to \(A\) by a light elastic string of natural length \(0.8\text{ m}\) and modulus of elasticity \(16\text{ N}\). The distance \(OA\) is \(1.6\text{ m}\) (see diagram). The motion of \(P\) is resisted by a force of magnitude \(24x^2\text{ N}\).
  1. Show that \(v\frac{\text{d}v}{\text{d}x} = 32 - 40x - 48x^2\) while \(P\) is in motion and the string is stretched. [3] The maximum value of \(v\) is \(4.5\).
  2. Find the initial value of \(v\). [5]
CAIE M2 2018 November Q6
8 marks Standard +0.3
\includegraphics{figure_6} Fig. 1 shows the cross-section \(ABCDE\) through the centre of mass \(G\) of a uniform prism. The cross-section consists of a rectangle \(ABCF\) from which a triangle \(DEF\) has been removed; \(AB = 0.6\text{ m}\), \(BC = 0.7\text{ m}\) and \(DF = EF = 0.3\text{ m}\).
  1. Show that the distance of \(G\) from \(BC\) is \(0.276\text{ m}\), and find the distance of \(G\) from \(AB\). [5] The prism is placed with \(CD\) on a rough horizontal surface. A force of magnitude \(2\text{ N}\) 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 \(DE\) (see Fig. 2). The prism is on the point of toppling about the edge through \(D\).
  2. Calculate the weight of the prism. [3]
CAIE M2 2018 November Q7
9 marks Standard +0.8
\includegraphics{figure_7} A small object is projected with speed \(24\text{ m s}^{-1}\) from a point \(O\) at the foot of a plane inclined at \(45°\) to the horizontal. The angle of projection of the object is \(15°\) above a line of greatest slope of the plane (see diagram). At time \(t\text{ s}\) after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x\text{ m}\) and \(y\text{ m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(t\), and hence find the value of \(t\) for the instant when the object strikes the plane. [4]
  2. 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. [5]
CAIE Further Paper 3 2020 June Q1
5 marks Standard +0.3
A particle \(P\) is projected with speed \(u\) at an angle of \(30°\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The particle reaches its greatest height at time \(T\) after projection. Find, in terms of \(u\), the speed of \(P\) at time \(\frac{2}{3}T\) after projection. [5]
CAIE Further Paper 3 2020 June Q2
5 marks Challenging +1.2
\includegraphics{figure_2} A light inextensible string of length \(a\) is threaded through a fixed smooth ring \(R\). One end of the string is attached to a particle \(A\) of mass \(3m\). The other end of the string is attached to a particle \(B\) of mass \(m\). The particle \(A\) hangs in equilibrium at a distance \(x\) vertically below the ring. The angle between \(AR\) and \(BR\) is \(\theta\) (see diagram). The particle \(B\) moves in a horizontal circle with constant angular speed \(2\sqrt{\frac{g}{a}}\). Show that \(\cos \theta = \frac{1}{3}\) and find \(x\) in terms of \(a\). [5]
CAIE Further Paper 3 2020 June Q3
7 marks Standard +0.3
One end of a light elastic spring, of natural length \(a\) and modulus of elasticity \(5mg\), is attached to a fixed point \(A\). The other end of the spring is attached to a particle \(P\) of mass \(m\). The spring hangs with \(P\) vertically below \(A\). The particle \(P\) is released from rest in the position where the extension of the spring is \(\frac{3}{5}a\).
  1. Show that the initial acceleration of \(P\) is \(\frac{3}{5}g\) upwards. [3]
  2. Find the speed of \(P\) when the spring first returns to its natural length. [4]
CAIE Further Paper 3 2020 June Q4
7 marks Challenging +1.2
\includegraphics{figure_4} A uniform square lamina \(ABCD\) has sides of length \(10\text{cm}\). The point \(E\) is on \(BC\) with \(EC = 7.5\text{cm}\), and the point \(F\) is on \(DC\) with \(CF = x\text{cm}\). The triangle \(EFC\) is removed from \(ABCD\) (see diagram). The centre of mass of the resulting shape \(ABEFD\) is a distance \(\bar{x}\text{cm}\) from \(CB\) and a distance \(\bar{y}\text{cm}\) from \(CD\).
  1. Show that \(\bar{x} = \frac{400 - x^2}{80 - 3x}\) and find a corresponding expression for \(\bar{y}\). [4]
The shape \(ABEFD\) is in equilibrium in a vertical plane with the edge \(DF\) resting on a smooth horizontal surface.
  1. Find the greatest possible value of \(x\), giving your answer in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are constants to be determined. [3]
CAIE Further Paper 3 2020 June Q5
8 marks Challenging +1.2
A particle \(P\) is moving along a straight line with acceleration \(3ku - kv\) where \(v\) is its velocity at time \(t\), \(u\) is its initial velocity and \(k\) is a constant. The velocity and acceleration of \(P\) are both in the direction of increasing displacement from the initial position.
  1. Find the time taken for \(P\) to achieve a velocity of \(2u\). [3]
  2. Find an expression for the displacement of \(P\) from its initial position when its velocity is \(2u\). [5]
CAIE Further Paper 3 2020 June Q6
8 marks Challenging +1.2
A particle \(P\) of mass \(m\) is moving with speed \(u\) on a fixed smooth horizontal surface. The particle strikes a fixed vertical barrier. At the instant of impact the direction of motion of \(P\) makes an angle \(\alpha\) with the barrier. The coefficient of restitution between \(P\) and the barrier is \(e\). As a result of the impact, the direction of motion of \(P\) is turned through \(90°\).
  1. Show that \(\tan^2 \alpha = \frac{1}{e}\). [3]
The particle \(P\) loses two-thirds of its kinetic energy in the impact.
  1. Find the value of \(\alpha\) and the value of \(e\). [5]
CAIE Further Paper 3 2020 June Q7
10 marks Challenging +1.8
A hollow cylinder of radius \(a\) is fixed with its axis horizontal. A particle \(P\), of mass \(m\), moves in part of a vertical circle of radius \(a\) and centre \(O\) on the smooth inner surface of the cylinder. The speed of \(P\) when it is at the lowest point \(A\) of its motion is \(\sqrt{\frac{7}{2}ga}\). The particle \(P\) loses contact with the surface of the cylinder when \(OP\) makes an angle \(\theta\) with the upward vertical through \(O\).
  1. Show that \(\theta = 60°\). [5]
  2. Show that in its subsequent motion \(P\) strikes the cylinder at the point \(A\). [5]