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Edexcel M2 2015 January Q4
9 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{25c503ad-94c7-4137-83b5-c3e0aea62f0c-07_887_707_269_621} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform plane lamina \(A B C D E F\) shown in Figure 1 is made from two identical rhombuses. Each rhombus has sides of length \(a\) and angle \(B A D =\) angle \(D A F = \theta\). The centre of mass of the lamina is \(0.9 a\) from \(A\).
  1. Show that \(\cos \theta = 0.8\) The weight of the lamina is \(W\). A particle of weight \(k W\) is fixed to the lamina at the point \(A\). The lamina is freely suspended from \(B\) and hangs in equilibrium with \(D A\) horizontal.
  2. Find the value of \(k\).
Edexcel M2 2015 January Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{25c503ad-94c7-4137-83b5-c3e0aea62f0c-09_636_1143_251_468} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is freely hinged to a fixed point \(A\). A particle of mass \(k m\) is fixed to the rod at \(B\). The rod is held in equilibrium, at an angle \(\theta\) to the horizontal, by a force of magnitude \(F\) acting at the point \(C\) on the rod, where \(A C = \frac { 5 } { 4 } a\), as shown in Figure 2. The line of action of the force at \(C\) is at right angles to \(A B\) and in the vertical plane containing \(A B\). Given that \(\tan \theta = \frac { 3 } { 4 }\)
  1. show that \(F = \frac { 16 } { 25 } m g ( 1 + 2 k )\),
  2. find, in terms of \(m , g\) and \(k\),
    1. the horizontal component of the force exerted by the hinge on the rod at \(A\),
    2. the vertical component of the force exerted by the hinge on the rod at \(A\). Given also that the force acting on the rod at \(A\) acts at \(45 ^ { \circ }\) above the horizontal,
  3. find the value of \(k\).
Edexcel M2 2015 January Q6
12 marks Moderate -0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{25c503ad-94c7-4137-83b5-c3e0aea62f0c-11_452_865_264_495} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A small ball \(P\) is projected with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A 10 \mathrm {~m}\) above horizontal ground. The angle of projection is \(55 ^ { \circ }\) above the horizontal. The ball moves freely under gravity and hits the ground at the point \(B\), as shown in Figure 3. Find
  1. the speed of \(P\) as it hits the ground at \(B\),
  2. the direction of motion of \(P\) as it hits the ground at \(B\),
  3. the time taken for \(P\) to move from \(A\) to \(B\).
Edexcel M2 2015 January Q7
14 marks Standard +0.8
7. Three particles \(P , Q\) and \(R\) lie at rest in a straight line on a smooth horizontal surface with \(Q\) between \(P\) and \(R\). Particle \(P\) has mass \(m\), particle \(Q\) has mass \(2 m\) and particle \(R\) has mass \(3 m\). The coefficient of restitution between each pair of particles is \(e\). Particle \(P\) is projected towards \(Q\) with speed \(3 u\) and collides directly with \(Q\).
  1. Find, in terms of \(u\) and \(e\),
    1. the speed of \(Q\) immediately after the collision,
    2. the speed of \(P\) immediately after the collision.
  2. Find the range of values of \(e\) for which the direction of motion of \(P\) is reversed as a result of the collision with \(Q\). Immediately after the collision between \(P\) and \(Q\), particle \(R\) is projected towards \(Q\) with speed \(u\) so that \(R\) and \(Q\) collide directly. Given that \(e = \frac { 2 } { 3 }\)
  3. show that there will be a second collision between \(P\) and \(Q\).
Edexcel M2 2017 January Q1
10 marks Standard +0.3
  1. A car of mass 1200 kg moves up a straight road. The road is inclined to the horizontal at an angle \(\alpha\) where \(\sin \alpha = \frac { 1 } { 15 }\). The car is moving up the road with constant speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine of the car is working at a constant rate of 11760 watts. The non-gravitational resistance to motion has a constant magnitude of \(R\) newtons.
    1. Find the value of \(R\).
    The rate of working of the car is now increased to 50 kW . At the instant when the speed of the car is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the magnitude of the non-gravitational resistance to the motion of the car is 700 N and the acceleration of the car is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the value of \(V\).
Edexcel M2 2017 January Q2
9 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{36cced0d-f982-4534-a3fe-13c32fb37f5b-04_538_625_251_657} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform lamina is in the shape of a trapezium \(A B C D\) with \(A B = a , D A = D C = 2 a\) and angle \(B A D =\) angle \(A D C = 90 ^ { \circ }\), as shown in Figure 1. The centre of mass of the lamina is at the point \(G\).
    1. Show that the distance of \(G\) from \(A B\) is \(\frac { 10 a } { 9 }\).
    2. Find the distance of \(G\) from \(A D\). The mass of the lamina is \(3 M\). A particle of mass \(k M\) is now attached to the lamina at \(B\). The lamina is freely suspended from the midpoint of \(A D\) and hangs in equilibrium with \(A D\) horizontal.
  2. Find the value of \(k\).
Edexcel M2 2017 January Q3
6 marks Standard +0.3
  1. A particle \(P\) moves along a straight line. At time \(t = 0 , P\) passes the point \(A\) on the line and at time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) where
$$v = ( 2 t - 3 ) ( t - 2 )$$ At \(t = 3 , P\) reaches the point \(B\). Find the total distance moved by \(P\) as it travels from \(A\) to \(B\).
(6)
Edexcel M2 2017 January Q4
9 marks Standard +0.3
4. A particle \(P\) of mass 0.2 kg is moving with velocity \(( 20 \mathbf { i } - 16 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse ( \(- 6 \mathbf { i } + 8 \mathbf { j }\) ) N s.
  1. Find the speed of \(P\) immediately after it receives the impulse.
    (5)
  2. Find the size of the angle between the direction of motion of \(P\) before the impulse is received and the direction of motion of \(P\) after the impulse is received.
    (4)
Edexcel M2 2017 January Q5
10 marks Standard +0.8
5. Two particles \(P\) and \(Q\), of masses \(2 m\) and \(3 m\) respectively, are moving in opposite directions along the same straight line on a smooth horizontal plane. The particles collide directly and, as a result of the collision, the direction of motion of \(P\) is reversed and the direction of motion of \(Q\) is reversed. Immediately after the collision, the speed of \(P\) is \(v\) and the speed of \(Q\) is \(\frac { 3 v } { 2 }\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 5 }\).
  1. Find
    1. the speed of \(P\) immediately before the collision,
    2. the speed of \(Q\) immediately before the collision. After the collision with \(P\), the particle \(Q\) moves on the plane and strikes at right angles a fixed smooth vertical wall and rebounds. The coefficient of restitution between \(Q\) and the wall is \(e\). Given that there is a further collision between the particles,
  2. find the range of possible values of \(e\).
Edexcel M2 2017 January Q6
5 marks Standard +0.3
6. A ball of mass 0.6 kg is projected vertically upwards with speed \(22.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point which is 1.5 m above horizontal ground. The ball moves freely under gravity until it reaches the ground. The ground is soft and the ball sinks 2.5 cm into the ground before coming to rest. The ball is modelled as a particle and the ground is assumed to exert a constant resistive force of magnitude \(R\) newtons on the ball. Using the work-energy principle, find, to 3 significant figures, the value of \(R\).
(5)
Edexcel M2 2017 January Q7
13 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{36cced0d-f982-4534-a3fe-13c32fb37f5b-11_513_429_123_762} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The end \(A\) is in contact with rough horizontal ground and the end \(B\) is in contact with a smooth vertical wall. The rod rests in equilibrium in a vertical plane perpendicular to the wall and makes an angle of \(30 ^ { \circ }\) with the wall, as shown in Figure 2. The coefficient of friction between the rod and the ground is \(\mu\).
  1. Find, in terms of \(m\) and \(g\), the magnitude of the force exerted on the rod by the wall.
  2. Show that \(\mu \geqslant \frac { \sqrt { 3 } } { 6 }\). A particle of mass \(k m\) is now attached to the rod at \(B\). Given that \(\mu = \frac { \sqrt { 3 } } { 5 }\) and that the rod is now in limiting equilibrium,
  3. find the value of \(k\).
Edexcel M2 2017 January Q8
13 marks Moderate -0.3
  1. At time \(t = 0\) seconds, a golf ball is hit from a point \(O\) on horizontal ground. The horizontal and vertical components of the initial velocity of the ball are \(3 U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The ball hits the ground at the point \(A\), where \(O A = 120 \mathrm {~m}\). The ball is modelled as a particle moving freely under gravity.
    1. Show that \(U = 14\)
    2. Find the speed of the ball immediately before it hits the ground at \(A\).
    3. Find the values of \(t\) when the ball is moving at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 1 } { 4 }\).
Edexcel M2 2018 January Q1
6 marks Standard +0.3
  1. A ball of mass 0.5 kg is moving with velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(( 4 \mathbf { i } + 5 \mathbf { j } ) \mathrm { Ns }\). Find the gain in kinetic energy of the ball due to the impulse.
    (6)
Edexcel M2 2018 January Q2
9 marks Moderate -0.8
2. A particle \(P\) moves in a straight line. At time \(t = 0 , P\) passes through a point \(O\) on the line. At time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) where
  1. Find the acceleration of \(P\) when \(t = \frac { 1 } { 2 }\)
  2. Find the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 1\) At time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) where $$v = ( 2 t - 1 ) ( 1 - t )$$
  3. Find the acceleration of \(P\) when \(t = \frac { 1 } { 2 }\)
Edexcel M2 2018 January Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{54112b4a-3727-4e5b-97e5-4291e7172438-06_479_608_246_667} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform lamina \(O A B C D\) is shown in Figure 1, with \(O A = 6 a , A B = 3 a , C D = 2 a\) and \(D O = 6 a\) and with right angles at \(O , A\) and \(D\).
  1. Find the distance of the centre of mass of the lamina
    1. from \(O D\),
    2. from \(O A\). The lamina is suspended from \(C\) and hangs freely in equilibrium with \(C B\) inclined at an angle \(\alpha\) to the vertical.
  2. Find, to the nearest degree, the size of the angle \(\alpha\).
Edexcel M2 2018 January Q4
13 marks Standard +0.3
  1. A particle \(P\) of mass \(2 m\) is moving in a straight line with speed \(u\) on a smooth horizontal plane. The particle \(P\) collides directly with a particle \(Q\), of mass \(m\), which is moving on the plane along the same straight line as \(P\) but in the opposite direction to \(P\). Immediately before the collision the speed of \(Q\) is \(3 u\). The coefficient of restitution between \(P\) and \(Q\) is \(e\), where \(e > \frac { 1 } { 8 }\)
    1. Find, in terms of \(u\) and \(e\),
      1. the speed of \(P\) immediately after the collision,
      2. the speed of \(Q\) immediately after the collision.
    2. Show that, for all possible values of \(e\), the direction of motion of \(P\) is reversed by the collision.
    After the collision, \(Q\) strikes a smooth fixed vertical wall, which is perpendicular to the direction of motion of \(Q\), and rebounds. The coefficient of restitution between \(Q\) and the wall is \(f\). Given that \(e = \frac { 3 } { 4 }\) and that there is a second collision between \(Q\) and \(P\),
  2. find the range of possible values of \(f\).
Edexcel M2 2018 January Q5
10 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{54112b4a-3727-4e5b-97e5-4291e7172438-14_472_789_253_575} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform rod, of weight \(W\) and length \(16 b\), has one end freely hinged to a fixed point \(A\). The rod rests against a smooth circular cylinder, of radius \(5 b\), fixed with its axis horizontal and at the same horizontal level as \(A\). The distance of \(A\) from the axis of the cylinder is 13b, as shown in Figure 2. The rod rests in a vertical plane which is perpendicular to the axis of the cylinder.
  1. Find, in terms of \(W\), the magnitude of the reaction on the rod at its point of contact with the cylinder.
  2. Show that the resultant force acting on the rod at \(A\) is inclined to the vertical at an angle \(\alpha\) where \(\tan \alpha = \frac { 40 } { 73 }\)
    5 continued \includegraphics[max width=\textwidth, alt={}, center]{54112b4a-3727-4e5b-97e5-4291e7172438-17_81_72_2631_1873}
Edexcel M2 2018 January Q6
10 marks Standard +0.3
6. A car of mass 800 kg pulls a trailer of mass 300 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 direction of motion of the car. The car's engine works at a constant rate of \(P \mathrm {~kW}\). The non-gravitational resistances to motion are constant and of magnitude 600 N on the car and 200 N on the trailer. At a given instant the car is moving at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is accelerating at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
  1. Find the value of \(P\). When the car is moving up the road at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the towbar breaks. The trailer comes to instantaneous rest after moving a distance \(d\) metres up the road from the point where the towbar broke. The non-gravitational resistance to the motion of the trailer remains constant and of magnitude 200 N .
  2. Find, using the work-energy principle, the value of \(d\).
Edexcel M2 2018 January Q7
17 marks Standard +0.3
7. A particle is projected from a point \(O\) with speed \(U\) at an angle of elevation \(\alpha\) to 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 $$y = x \tan \alpha - \frac { g x ^ { 2 } \left( 1 + \tan ^ { 2 } \alpha \right) } { 2 U ^ { 2 } }$$ \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{54112b4a-3727-4e5b-97e5-4291e7172438-22_330_857_632_548} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A small stone is projected horizontally with speed \(U\) from a point \(C\) at the top of a vertical cliff \(A C\) so as to hit a fixed target \(B\) on the horizontal ground. The point \(C\) is a height \(h\) above the ground, as shown in Figure 3. The time of flight of the stone from \(C\) to \(B\) is \(T\), and the stone is modelled as a particle moving freely under gravity.
  2. Find, in terms of \(U , g\) and \(T\), the speed of the stone as it hits the target at \(B\). It is found that, using the same initial speed \(U\), the target can also be hit by projecting the stone from \(C\) at an angle \(\alpha\) above the horizontal. The stone is again modelled as a particle moving freely under gravity and the distance \(A B = d\).
  3. Using the result in part (a), or otherwise, show that $$d = \frac { 1 } { 2 } g T ^ { 2 } \tan \alpha$$
Edexcel M2 2019 January Q1
5 marks Moderate -0.8
  1. Three particles of masses \(3 m , m\) and \(2 m\) are positioned at the points with coordinates \(( a , 8 ) , ( - 4,0 )\) and \(( 5 , - 2 )\) respectively.
Given that the centre of mass of the three particles is at the point with coordinates \(( k , 2 k )\), where \(k\) is a constant, find the value of \(a\).
(5)
Edexcel M2 2019 January Q2
6 marks Moderate -0.3
  1. A particle of mass 0.75 kg is moving with velocity ( \(4 \mathbf { i } + \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse ( \(- 6 \mathbf { i } + 4 \mathbf { j }\) ) N s. impulse \(( - 6 \mathbf { i } + 4 \mathbf { j } )\) N s.
\section*{Find
Find} $$\begin{aligned} & \text { (a) the velocity of the particle immediately after receiving the impulse, } \\ & \text { (b) the size of the angle through which the path of the particle is deflected as a result of } \\ & \text { the impulse. } \end{aligned}$$ (3)
Edexcel M2 2019 January Q3
8 marks Standard +0.3
  1. A car of mass 900 kg is moving on a straight road that is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 49 }\). When the car is moving up the road, with the engine of the car working at a constant rate of 10.8 kW , the car has a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons.
When the car is moving down the road, with the engine of the car working at a constant rate of 10.8 kW , the car has a constant speed of \(2 v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the motion of the car is still modelled as a constant force of magnitude \(R\) newtons. Find
  1. the value of \(R\),
  2. the value of \(v\).
Edexcel M2 2019 January Q4
9 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4065fe1-55fa-4a01-8ae2-006e0d529c50-10_787_814_246_566} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform lamina \(L\), shown shaded in Figure 1, is formed by removing a circular disc of radius \(2 a\) from a uniform circular disc of radius \(4 a\). The larger disc has centre \(O\) and diameter \(A B\). The radius \(O D\) is perpendicular to \(A B\). The smaller disc has centre \(C\), where \(C\) is on \(A B\) and \(B C = 3 a\)
  1. Show that the centre of mass of \(L\) is \(\frac { 13 } { 3 } a\) from \(B\). The mass of \(L\) is \(M\) and a particle of mass \(k M\) is attached to \(L\) at \(B\). When \(L\), with the particle attached, is freely suspended from point \(D\), it hangs in equilibrium with \(A\) higher than \(B\) and \(A B\) at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 3 } { 4 }\)
  2. Find the value of \(k\).
Edexcel M2 2019 January Q5
8 marks Standard +0.8
5. A particle moves along the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the velocity of the particle is \(v \mathrm {~ms} ^ { - 1 }\) in the direction of \(x\) increasing, where \(v = 2 t ^ { \frac { 3 } { 2 } } - 6 t + 2\) At time \(t = 0\) the particle passes through the origin \(O\). At the instant when the acceleration of the particle is zero, the particle is at the point \(A\). Find the distance \(O A\).
(8)
Edexcel M2 2019 January Q6
11 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4065fe1-55fa-4a01-8ae2-006e0d529c50-16_449_974_237_445} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A plank \(A B\) rests in equilibrium against a fixed horizontal pole. The plank has length 4 m and weight 20 N and rests on the pole at \(C\), where \(A C = 2.5 \mathrm {~m}\). The end \(A\) of the plank rests on rough horizontal ground and \(A B\) makes an angle \(\theta\) with the ground, as shown in
Figure 2. The coefficient of friction between the plank and the ground is \(\frac { 1 } { 4 }\).
The plank is modelled as a uniform rod and the pole as a rough horizontal peg that is perpendicular to the vertical plane containing \(A B\). Given that \(\cos \theta = \frac { 4 } { 5 }\) and that the friction is limiting at both \(A\) and \(C\),
  1. find the magnitude of the normal reaction on the plank at \(C\),
  2. find the coefficient of friction between the plank and the pole.