Questions — Edexcel M2 (551 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel M2 2010 June Q4
  1. A car of mass 750 kg is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 15 }\). The resistance to motion of the car from non-gravitational forces has constant magnitude \(R\) newtons. The power developed by the car's engine is 15 kW and the car is moving at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Show that \(R = 260\).
    The power developed by the car's engine is now increased to 18 kW . The magnitude of the resistance to motion from non-gravitational forces remains at 260 N . At the instant when the car is moving up the road at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the car's acceleration is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the value of \(a\).
Edexcel M2 2010 June Q5
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]
A ball of mass 0.5 kg is moving with velocity \(( 10 \mathbf { i } + 24 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it is struck by a bat. Immediately after the impact the ball is moving with velocity \(20 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
  1. the magnitude of the impulse of the bat on the ball,
  2. the size of the angle between the vector \(\mathbf { i }\) and the impulse exerted by the bat on the ball,
  3. the kinetic energy lost by the ball in the impact.
Edexcel M2 2010 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0e4552e0-7737-439b-a337-789c83c5258c-10_527_966_310_486} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a uniform rod \(A B\) of mass \(m\) and length \(4 a\). The end \(A\) of the rod is freely hinged to a point on a vertical wall. A particle of mass \(m\) is attached to the rod at \(B\). One end of a light inextensible string is attached to the rod at \(C\), where \(A C = 3 a\). The other end of the string is attached to the wall at \(D\), where \(A D = 2 a\) and \(D\) is vertically above \(A\). The rod rests horizontally in equilibrium in a vertical plane perpendicular to the wall and the tension in the string is \(T\).
  1. Show that \(T = m g \sqrt { } 13\). The particle of mass \(m\) at \(B\) is removed from the rod and replaced by a particle of mass \(M\) which is attached to the rod at \(B\). The string breaks if the tension exceeds \(2 m g \sqrt { } 13\). Given that the string does not break,
  2. show that \(M \leqslant \frac { 5 } { 2 } m\).
Edexcel M2 2010 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0e4552e0-7737-439b-a337-789c83c5258c-12_631_1041_242_447} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A ball is projected with speed \(40 \mathrm {~ms} ^ { - 1 }\) from a point \(P\) on a cliff above horizontal ground. The point \(O\) on the ground is vertically below \(P\) and \(O P\) is 36 m . The ball is projected at an angle \(\theta ^ { \circ }\) to the horizontal. The point \(Q\) is the highest point of the path of the ball and is 12 m above the level of \(P\). The ball moves freely under gravity and hits the ground at the point \(R\), as shown in Figure 3. Find
  1. the value of \(\theta\),
  2. the distance \(O R\),
  3. the speed of the ball as it hits the ground at \(R\).
Edexcel M2 2010 June Q8
  1. A small ball \(A\) of mass \(3 m\) is moving with speed \(u\) in a straight line on a smooth horizontal table. The ball collides directly with another small ball \(B\) of mass \(m\) moving with speed \(u\) towards \(A\) along the same straight line. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\). The balls have the same radius and can be modelled as particles.
    1. Find
      1. the speed of \(A\) immediately after the collision,
      2. the speed of \(B\) immediately after the collision.
    After the collision \(B\) hits a smooth vertical wall which is perpendicular to the direction of motion of \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 5 }\).
  2. Find the speed of \(B\) immediately after hitting the wall. The first collision between \(A\) and \(B\) occurred at a distance 4a from the wall. The balls collide again \(T\) seconds after the first collision.
  3. Show that \(T = \frac { 112 a } { 15 u }\).
Edexcel M2 2011 June Q1
  1. A car of mass 1000 kg moves with constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 30 }\). The engine of the car is working at a rate of 12 kW . The resistance to motion from non-gravitational forces has magnitude 500 N . Find the value of \(V\).
  2. A particle \(P\) of mass \(m\) is moving in a straight line on a smooth horizontal surface with speed \(4 u\). The particle \(P\) collides directly with a particle \(Q\) of mass \(3 m\) which is at rest on the surface. The coefficient of restitution between \(P\) and \(Q\) is \(e\). The direction of motion of \(P\) is reversed by the collision.
Show that \(e > \frac { 1 } { 3 }\).
Edexcel M2 2011 June Q3
3. A ball of mass 0.5 kg is moving with velocity \(12 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is struck by a bat. The impulse received by the ball is \(( - 4 \mathbf { i } + 7 \mathbf { j } ) \mathrm { N } \mathrm { s }\). By modelling the ball as a particle, find
  1. the speed of the ball immediately after the impact,
  2. the angle, in degrees, between the velocity of the ball immediately after the impact and the vector \(\mathbf { i }\),
  3. the kinetic energy gained by the ball as a result of the impact.
Edexcel M2 2011 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e8329378-c976-4068-95ff-e2d254546d6d-05_394_846_239_555} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a uniform lamina \(A B C D E\) such that \(A B D E\) is a rectangle, \(B C = C D\), \(A B = 4 a\) and \(A E = 2 a\). The point \(F\) is the midpoint of \(B D\) and \(F C = a\).
  1. Find, in terms of \(a\), the distance of the centre of mass of the lamina from \(A E\). The lamina is freely suspended from \(A\) and hangs in equilibrium.
  2. Find the angle between \(A B\) and the downward vertical.
Edexcel M2 2011 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e8329378-c976-4068-95ff-e2d254546d6d-07_366_771_267_589} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of mass 0.5 kg is projected from a point \(A\) up a line of greatest slope \(A B\) of a fixed plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal and \(A B = 2 \mathrm {~m}\) with \(B\) above \(A\), as shown in Figure 2. The particle \(P\) passes through \(B\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The plane is smooth from \(A\) to \(B\).
  1. Find the speed of projection. The particle \(P\) comes to instantaneous rest at the point \(C\) on the plane, where \(C\) is above \(B\) and \(B C = 1.5 \mathrm {~m}\). From \(B\) to \(C\) the plane is rough and the coefficient of friction between \(P\) and the plane is \(\mu\). By using the work-energy principle,
  2. find the value of \(\mu\).
Edexcel M2 2011 June Q6
  1. A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\) at time \(t\) seconds is \(( t - 4 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the positive \(x\)-direction. The velocity of \(P\) at time \(t\) seconds is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 0 , v = 6\).
Find
  1. \(v\) in terms of \(t\),
  2. the values of \(t\) when \(P\) is instantaneously at rest,
  3. the distance between the two points at which \(P\) is instantaneously at rest.
Edexcel M2 2011 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e8329378-c976-4068-95ff-e2d254546d6d-11_609_773_244_589} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform rod \(A B\), of mass \(3 m\) and length \(4 a\), is held in a horizontal position with the end \(A\) against a rough vertical wall. One end of a light inextensible string \(B D\) is attached to the rod at \(B\) and the other end of the string is attached to the wall at the point \(D\) vertically above \(A\), where \(A D = 3 a\). A particle of mass \(3 m\) is attached to the rod at \(C\), where \(A C = x\). The rod is in equilibrium in a vertical plane perpendicular to the wall as shown in Figure 3. The tension in the string is \(\frac { 25 } { 4 } m g\). Show that
  1. \(x = 3 a\),
  2. the horizontal component of the force exerted by the wall on the rod has magnitude 5 mg . The coefficient of friction between the wall and the rod is \(\mu\). Given that the rod is about to slip,
  3. find the value of \(\mu\).
Edexcel M2 2011 June Q8
  1. A particle is projected from a point \(O\) with speed \(u\) at an angle of elevation \(\alpha\) above the horizontal and moves freely under gravity. When the particle has moved a horizontal distance \(x\), its height above \(O\) is \(y\).
    1. Show that
    $$y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } \cos ^ { 2 } \alpha }$$ A girl throws a ball from a point \(A\) at the top of a cliff. The point \(A\) is 8 m above a horizontal beach. The ball is projected with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(45 ^ { \circ }\). By modelling the ball as a particle moving freely under gravity,
  2. find the horizontal distance of the ball from \(A\) when the ball is 1 m above the beach. A boy is standing on the beach at the point \(B\) vertically below \(A\). He starts to run in a straight line with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), leaving \(B 0.4\) seconds after the ball is thrown. He catches the ball when it is 1 m above the beach.
  3. Find the value of \(v\).
Edexcel M2 2012 June Q1
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]
A particle \(P\) moves in such a way that its velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) at time \(t\) seconds is given by $$\mathbf { v } = \left( 3 t ^ { 2 } - 1 \right) \mathbf { i } + \left( 4 t - t ^ { 2 } \right) \mathbf { j }$$
  1. Find the magnitude of the acceleration of \(P\) when \(t = 1\) Given that, when \(t = 0\), the position vector of \(P\) is i metres,
  2. find the position vector of \(P\) when \(t = 3\)
Edexcel M2 2012 June Q2
2. A particle \(P\) of mass \(3 m\) is moving with speed \(2 u\) in a straight line on a smooth horizontal plane. The particle \(P\) collides directly with a particle \(Q\) of mass \(4 m\) moving on the plane with speed \(u\) in the opposite direction to \(P\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  1. Find the speed of \(Q\) immediately after the collision. Given that the direction of motion of \(P\) is reversed by the collision,
  2. find the range of possible values of \(e\).
Edexcel M2 2012 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12cd7355-f632-4a84-825f-a269851c6ec4-04_374_798_255_559} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform rod \(A B\), of mass 5 kg and length 4 m , has its end \(A\) smoothly hinged at a fixed point. The rod is held in equilibrium at an angle of \(25 ^ { \circ }\) above the horizontal by a force of magnitude \(F\) newtons applied to its end \(B\). The force acts in the vertical plane containing the rod and in a direction which makes an angle of \(40 ^ { \circ }\) with the rod, as shown in Figure 1.
  1. Find the value of \(F\).
  2. Find the magnitude and direction of the vertical component of the force acting on the rod at \(A\).
Edexcel M2 2012 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12cd7355-f632-4a84-825f-a269851c6ec4-06_796_789_276_566} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform circular disc has centre \(O\) and radius 4a. The lines \(P Q\) and \(S T\) are perpendicular diameters of the disc. A circular hole of radius \(2 a\) is made in the disc, with the centre of the hole at the point \(R\) on \(O P\) where \(O R = 2 a\), to form the lamina \(L\), shown shaded in Figure 2.
  1. Show that the distance of the centre of mass of \(L\) from \(P\) is \(\frac { 14 a } { 3 }\). The mass of \(L\) is \(m\) and a particle of mass \(k m\) is now fixed to \(L\) at the point \(P\). The system is now suspended from the point \(S\) and hangs freely in equilibrium. The diameter \(S T\) makes an angle \(\alpha\) with the downward vertical through \(S\), where \(\tan \alpha = \frac { 5 } { 6 }\).
  2. Find the value of \(k\).
Edexcel M2 2012 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12cd7355-f632-4a84-825f-a269851c6ec4-08_330_570_242_657} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A small ball \(B\) of mass 0.25 kg is moving in a straight line with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane when it is given an impulse. The impulse has magnitude 12.5 N s and is applied in a horizontal direction making an angle of \(\left( 90 ^ { \circ } + \alpha \right)\), where \(\tan \alpha = \frac { 3 } { 4 }\), with the initial direction of motion of the ball, as shown in Figure 3.
  1. Find the speed of \(B\) immediately after the impulse is applied.
  2. Find the direction of motion of \(B\) immediately after the impulse is applied.
Edexcel M2 2012 June Q6
6. A car of mass 1200 kg pulls a trailer of mass 400 kg up a straight road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 14 }\). The trailer is attached to the car by a light inextensible towbar which is parallel to the road. The car's engine works at a constant rate of 60 kW . The non-gravitational resistances to motion are constant and of magnitude 1000 N on the car and 200 N on the trailer. At a given instant, the car is moving at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the acceleration of the car at this instant,
  2. the tension in the towbar at this instant. The towbar breaks when the car is moving at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find, using the work-energy principle, the further distance that the trailer travels before coming instantaneously to rest.
Edexcel M2 2012 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{12cd7355-f632-4a84-825f-a269851c6ec4-12_602_1175_237_386} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A small stone is projected from a point \(O\) at the top of a vertical cliff \(O A\). The point \(O\) is 52.5 m above the sea. The stone rises to a maximum height of 10 m above the level of \(O\) before hitting the sea at the point \(B\), where \(A B = 50 \mathrm {~m}\), as shown in Figure 4. The stone is modelled as a particle moving freely under gravity.
  1. Show that the vertical component of the velocity of projection of the stone is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the speed of projection.
  3. Find the time after projection when the stone is moving parallel to \(O B\).
Edexcel M2 2013 June Q1
  1. A caravan of mass 600 kg is towed by a car of mass 900 kg along a straight horizontal road. The towbar joining the car to the caravan is modelled as a light rod parallel to the road. The total resistance to motion of the car is modelled as having magnitude 300 N . The total resistance to motion of the caravan is modelled as having magnitude 150 N . At a given instant the car and the caravan are moving with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and acceleration \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    1. Find the power being developed by the car's engine at this instant.
    2. Find the tension in the towbar at this instant.
    3. A ball of mass 0.2 kg is projected vertically upwards from a point \(O\) with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The non-gravitational resistance acting on the ball is modelled as a force of constant magnitude 1.24 N and the ball is modelled as a particle. Find, using the work-energy principle, the speed of the ball when it first reaches the point which is 8 m vertically above \(O\).
      (6)
    4. A particle \(P\) moves along a straight line in such a way that at time \(t\) seconds its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is given by
    $$v = \frac { 1 } { 2 } t ^ { 2 } - 3 t + 4$$ Find
  2. the times when \(P\) is at rest,
  3. the total distance travelled by \(P\) between \(t = 0\) and \(t = 4\).
Edexcel M2 2013 June Q4
  1. A rough circular cylinder of radius \(4 a\) is fixed to a rough horizontal plane with its axis horizontal. A uniform rod \(A B\), of weight \(W\) and length \(6 a \sqrt { 3 }\), rests with its lower end \(A\) on the plane and a point \(C\) of the rod against the cylinder. The vertical plane through the rod is perpendicular to the axis of the cylinder. The rod is inclined at \(60 ^ { \circ }\) to the horizontal, as shown in Figure 1.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2226b09b-a19d-4253-baaa-3c3d602d0d6d-06_389_862_482_550} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
  1. Show that \(A C = 4 a \sqrt { } 3\) The coefficient of friction between the rod and the cylinder is \(\frac { \sqrt { } 3 } { 3 }\) and the coefficient of friction between the rod and the plane is \(\mu\). Given that friction is limiting at both \(A\) and \(C\),
  2. find the value of \(\mu\).
Edexcel M2 2013 June Q5
5. Two particles \(P\) and \(Q\), of masses \(2 m\) and \(m\) respectively, are on a smooth horizontal table. Particle \(Q\) is at rest and particle \(P\) collides directly with it when moving with speed \(u\). After the collision the total kinetic energy of the two particles is \(\frac { 3 } { 4 } m u ^ { 2 }\). Find
  1. the speed of \(Q\) immediately after the collision,
  2. the coefficient of restitution between the particles.
Edexcel M2 2013 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2226b09b-a19d-4253-baaa-3c3d602d0d6d-10_442_871_264_525} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform triangular lamina \(A B C\) of mass \(M\) is such that \(A B = A C , B C = 2 a\) and the distance of \(A\) from \(B C\) is \(h\). A line, parallel to \(B C\) and at a distance \(\frac { 2 h } { 3 }\) from \(A\), cuts \(A B\) at \(D\) and cuts \(A C\) at \(E\), as shown in Figure 2.
It is given that the mass of the trapezium \(B C E D\) is \(\frac { 5 M } { 9 }\).
  1. Show that the centre of mass of the trapezium \(B C E D\) is \(\frac { 7 h } { 45 }\) from \(B C\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2226b09b-a19d-4253-baaa-3c3d602d0d6d-10_357_679_1354_630} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The portion \(A D E\) of the lamina is folded through \(180 ^ { \circ }\) about \(D E\) to form the folded lamina shown in Figure 3.
  2. Find the distance of the centre of mass of the folded lamina from \(B C\). The folded lamina is freely suspended from \(D\) and hangs in equilibrium. The angle between \(D E\) and the downward vertical is \(\alpha\).
  3. Find \(\tan \alpha\) in terms of \(a\) and \(h\).
Edexcel M2 2013 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2226b09b-a19d-4253-baaa-3c3d602d0d6d-13_520_1027_296_447} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A small ball is projected from a fixed point \(O\) so as to hit a target \(T\) which is at a horizontal distance \(9 a\) from \(O\) and at a height \(6 a\) above the level of \(O\). The ball is projected with speed \(\sqrt { } ( 27 a g )\) at an angle \(\theta\) to the horizontal, as shown in Figure 4. The ball is modelled as a particle moving freely under gravity.
  1. Show that \(\tan ^ { 2 } \theta - 6 \tan \theta + 5 = 0\) The two possible angles of projection are \(\theta _ { 1 }\) and \(\theta _ { 2 }\), where \(\theta _ { 1 } > \theta _ { 2 }\).
  2. Find \(\tan \theta _ { 1 }\) and \(\tan \theta _ { 2 }\). The particle is projected at the larger angle \(\theta _ { 1 }\).
  3. Show that the time of flight from \(O\) to \(T\) is \(\sqrt { } \left( \frac { 78 a } { g } \right)\).
  4. Find the speed of the particle immediately before it hits \(T\).
Edexcel M2 2013 June Q1
  1. Three particles of masses \(2 \mathrm {~kg} , 3 \mathrm {~kg}\) and \(m \mathrm {~kg}\) are positioned at the points with coordinates \(( a , 3 ) , ( 3 , - 1 )\) and \(( - 2,4 )\) respectively. Given that the centre of mass of the particles is at the point with coordinates \(( 0,2 )\), find
    1. the value of \(m\),
    2. the value of \(a\).
    3. A car has mass 1200 kg . The maximum power of the car's engine is 32 kW . The resistance to motion due to non-gravitational forces is modelled as a force of constant magnitude 800 N . When the car is travelling on a horizontal road at constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the engine of the car is working at maximum power.
    4. Find the value of \(V\).
    The car now travels downhill on a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 40 }\). The resistance to motion due to non-gravitational forces is still modelled as a force of constant magnitude 800 N . Given that the engine of the car is again working at maximum power,
  2. find the acceleration of the car when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).