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Edexcel M2 2014 June Q7
14 marks Standard +0.3
7. A particle \(P\) of mass \(2 m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal table. A second particle \(Q\) of mass \(3 m\) is moving in the opposite direction to \(P\) along the same straight line with speed \(u\). The particle \(P\) collides directly with \(Q\). The direction of motion of \(P\) is reversed by the collision. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  1. Show that the speed of \(Q\) immediately after the collision is \(\frac { u } { 5 } ( 8 e + 3 )\)
  2. Find the range of possible values of \(e\). The total kinetic energy of the particles before the collision is \(T\). The total kinetic energy of the particles after the collision is \(k T\). Given that \(e = \frac { 1 } { 2 }\)
  3. find the value of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{82cadc37-4cb0-455e-9531-e09ec0c19533-14_104_61_2407_1836}
Edexcel M2 2014 June Q1
6 marks Moderate -0.8
  1. Three particles of mass \(3 m , 2 m\) and \(k m\) are placed at the points whose coordinates are \(( 1,5 ) , ( 6,4 )\) and \(( a , 1 )\) respectively. The centre of mass of the three particles is at the point with coordinates \(( 3,3 )\).
Find
  1. the value of \(k\),
  2. the value of \(a\).
Edexcel M2 2014 June Q2
9 marks Moderate -0.3
2. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) is moving on a horizontal plane with acceleration \(\left[ \left( 3 t ^ { 2 } - 4 t \right) \mathbf { i } + ( 6 t - 5 ) \mathbf { j } \right] \mathrm { m } \mathrm { s } ^ { - 2 }\). When \(t = 3\) the velocity of \(P\) is \(( 11 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
  1. the velocity of \(P\) at time \(t\) seconds,
  2. the speed of \(P\) when it is moving parallel to the vector \(\mathbf { i }\).
Edexcel M2 2014 June Q3
9 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{47420c50-c232-41e9-8c4d-a890d86ea933-04_814_1127_219_411} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform lamina \(A B C D E F\), shown shaded in Figure 1, is symmetrical about the line through \(B\) and \(E\). It is formed by removing the isosceles triangle \(F E D\), of height \(6 a\) and base \(8 a\), from the isosceles triangle \(A B C\) of height \(9 a\) and base \(12 a\).
  1. Find, in terms of \(a\), the distance of the centre of mass of the lamina from \(A C\). The lamina is freely suspended from \(A\) and hangs in equilibrium.
  2. Find, to the nearest degree, the size of the angle between \(A B\) and the downward vertical.
Edexcel M2 2014 June Q4
9 marks Standard +0.3
  1. A truck of mass 1800 kg is towing a trailer of mass 800 kg up a straight road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 20 }\). The truck is connected to the trailer by a light inextensible rope which is parallel to the direction of motion of the truck. The resistances to motion of the truck and the trailer from non-gravitational forces are modelled as constant forces of magnitudes 300 N and 200 N respectively. The truck is moving at constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine of the truck is working at a rate of 40 kW .
    1. Find the value of \(v\).
    As the truck is moving up the road the rope breaks.
  2. Find the acceleration of the truck immediately after the rope breaks.
Edexcel M2 2014 June Q5
9 marks Standard +0.3
5. A particle of mass \(m \mathrm {~kg}\) lies on a smooth horizontal surface. Initially the particle is at rest at a point \(O\) midway between a pair of fixed parallel vertical walls. The walls are 2 m apart. At time \(t = 0\) the particle is projected from \(O\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the walls. The coefficient of restitution between the particle and each wall is \(\frac { 2 } { 3 }\). The magnitude of the impulse on the particle due to the first impact with a wall is \(\lambda m u \mathrm {~N} \mathrm {~s}\).
  1. Find the value of \(\lambda\). The particle returns to \(O\), having bounced off each wall once, at time \(t = 3\) seconds.
  2. Find the value of \(u\).
Edexcel M2 2014 June Q6
12 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{47420c50-c232-41e9-8c4d-a890d86ea933-10_645_1196_125_351} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A small ball is projected with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) on horizontal ground. The angle of projection is \(\alpha\) above the horizontal. A horizontal platform is at height \(h\) metres above the ground. The ball moves freely under gravity until it hits the platform at the point B, as shown in Figure 2. The speed of the ball immediately before it hits the platform at \(B\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the value of \(h\). Given that \(\sin \alpha = 0.85\),
  2. find the horizontal distance from \(A\) to \(B\).
Edexcel M2 2014 June Q7
12 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{47420c50-c232-41e9-8c4d-a890d86ea933-12_837_565_226_694} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform rod \(A B\) of weight \(W\) has its end \(A\) freely hinged to a point on a fixed vertical wall. The rod is held in equilibrium, at angle \(\theta\) to the horizontal, by a force of magnitude \(P\). The force acts perpendicular to the rod at \(B\) and in the same vertical plane as the rod, as shown in Figure 3. The rod is in a vertical plane perpendicular to the wall. The magnitude of the vertical component of the force exerted on the rod by the wall at \(A\) is \(Y\).
  1. Show that \(Y = \frac { W } { 2 } \left( 2 - \cos ^ { 2 } \theta \right)\). Given that \(\theta = 45 ^ { \circ }\)
  2. find the magnitude of the force exerted on the rod by the wall at \(A\), giving your answer in terms of \(W\).
Edexcel M2 2014 June Q8
9 marks Standard +0.3
8. The points \(A\) and \(B\) are 10 m apart on a line of greatest slope of a fixed rough inclined plane, with \(A\) above \(B\). The plane is inclined at \(25 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 5 kg is released from rest at \(A\) and slides down the slope. As \(P\) passes \(B\), it is moving with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find, using the work-energy principle, the work done against friction as \(P\) moves from \(A\) to \(B\).
  2. Find the coefficient of friction between the particle and the plane.
Edexcel M2 2015 June Q1
5 marks Standard +0.3
  1. A van of mass 900 kg is moving down a straight road that is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 30 }\). The resistance to motion of the van has constant magnitude 570 N . The engine of the van is working at a constant rate of 12.5 kW .
At the instant when the van is moving down the road at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the van is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the value of \(a\).
Edexcel M2 2015 June Q2
8 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1822f86a-9089-44af-ab36-6006adfeb5b9-03_709_620_116_667} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform lamina \(O A B C D\), shown in Figure 1, is formed by removing the triangle \(O A D\) from the square \(A B C D\) with centre \(O\). The square has sides of length \(2 a\).
  1. Show that the centre of mass of \(O A B C D\) is \(\frac { 2 } { 9 } a\) from \(O\). The mass of the lamina is \(M\). A particle of mass \(k M\) is attached to the lamina at \(D\) to form the system \(S\). The system \(S\) is freely suspended from \(A\) and hangs in equilibrium with \(A O\) vertical.
  2. Find the value of \(k\).
Edexcel M2 2015 June Q3
8 marks Moderate -0.3
  1. A particle \(P\) of mass 0.75 kg is moving with velocity \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse \(( 6 \mathbf { i } + 6 \mathbf { j } ) \mathrm { Ns }\). The angle between the velocity of \(P\) before the impulse and the velocity of \(P\) after the impulse is \(\theta ^ { \circ }\).
Find
  1. the value of \(\theta\),
  2. the kinetic energy gained by \(P\) as a result of the impulse.
Edexcel M2 2015 June Q4
9 marks Standard +0.3
  1. A ladder \(A B\), of weight \(W\) and length \(2 l\), has one end \(A\) resting on rough horizontal ground. The other end \(B\) rests against a rough vertical wall. The coefficient of friction between the ladder and the wall is \(\frac { 1 } { 3 }\). The coefficient of friction between the ladder and the ground is \(\mu\). Friction is limiting at both \(A\) and \(B\). The ladder is at an angle \(\theta\) to the ground, where \(\tan \theta = \frac { 5 } { 3 }\). The ladder is modelled as a uniform rod which lies in a vertical plane perpendicular to the wall.
Find the value of \(\mu\).
Edexcel M2 2015 June Q5
9 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1822f86a-9089-44af-ab36-6006adfeb5b9-09_538_1147_114_402} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of mass 10 kg is projected from a point \(A\) up a line of greatest slope \(A B\) of a fixed rough plane. The plane is inclined at angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) and \(A B = 6.5 \mathrm {~m}\), as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\mu\). The work done against friction as \(P\) moves from \(A\) to \(B\) is 245 J .
  1. Find the value of \(\mu\). The particle is projected from \(A\) with speed \(11.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By using the work-energy principle,
  2. find the speed of the particle as it passes through \(B\).
Edexcel M2 2015 June Q6
11 marks Standard +0.3
  1. A particle \(P\) moves on the positive \(x\)-axis. The velocity of \(P\) at time \(t\) seconds is \(\left( 2 t ^ { 2 } - 9 t + 4 \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , P\) is 15 m from the origin \(O\).
Find
  1. the values of \(t\) when \(P\) is instantaneously at rest,
  2. the acceleration of \(P\) when \(t = 5\)
  3. the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 5\)
Edexcel M2 2015 June Q7
12 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1822f86a-9089-44af-ab36-6006adfeb5b9-13_506_1379_287_280} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} At time \(t = 0\), a particle is projected from a fixed point \(O\) on horizontal ground with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) to the horizontal. The particle moves freely under gravity and passes through the point \(A\) when \(t = 4 \mathrm {~s}\). As it passes through \(A\), the particle is moving upwards at \(20 ^ { \circ }\) to the horizontal with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Figure 3.
  1. Find the value of \(u\) and the value of \(\theta\). At the point \(B\) on its path the particle is moving downwards at \(20 ^ { \circ }\) to the horizontal with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the time taken for the particle to move from \(A\) to \(B\). The particle reaches the ground at the point \(C\).
  3. Find the distance \(O C\).
Edexcel M2 2015 June Q8
13 marks Standard +0.8
  1. Three identical particles \(P , Q\) and \(R\), each of mass \(m\), lie in a straight line on a smooth horizontal plane with \(Q\) between \(P\) and \(R\). Particles \(P\) and \(Q\) are projected directly towards each other with speeds \(4 u\) and \(2 u\) respectively, and at the same time particle \(R\) is projected along the line away from \(Q\) with speed \(3 u\). The coefficient of restitution between each pair of particles is \(e\). After the collision between \(P\) and \(Q\) there is a collision between \(Q\) and \(R\).
    1. Show that \(e > \frac { 2 } { 3 }\)
    It is given that \(e = \frac { 3 } { 4 }\)
  2. Show that there will not be a further collision between \(P\) and \(Q\).
Edexcel M2 2017 June Q1
6 marks Moderate -0.8
  1. A particle \(P\) of mass 0.5 kg is moving with velocity \(4 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse I Ns. Immediately after \(P\) receives the impulse, the velocity of \(P\) is \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
Find
  1. the magnitude of \(\mathbf { I }\),
  2. the angle between \(\mathbf { I }\) and \(\mathbf { j }\).
Edexcel M2 2017 June Q2
12 marks Standard +0.3
  1. A truck of mass 900 kg is towing a trailer of mass 150 kg up an inclined straight road with constant speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The trailer is attached to the truck by a light inextensible towbar which is parallel to the road. The road is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 9 }\). The resistance to motion of the truck from non-gravitational forces has constant magnitude 200 N and the resistance to motion of the trailer from non-gravitational forces has constant magnitude 50 N .
    1. Find the rate at which the engine of the truck is working.
    When the truck and trailer are moving up the road at \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the towbar breaks, and the trailer is no longer attached to the truck. The rate at which the engine of the truck is working is unchanged. The resistance to motion of the truck from non-gravitational forces and the resistance to motion of the trailer from non-gravitational forces are still forces of constant magnitudes 200 N and 50 N respectively.
  2. Find the acceleration of the truck at the instant after the towbar breaks.
  3. Use the work-energy principle to find out how much further up the road the trailer travels before coming to instantaneous rest.
Edexcel M2 2017 June Q3
9 marks Standard +0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{64b0abc9-4021-44e6-8bf7-1a5862617085-08_744_369_246_447} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{64b0abc9-4021-44e6-8bf7-1a5862617085-08_538_593_452_1023} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The uniform rectangular lamina \(A B D E\), shown in Figure 1, has side \(A B\) of length \(2 a\) and side \(B D\) of length \(6 a\). The point \(C\) divides \(B D\) in the ratio 1:2 and the point \(F\) divides \(E A\) in the ratio \(1 : 2\). The rectangular lamina is folded along \(F C\) to produce the folded lamina \(L\), shown in Figure 2.
  1. Show that the centre of mass of \(L\) is \(\frac { 16 } { 9 } a\) from \(E F\). The folded lamina, \(L\), is freely suspended from \(C\) and hangs in equilibrium.
  2. Find the size of the angle between \(C F\) and the downward vertical.
Edexcel M2 2017 June Q4
12 marks Standard +0.3
  1. At time \(t = 0\) a particle \(P\) leaves the origin \(O\) and moves along the \(x\)-axis. At time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction, where
$$v = 3 t ^ { 2 } - 16 t + 21$$ The particle is instantaneously at rest when \(t = t _ { 1 }\) and when \(t = t _ { 2 } \left( t _ { 1 } < t _ { 2 } \right)\).
  1. Find the value of \(t _ { 1 }\) and the value of \(t _ { 2 }\).
  2. Find the magnitude of the acceleration of \(P\) at the instant when \(t = t _ { 1 }\).
  3. Find the distance travelled by \(P\) in the interval \(t _ { 1 } \leqslant t \leqslant t _ { 2 }\).
  4. Show that \(P\) does not return to \(O\).
Edexcel M2 2017 June Q5
11 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{64b0abc9-4021-44e6-8bf7-1a5862617085-16_606_1287_260_331} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform rod \(A B\), of mass 5 kg and length 8 m , has its end \(B\) resting on rough horizontal ground. The rod is held in limiting equilibrium at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\), by a rope attached to the rod at \(C\). The distance \(A C = 1 \mathrm {~m}\). The rope is in the same vertical plane as the rod. The angle between the rope and the rod is \(\beta\) and the tension in the rope is \(T\) newtons, as shown in Figure 3. The coefficient of friction between the rod and the ground is \(\frac { 2 } { 3 }\). The vertical component of the force exerted on the rod at \(B\) by the ground is \(R\) newtons.
  1. Find the value of \(R\).
  2. Find the size of angle \(\beta\).
Edexcel M2 2017 June Q6
11 marks Challenging +1.2
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{64b0abc9-4021-44e6-8bf7-1a5862617085-20_248_1063_260_443} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The points \(A\) and \(B\) lie 40 m apart on horizontal ground. At time \(t = 0\) the particles \(P\) and \(Q\) are projected in the vertical plane containing \(A B\) and move freely under gravity. Particle \(P\) is projected from \(A\) with speed \(30 \mathrm {~ms} ^ { - 1 }\) at \(60 ^ { \circ }\) to \(A B\) and particle \(Q\) is projected from \(B\) with speed \(q \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at angle \(\theta\) to \(B A\), as shown in Figure 4. At \(t = 2\) seconds, \(P\) and \(Q\) collide.
  1. Find
    1. the size of angle \(\theta\),
    2. the value of \(q\).
  2. Find the speed of \(P\) at the instant before it collides with \(Q\).
Edexcel M2 2017 June Q7
14 marks Standard +0.3
  1. Two particles \(A\) and \(B\), of masses \(3 m\) and \(4 m\) respectively, lie at rest on a smooth horizontal surface. Particle \(B\) lies between \(A\) and a smooth vertical wall which is perpendicular to the line joining \(A\) and \(B\). Particle \(B\) is projected with speed \(5 u\) in a direction perpendicular to the wall and collides with the wall. The coefficient of restitution between \(B\) and the wall is \(\frac { 3 } { 5 }\).
    1. Find the magnitude of the impulse received by \(B\) in the collision with the wall.
    After the collision with the wall, \(B\) rebounds from the wall and collides directly with \(A\). The coefficient of restitution between \(A\) and \(B\) is \(e\).
  2. Show that, immediately after they collide, \(A\) and \(B\) are both moving in the same direction. The kinetic energy of \(B\) immediately after it collides with \(A\) is one quarter of the kinetic energy of \(B\) immediately before it collides with \(A\).
  3. Find the value of \(e\).
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Edexcel M2 2018 June Q1
8 marks Moderate -0.3
  1. A truck of mass 750 kg is moving with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 49 }\). The resistance to motion of the truck is modelled as a constant force of magnitude 1200 N . The engine of the truck is working at a constant rate of 9 kW .
    1. Find the value of \(v\).
    On another occasion the truck is moving up the same straight road. The resistance to motion of the truck from non-gravitational forces is modelled as a constant force of magnitude 1200 N . The engine of the truck is working at a constant rate of 9 kW .
  2. Find the acceleration of the truck at the instant when it is moving with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).