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Edexcel M2 2021 June Q2
8 marks Standard +0.3
2. A particle \(P\) of mass 1.5 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, \(t \geqslant 0 , P\) has velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where $$\mathbf { v } = \left( 5 t ^ { 2 } - t ^ { 3 } \right) \mathbf { i } + \left( 2 t ^ { 3 } - 8 t \right) \mathbf { j }$$
  1. Find \(\mathbf { F }\) when \(t = 2\) At time \(t = 0 , P\) is at the origin \(O\).
  2. Find the position vector of \(P\) relative to \(O\) at the instant when \(P\) is moving in the direction of the vector \(\mathbf { j }\)
Edexcel M2 2021 June Q3
8 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6e37d85-f8de-490a-82a9-8a3c16e2fdd0-06_645_684_260_639} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform lamina \(A B C D\) is a square of side \(6 a\). The template \(T\), shown shaded in Figure 1, is formed by removing the right-angled triangle \(E F G\) and the circle, centre \(H\) and radius \(a\), from the square lamina. Triangle \(E F G\) has \(E F = E G = 4 a\), with \(E F\) parallel to \(A B\) and \(E G\) parallel to \(A D\). The distance between \(A B\) and \(E F\) is \(a\) and the distance between \(A D\) and \(E G\) is \(a\). The point \(H\) lies on \(A C\) and the distance of \(H\) from \(B C\) is \(2 a\).
  1. Show that the centre of mass of \(T\) is a distance \(\frac { 4 ( 67 - 3 \pi ) } { 3 ( 28 - \pi ) } a\) from \(A D\). The template \(T\) is suspended from the ceiling by two light inextensible vertical strings. One string is attached to \(T\) at \(A\) and the other string is attached to \(T\) at \(B\) so that \(T\) hangs in equilibrium with \(A B\) horizontal. The weight of \(T\) is \(W\). The tension in the string attached to \(T\) at \(B\) is \(k W\), where \(k\) is a constant.
  2. Find the value of \(k\), giving your answer to 2 decimal places.
Edexcel M2 2021 June Q4
6 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6e37d85-f8de-490a-82a9-8a3c16e2fdd0-10_410_369_251_790} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of mass 0.3 kg is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight line on a smooth horizontal plane. The particle receives a horizontal impulse of magnitude \(J\) Ns. The speed of \(P\) immediately after receiving the impulse is \(8 \mathrm {~ms} ^ { - 1 }\). The angle between the direction of motion of \(P\) before it receives the impulse and the direction of the impulse is \(60 ^ { \circ }\), as shown in Figure 2. Find the value of \(J\).
(6)
Edexcel M2 2021 June Q5
9 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6e37d85-f8de-490a-82a9-8a3c16e2fdd0-12_638_595_251_676} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform rod, of length \(8 a\) and mass \(M\), has one end freely hinged to a fixed point \(A\) on a vertical wall. One end of a light inextensible string is attached to the rod at the point \(B\), where \(A B = 5 a\). The other end of the string is attached to the wall at the point \(C\), where \(A C = 5 a\) and \(C\) is vertically above \(A\). The rod rests in equilibrium in a vertical plane perpendicular to the wall with angle \(B A C = 70 ^ { \circ }\), as shown in Figure 3.
  1. Find, in terms of \(M\) and \(g\), the tension in the string. The magnitude of the force acting on the rod at \(A\) is \(\lambda M g\), where \(\lambda\) is a constant.
  2. Find, to 2 significant figures, the value of \(\lambda\).
Edexcel M2 2021 June Q6
15 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6e37d85-f8de-490a-82a9-8a3c16e2fdd0-16_273_819_260_566} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Two particles, \(A\) and \(B\), of mass 2 kg and 3 kg respectively, are connected by a light inextensible string. Particle \(A\) is held at rest at the point \(X\) on a fixed rough ramp that is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\). The string passes over a small smooth pulley \(P\) that is fixed at the top of the ramp. Particle \(B\) hangs vertically below \(P\), 2 m above the ground, as shown in Figure 4. The particles are released from rest with the string taut so that \(A\) moves up the ramp and the section of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the ramp. The coefficient of friction between \(A\) and the ramp is \(\frac { 3 } { 8 }\) Air resistance is ignored.
  1. Find the potential energy lost by the system as \(A\) moves 2 m up the ramp.
  2. Find the work done against friction as \(A\) moves 2 m up the ramp. When \(B\) hits the ground, \(B\) is brought to rest by the impact and does not rebound and \(A\) continues to move up the ramp.
  3. Use the work-energy principle to find the speed of \(B\) at the instant before it hits the ground. Particle \(A\) comes to instantaneous rest at the point \(Y\) on the ramp, where \(X Y = ( 2 + d ) \mathrm { m }\).
  4. Use the work-energy principle to find the value of \(d\).
Edexcel M2 2021 June Q7
11 marks Standard +0.3
  1. \hspace{0pt} [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e6e37d85-f8de-490a-82a9-8a3c16e2fdd0-20_289_837_347_486} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A small ball is projected with velocity \(( 6 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) from a fixed point \(A\) on horizontal ground. The ball hits the ground at the point \(B\), as shown in Figure 5. The motion of the ball is modelled as a particle moving freely under gravity.
  1. Find the distance \(A B\). When the height of the ball above the ground is more than \(h\) metres, the speed of the ball is less than \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  2. Find the smallest possible value of \(h\). When the ball is at the point \(C\) on its path, the direction of motion of the ball is perpendicular to the direction of motion of the ball at the instant before it hits the ground at \(B\).
  3. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of the ball when it is at \(C\).
Edexcel M2 2021 June Q8
13 marks Standard +0.8
  1. Particles \(A , B\) and \(C\), of masses \(2 m , m\) and \(3 m\) respectively, lie at rest in a straight line on a smooth horizontal plane with \(B\) between \(A\) and \(C\). Particle \(A\) is projected towards particle \(B\) with speed \(2 u\) and collides directly with \(B\).
The coefficient of restitution between each pair of particles is \(e\).
    1. Show that the speed of \(B\) immediately after the collision with \(A\) is \(\frac { 4 } { 3 } u ( 1 + e )\)
    2. Find the speed of \(A\) immediately after the collision with \(B\). At the instant when \(A\) collides with \(B\), particle \(C\) is projected with speed \(u\) towards \(B\) so that \(B\) and \(C\) collide directly.
  2. Show that there will be a second collision between \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{e6e37d85-f8de-490a-82a9-8a3c16e2fdd0-27_2644_1840_118_111}
Edexcel M2 2022 June Q1
8 marks Standard +0.3
  1. At time \(t\) seconds, \(t \geqslant 0\), a particle \(P\) has position vector \(\mathbf { r }\) metres with respect to a fixed origin \(O\), where
$$\mathbf { r } = \left( t ^ { 3 } - 8 t \right) \mathbf { i } + \left( \frac { 1 } { 3 } t ^ { 3 } - t ^ { 2 } + 2 t \right) \mathbf { j }$$
  1. Find the acceleration of \(P\) when \(t = 4\) At time \(T\) seconds, \(T \geqslant 0 , P\) is moving in the direction of ( \(2 \mathbf { i } + \mathbf { j }\) )
  2. Find the value of \(T\)
Edexcel M2 2022 June Q2
6 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7eedd755-0dfd-4506-b7fd-23b9def4ebc8-04_508_780_258_584} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The point \(A\) lies on a smooth horizontal floor between two fixed smooth parallel vertical walls \(W X\) and \(Y Z\), as shown in the plan view in Figure 1.
The distance between \(W X\) and \(Y Z\) is \(3 d\).
The distance of \(A\) from \(Y Z\) is \(d\).
A particle is projected from \(A\) along the floor with speed \(u\) towards \(Y Z\) in a direction perpendicular to \(Y Z\). The coefficient of restitution between the particle and each wall is \(\frac { 2 } { 3 }\) The time taken for the particle to move from \(A\), bounce off each wall once and return to A for the first time is \(T _ { 1 }\)
  1. Find \(T _ { 1 }\) in terms of \(d\) and \(u\). The ball returns to \(A\) for the first time after bouncing off each wall once. The further time taken for the particle to move from \(A\), bounce off each wall once and return to \(A\) for the second time is \(T _ { 2 }\)
  2. Find \(T _ { 2 }\) in terms of \(d\) and \(u\).
Edexcel M2 2022 June Q3
6 marks Standard +0.3
3. A particle \(P\) of mass 0.5 kg is moving with velocity \(\lambda ( \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when \(P\) receives an impulse of magnitude \(\sqrt { \frac { 5 } { 2 } } \mathrm { Ns }\) Immediately after \(P\) receives the impulse, the velocity of \(P\) is \(4 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) Given that \(\lambda\) is a constant, find the two possible values of \(\lambda\)
Edexcel M2 2022 June Q4
8 marks Standard +0.3
4. A truck of mass 900 kg is moving along a straight horizontal road with the engine of the truck working at a constant rate of \(P\) watts. The resistance to the motion of the truck is modelled as a constant force of magnitude \(R\) newtons.
At the instant when the speed of the truck is \(15 \mathrm {~ms} ^ { - 1 }\), the deceleration of the truck is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Later the same truck is moving down a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 30 }\). The resistance to the motion of the truck is again modelled as a constant force of magnitude \(R\) newtons. The engine of the truck is again working at a constant rate of \(P\) watts.
At the instant when the speed of the truck is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the truck is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Find the value of \(R\).
Edexcel M2 2022 June Q5
9 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7eedd755-0dfd-4506-b7fd-23b9def4ebc8-12_470_876_255_529} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform rod \(A B\) has length 4 m and weight 50 N .
The rod has its end \(A\) on rough horizontal ground. The rod is held in equilibrium at an angle \(\alpha\) to the ground by a light inextensible cable attached to the rod at \(B\), as shown in Figure 2. The cable and the rod lie in the same vertical plane and the cable is perpendicular to the rod. The tension in the cable is \(T\) newtons. Given that \(\sin \alpha = \frac { 3 } { 5 }\)
  1. show that \(T = 20\) Given also that the rod is in limiting equilibrium,
  2. find the value of the coefficient of friction between the rod and the ground.
Edexcel M2 2022 June Q6
12 marks Standard +0.3
6. Two particles, \(P\) and \(Q\), are moving in opposite directions along the same straight line on a smooth horizontal surface so that the particles collide directly.
The mass of \(P\) is \(k m\) and the mass of \(Q\) is \(m\).
Immediately before the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\). Immediately after the collision, \(P\) and \(Q\) are moving in the same direction, the speed of \(P\) is \(v\) and the speed of \(Q\) is \(2 v\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 5 }\) The magnitude of the impulse received by \(Q\) in the collision is \(5 m v\)
  1. Find (i) \(y\) in terms of \(v\) (ii) \(x\) in terms of \(v\) (iii) the value of \(k\)
  2. Find, in terms of \(m\) and \(v\), the total kinetic energy lost in the collision between \(P\) and \(Q\).
Edexcel M2 2022 June Q7
12 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7eedd755-0dfd-4506-b7fd-23b9def4ebc8-20_679_695_260_628} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The template shown in Figure 3 is formed by joining together three separate laminas. All three laminas lie in the same plane.
  • PQUV is a uniform square lamina with sides of length \(3 a\)
  • URST is a uniform square lamina with sides of length \(6 a\)
  • \(Q R U\) is a uniform triangular lamina with \(U Q = 3 a , U R = 6 a\) and angle \(Q U R = 90 ^ { \circ }\)
The mass per unit area of \(P Q U V\) is \(k\), where \(k\) is a constant.
The mass per unit area of URST is \(k\).
The mass per unit area of \(Q R U\) is \(2 k\).
The distance of the centre of mass of the template from \(Q T\) is \(d\).
  1. Show that \(d = \frac { 29 } { 14 } a\) The template is freely suspended from the point \(Q\) and hangs in equilibrium with \(Q R\) at \(\theta ^ { \circ }\) to the downward vertical.
  2. Find the value of \(\theta\)
Edexcel M2 2022 June Q8
14 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7eedd755-0dfd-4506-b7fd-23b9def4ebc8-24_259_1045_255_456} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a rough ramp fixed to horizontal ground.
The ramp is inclined at angle \(\alpha\) to the ground, where \(\tan \alpha = \frac { 1 } { 6 }\) The point \(A\) is on the ground at the bottom of the ramp.
The point \(B\) is at the top of the ramp.
The line \(A B\) is a line of greatest slope of the ramp and \(A B = 4 \mathrm {~m}\).
A particle \(P\) of mass 3 kg is projected with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) directly towards \(B\).
The coefficient of friction between the particle and the ramp is \(\frac { 3 } { 4 }\)
  1. Find the work done against friction as \(P\) moves from \(A\) to \(B\). Given that at the instant \(P\) reaches the point \(B\), the speed of \(P\) is \(5 \mathrm {~ms} ^ { - 1 }\)
  2. use the work-energy principle to find the value of \(U\). The particle leaves the ramp at \(B\), and moves freely under gravity until it hits the ground at the point \(C\).
  3. Find the horizontal distance from \(B\) to \(C\).
Edexcel M2 2023 June Q1
7 marks Moderate -0.3
  1. A particle \(P\) of mass 0.3 kg is moving with velocity \(5 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
The particle receives an impulse I Ns.
Immediately after receiving the impulse, the velocity of \(P\) is \(( 7 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)
  1. Find the magnitude of \(\mathbf { I }\)
  2. Find the angle between the direction of \(\mathbf { I }\) and the direction of motion of \(P\) immediately before receiving the impulse.
Edexcel M2 2023 June Q2
10 marks Moderate -0.3
  1. \hspace{0pt} [In this question, the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a horizontal plane.]
In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A particle \(P\) is moving on a smooth horizontal plane.
At time \(t\) seconds \(( t \geqslant 0 )\), the position vector of \(P\), relative to a fixed point \(O\), is \(\mathbf { r }\) metres and the velocity of \(P\) is \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) where $$\mathbf { v } = \left( 4 t ^ { 2 } - 5 t \right) \mathbf { i } + ( - 10 t - 12 ) \mathbf { j }$$ When \(t = 0 , \mathbf { r } = 2 \mathbf { i } + 6 \mathbf { j }\)
  1. Find \(\mathbf { r }\) when \(t = 2\) When \(t = T\) particle \(P\) is moving in the direction of the vector \(\mathbf { i } - 2 \mathbf { j }\)
  2. Find the value of \(T\)
  3. Find the exact magnitude of the acceleration of \(P\) when \(t = 2.5\)
Edexcel M2 2023 June Q3
8 marks Challenging +1.2
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{52966963-2e62-4361-bcd5-a76322f8621e-08_1141_810_287_148} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{52966963-2e62-4361-bcd5-a76322f8621e-08_752_803_484_1114} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The uniform triangular lamina \(A B C\), shown in Figure 1, has height \(9 y\), base \(B C = 6 x\), and \(A B = A C\) The points \(P\) and \(Q\) are such that \(A P : P C = A Q : Q B = 2 : 1\) The lamina is folded along \(P Q\) to form the folded lamina \(F\) The distance of the centre of mass of \(F\) from \(P Q\) is \(d\)
  1. Show that \(d = \frac { 16 } { 9 } y\) The folded lamina is suspended from \(P\) and hangs freely in equilibrium with \(P Q\) at an angle \(\alpha\) to the downward vertical.
    Given that \(\tan \alpha = \frac { 64 } { 81 }\)
  2. find \(x\) in terms of \(y\)
Edexcel M2 2023 June Q4
12 marks Standard +0.3
  1. A particle \(P\) of mass \(3 m\) and a particle \(Q\) of mass \(5 m\) are moving towards each other along the same straight line on a smooth horizontal surface. The particles collide directly.
Immediately before the collision, the speed of \(P\) is \(u\) and the speed of \(Q\) is \(k u\).
Immediately after the collision, the speed of \(P\) is \(2 v\) and the speed of \(Q\) is \(v\).
The direction of motion of each particle is reversed by the collision.
In the collision, \(P\) receives an impulse of magnitude \(15 m v\).
  1. Show that \(u = 3 v\).
  2. Find the value of \(k\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  3. Find the value of \(e\). The total kinetic energy lost in the collision is \(\lambda m v ^ { 2 }\)
  4. Find the value of \(\lambda\).
Edexcel M2 2023 June Q5
11 marks Standard +0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{52966963-2e62-4361-bcd5-a76322f8621e-16_825_670_283_699} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform beam \(A B\), of mass 15 kg and length 6 m , rests with end \(A\) on rough horizontal ground. The end \(B\) of the beam rests against a rough vertical wall. The beam is inclined at \(75 ^ { \circ }\) to the ground, as shown in Figure 2.
The coefficient of friction between the beam and the wall is 0.2
The coefficient of friction between the beam and the ground is \(\mu\) The beam is modelled as a uniform rod which lies in a vertical plane perpendicular to the wall. The beam rests in limiting equilibrium.
  1. Find the magnitude of the normal reaction between the beam and the wall at \(B\).
  2. Find the value of \(\mu\)
Edexcel M2 2023 June Q6
12 marks Standard +0.3
  1. A van of mass 900 kg is moving along a straight horizontal road.
The resistance to the motion of the van is modelled as a constant force of magnitude 600 N . The engine of the van is working at a constant rate of 24 kW .
At the instant when the speed of the van is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the van is \(2 \mathrm {~ms} ^ { - 2 }\)
  1. Find the value of \(V\) Later on, the van is towing a trailer of mass 700 kg up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\) The trailer is attached to the van by a towbar, as shown in Figure 3.
    The towbar is parallel to the direction of motion of the van and the trailer. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{52966963-2e62-4361-bcd5-a76322f8621e-20_367_1194_1091_438} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 600 N . The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 550 N . The towbar is modelled as a light rod.
    The engine of the van is working at a constant rate of 24 kW .
  2. Find the acceleration of the van at the instant when the van and the trailer are moving with speed \(8 \mathrm {~ms} ^ { - 1 }\) At the instant when the van and the trailer are moving up the road at \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the towbar breaks. The trailer continues to move in a straight line up the road until it comes to instantaneous rest. The distance moved by the trailer as it slows from a speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to instantaneous rest is \(d\) metres.
  3. Use the work-energy principle to find the value of \(d\).
Edexcel M2 2023 June Q7
15 marks Standard +0.3
  1. \hspace{0pt} [In this question, the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane with \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{52966963-2e62-4361-bcd5-a76322f8621e-24_679_1009_347_529} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A small ball is projected with velocity \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) from the fixed point \(A\).
The point \(A\) is 20 m above horizontal ground.
The ball hits the ground at the point \(B\), as shown in Figure 4.
The ball is modelled as a particle moving freely under gravity.
  1. By considering energy, find the speed of the ball at the instant immediately before it hits the ground.
  2. Find the direction of motion of the ball at the instant immediately before it hits the ground.
  3. Find the time taken for the ball to travel from \(A\) to \(B\). At the instant when the direction of motion of the ball is perpendicular to ( \(3 \mathbf { i } + 2 \mathbf { j }\) ) the ball is \(h\) metres above the ground.
  4. Find the value of \(h\).
Edexcel M2 2024 June Q1
8 marks Moderate -0.8
  1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.]
A particle \(A\) has mass 2 kg and a particle \(B\) has mass 3 kg . The particles are moving on a smooth horizontal plane when they collide. Immediately before the collision, the velocity of \(A\) is \(5 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the velocity of \(B\) is \(( 3 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) Immediately after the collision, the velocity of \(A\) is \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)
  1. Find the total kinetic energy of the two particles before the collision.
  2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the impulse received by \(A\) in the collision. Given that, in the collision, the impulse of \(A\) on \(B\) is equal and opposite to the impulse of \(B\) on \(A\),
  3. find the velocity of \(B\) immediately after the collision.
Edexcel M2 2024 June Q2
13 marks Standard +0.3
  1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
A particle \(P\) is moving in a straight line.
At time \(t\) seconds, the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is given by the continuous function $$v = \begin{cases} \sqrt { 2 t + 1 } & 0 \leqslant t \leqslant k \\ \frac { 3 } { 4 } t & t > k \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = 4\), explaining your method carefully.
  2. Find the acceleration of \(P\) when \(t = 1.5\) At time \(t = 0 , P\) passes through the point \(O\)
  3. Find the distance of \(P\) from \(O\) when \(t = 8\)
Edexcel M2 2024 June Q3
12 marks Challenging +1.2
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b6e93edf-1b9f-4ea9-bb41-f46f380bc623-06_990_985_244_539} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform circular disc \(C\) has centre \(X\) and radius \(R\).
A disc with centre \(Y\) and radius \(r\), where \(0 < r < R\) and \(X Y = R - r\), is removed from \(C\) to form the template shown shaded in Figure 1. The centre of mass of the template is a distance \(k r\) from \(X\).
  1. Show that \(r = \frac { k } { 1 - k } R\)
  2. Hence find the range of possible values of \(k\). The point \(P\) is on the outer edge of the template and \(P X\) is perpendicular to \(X Y\).
    The template is freely suspended from \(P\) and hangs in equilibrium.
    Given that \(k = \frac { 4 } { 9 }\)
  3. find the angle that \(X Y\) makes with the vertical. The mass of the template is \(M\).
  4. Find, in terms of \(M\), the mass of the lightest particle that could be attached to the template so that it would hang in equilibrium from \(P\) with \(X Y\) horizontal.