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CAIE M2 2013 November Q1
Standard +0.3
\includegraphics{figure_1}
  1. \(A\) has velocity \(\vec{x}\) and \(C\) has velocity \(\vec{v}\)
CAIE M2 2013 November Q2
Moderate -0.5
\includegraphics{figure_2}
    1. \(C\)
    2. \(C\) has velocity \(\vec{v}\)
CAIE M2 2013 November Q3
Moderate -1.0
\(A\) has velocity \(\vec{v}\), there are velocities \(\vec{x}\), \(\vec{v}\), \(\vec{v}\) around point \(O\), and velocity \(\vec{v}\)
    1. —
    2. \(v\) and \(v\)
CAIE M2 2013 November Q4
Moderate -0.3
\(A\) has velocity \(\vec{v}\)
    1. \(C\) with velocity \(\vec{v}\)
    2. \(C\)
CAIE M2 2013 November Q5
Moderate -0.5
\includegraphics{figure_5} \(A\) has velocity \(\vec{x}\), there are velocities \(\vec{x}\), \(\vec{x}\), \(\vec{v}\) and \(\vec{v}\)
    1. \(v\)
    2. \(v\) and \(v\)
CAIE M2 2013 November Q6
Moderate -0.5
\includegraphics{figure_6} \(E\) has velocity \(\vec{v}\)
    1. \(B\)
    2. \(v\)
CAIE M2 2013 November Q7
Moderate -0.5
\(A\) has velocity \(\vec{x}\)
    1. \(C\)
  2. \(A\) has velocity \(\vec{x}\)
    1. \(C\)
  3. \(C\) with velocities \(v \vec{v}\)
CAIE M2 2013 November Q1
8 marks Moderate -0.8
A particle moves in a straight line. Its displacement from a fixed point \(O\) at time \(t\) seconds is \(s\) metres, where \(s = t^3 - 9t^2 + 24t\).
  1. Find expressions for the velocity \(v\) and acceleration \(a\) of the particle at time \(t\).
  2. Find the values of \(t\) for which the particle is at rest.
  3. Find the total distance travelled by the particle in the first \(6\) seconds.
[8]
CAIE M2 2013 November Q2
6 marks Moderate -0.5
A particle moves in a straight line. At time \(t\) seconds its velocity is \(v\) ms\(^{-1}\) and its acceleration is \(a\) ms\(^{-2}\).
  1. Given that \(a = —\), express \(v\) in terms of \(t\).
  2. Given that \(v = tv\) when \(t = 0\), find \(v\) in terms of \(t\).
  3. Find the displacement from the starting point when \(t = v\).
[6]
CAIE M2 2013 November Q3
8 marks Standard +0.3
\includegraphics{figure_3} A particle moves on the inner surface of a smooth hollow cone of semi-vertical angle \(\alpha\). The axis of the cone is vertical with the vertex at the bottom. The particle moves in a horizontal circle of radius \(r\) with constant speed \(v\). Find expressions for the normal reactions on the particle from the cone surface, and show that the height of the particle above the vertex is \(\frac{v^2}{g \tan \alpha}\). [8]
CAIE M2 2013 November Q4
14 marks Standard +0.8
A particle of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point. The particle moves in a vertical circle.
  1. Show that the speed \(v\) at the lowest point of the circle must satisfy \(v^2 \geq 5gl\) for the particle to complete the circle.
  2. Given that the particle just completes the circle, find the tensions in the string at the highest and lowest points of the circle.
  3. Given that \(v^2 = 6gl\) at the lowest point, find the tension in the string when the particle has risen through an angle \(\theta\) from the lowest point.
[14]
CAIE M2 2013 November Q5
8 marks Standard +0.3
A smooth sphere of mass \(M\) and radius \(a\) rests in contact with a smooth vertical wall and a smooth inclined plane. The plane makes an angle \(\alpha\) with the horizontal.
  1. Find the magnitude of each of the contact forces acting on the sphere.
  2. Find the range of values of \(\alpha\) for which this equilibrium is possible.
[8]
CAIE M2 2013 November Q6
8 marks Moderate -0.3
Two particles \(A\) and \(B\) have masses \(3m\) and \(2m\) respectively. Initially \(A\) is at rest and \(B\) is moving with speed \(u\) in a straight line towards \(A\). The coefficient of restitution between the particles is \(e\).
  1. Find the speeds of the particles immediately after the collision.
  2. Find the condition on \(e\) for \(A\) to be moving faster than \(B\) after the collision.
[8]
CAIE M2 2013 November Q7
16 marks Challenging +1.8
\includegraphics{figure_7} A uniform solid hemisphere of mass \(M\) and radius \(a\) is placed with its curved surface on rough horizontal ground. A horizontal force \(P\) is applied to the hemisphere at the centre of its flat circular face.
  1. Find the minimum value of the coefficient of friction \(\mu\) between the hemisphere and the ground for the hemisphere to slide without toppling.
  2. Show that if \(\mu < \frac{3}{8}\), the hemisphere will topple.
  3. Find the maximum horizontal distance that the centre of mass of the hemisphere moves before toppling begins, given that \(\mu = \frac{1}{4}\) and the hemisphere starts from rest.
  4. Find the angular acceleration of the hemisphere about its point of contact with the ground at the instant when toppling begins.
[16]
CAIE M2 2014 November Q1
7 marks Standard +0.8
A particle of mass \(m\) moves in a straight line. At time \(t\), its displacement from a fixed point on the line is \(s\) and its velocity is \(v\). The particle experiences a retarding force of magnitude \(mkv^2\), where \(k\) is a positive constant. Find the relationship between \(v\) and \(t\). [7]
CAIE M2 2014 November Q2
6 marks Standard +0.3
\includegraphics{figure_2} A uniform rod \(AB\) of mass \(3m\) and length \(4a\) rests in equilibrium in a vertical plane with the end \(A\) on rough horizontal ground and the end \(B\) against a smooth vertical wall. The rod makes an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac{3}{5}\).
  1. Find the normal reaction at \(A\) and the normal reaction at \(B\). [4]
  2. Find the coefficient of friction between the rod and the ground. [2]
CAIE M2 2014 November Q3
6 marks Moderate -0.3
A particle \(P\) of mass \(0.2\) kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). The particle moves in a horizontal circle of radius \(0.8\) m with the string making a constant angle of \(60°\) with the vertical. Calculate the speed of the particle and the tension in the string. [6]
CAIE M2 2014 November Q4
8 marks Standard +0.3
\includegraphics{figure_4} The diagram shows the cross-section of a uniform solid consisting of a cylinder of radius \(0.4\) m and height \(1.5\) m with a hemisphere of radius \(0.4\) m on top.
  1. Find the distance of the centre of mass above the base of the cylinder. [5]
  2. The solid can just rest in equilibrium on a plane inclined at angle \(\alpha\) to the horizontal. Find \(\alpha\). [3]
CAIE M2 2014 November Q5
7 marks Moderate -0.3
The position vector of a particle at time \(t\) is given by \(\mathbf{r} = t^2\mathbf{i} + (3t - 1)\mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. Find the velocity and acceleration of the particle when \(t = 2\).
  1. Hence find the angle between the velocity and acceleration vectors when \(t = 2\). [3]
  2. Find the value of \(t\) for which the velocity and acceleration vectors are perpendicular. [4]
CAIE M2 2014 November Q6
12 marks Standard +0.3
A particle of mass \(2\) kg moves under the action of a variable force. At time \(t\) seconds the force is \((6t - 3)\mathbf{i} + 4\mathbf{j}\) newtons, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. When \(t = 0\), the particle is at rest at the origin.
  1. Find the velocity of the particle when \(t = 4\). [4]
  2. Find the kinetic energy of the particle when \(t = 4\). [2]
  3. Find the distance of the particle from the origin when \(t = 2\). [6]
CAIE M2 2014 November Q7
8 marks Standard +0.3
\includegraphics{figure_7} A particle of mass \(0.4\) kg is attached to one end of a light inextensible string of length \(2\) m. The other end of the string is attached to a fixed point \(O\). The particle moves in a vertical circle and passes through the lowest point of the circle with speed \(6\) m s\(^{-1}\).
  1. Find the tension in the string when the particle is at the lowest point. [2]
  2. Find the speed of the particle when the string makes an angle of \(60°\) with the downward vertical. [4]
  3. Hence find the tension in the string at this position. [2]
CAIE M2 2014 November Q1
4 marks Standard +0.8
A particle \(P\) is projected with speed \(V\) m s\(^{-1}\) at an angle of \(30°\) above the horizontal from a point \(O\) on horizontal ground. At the instant \(2\) s after projection, \(OP\) makes an angle of \(15°\) above the horizontal. Calculate \(V\). [4]
CAIE M2 2014 November Q2
4 marks Standard +0.8
\includegraphics{figure_2} A uniform solid cone with height \(0.8\) m and semi-vertical angle \(30°\) has weight \(20\) N. The cone rests in equilibrium with a single point \(P\) of its base in contact with a rough horizontal surface, and its vertex \(V\) vertically above \(P\). Equilibrium is maintained by a force of magnitude \(F\) N acting along the axis of symmetry of the cone and applied to \(V\) (see diagram).
  1. Show that the moment of the weight of the cone about \(P\) is \(6\) N m. [2]
  2. Hence find \(F\). [2]
CAIE M2 2014 November Q3
5 marks Standard +0.3
One end of a light elastic string of natural length \(1.6\) m and modulus of elasticity \(28\) N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.35\) kg which hangs in equilibrium vertically below \(O\). The particle \(P\) is projected vertically upwards from the equilibrium position with speed \(1.8\) m s\(^{-1}\). Calculate the speed of \(P\) at the instant the string first becomes slack. [5]
CAIE M2 2014 November Q4
7 marks Standard +0.8
\includegraphics{figure_4} \(ABCDEF\) is the cross-section through the centre of mass of a uniform solid prism. \(ABCF\) is a rectangle in which \(AB = CF = 1.6\) m, and \(BC = AF = 0.4\) m. \(CDE\) is a triangle in which \(CD = 1.8\) m, \(CE = 0.4\) m, and angle \(DCE = 90°\). The prism stands on a rough horizontal surface. A horizontal force of magnitude \(T\) N acts at \(B\) in the direction \(CB\) (see diagram). The prism is in equilibrium.
  1. Show that the distance of the centre of mass of the prism from \(AB\) is \(0.488\) m. [4]
  2. Given that the weight of the prism is \(100\) N, find the greatest and least possible values of \(T\). [3]