Questions — Edexcel M2 (551 questions)

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Edexcel M2 2022 June Q1
  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
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
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
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
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
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
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
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
  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
  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
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
  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
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
  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
  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
  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
  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
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.
Edexcel M2 2024 June Q4
  1. A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\)
A particle \(P\) of mass \(m\) is held at rest at a point \(A\) on the plane.
The particle is then projected with speed \(u\) up a line of greatest slope of the plane and comes to instantaneous rest at the point \(B\). The coefficient of friction between the particle and the plane is \(\frac { 1 } { 7 }\)
  1. Show that the magnitude of the frictional force acting on the particle, as it moves from \(A\) to \(B\), is \(\frac { 4 m g } { 35 }\) Given that \(u = \sqrt { 10 a g }\), use the work-energy principle
  2. to find \(A B\) in terms of \(a\),
  3. to find, in terms of \(a\) and \(g\), the speed of \(P\) when it returns to \(A\).
Edexcel M2 2024 June Q5
  1. A particle \(P\) of mass \(m\) and a particle \(Q\) of mass \(2 m\) are at rest on a smooth horizontal plane.
Particle \(P\) is projected with speed \(u\) along the plane towards \(Q\) and the particles collide. The coefficient of restitution between the particles is \(e\). As a result of the collision, the direction of motion of \(P\) is reversed.
  1. Find, in terms of \(u\) and \(e\), the speed of \(P\) after the collision. After the collision, \(Q\) goes on to hit a vertical wall which is fixed at right angles to the direction of motion of \(Q\). The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\) Given that there is a second collision between \(P\) and \(Q\)
  2. find the full range of possible values of \(e\).
Edexcel M2 2024 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b6e93edf-1b9f-4ea9-bb41-f46f380bc623-18_625_803_246_632} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform rod, \(A B\), of mass \(m\) and length \(2 a\), rests in limiting equilibrium with its end \(A\) on rough horizontal ground and its end \(B\) against a smooth vertical wall.
The vertical plane containing the rod is at right angles to the wall.
The rod is inclined to the wall at an angle \(\alpha\), as shown in Figure 2.
The coefficient of friction between the rod and the ground is \(\frac { 1 } { 3 }\)
  1. Show that \(\tan \alpha = \frac { 2 } { 3 }\) With the rod in the same position, a horizontal force of magnitude \(k m g\) is applied to the \(\operatorname { rod }\) at \(A\), towards the wall. The line of action of this force is at right angles to the wall. The rod remains in equilibrium.
  2. Find the largest possible value of \(k\).
Edexcel M2 2024 June Q7
  1. \hspace{0pt} [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and j being vertically upwards.]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b6e93edf-1b9f-4ea9-bb41-f46f380bc623-22_398_1438_420_267} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A golf ball is hit from a point \(O\) on horizontal ground and is modelled as a particle moving freely under gravity. The initial velocity of the ball is \(( 2 u \mathbf { i } + u \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) The ball first hits the horizontal ground at a point which is 80 m from \(O\), as shown in Figure 3. Use the model to
  1. show that \(u = 14\)
  2. find the total time, while the ball is in the air, for which the speed of the ball is greater than \(7 \sqrt { 17 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
Edexcel M2 2016 October Q1
  1. Three particles of masses \(m , 4 m\) and \(k m\) are placed at the points whose coordinates are \(( - 3,2 ) , ( 4,3 )\) and \(( 6 , - 4 )\) respectively. The centre of mass of the three particles is at the point with coordinates \(( c , 0 )\).
Find
  1. the value of \(k\),
  2. the value of \(c\).
Edexcel M2 2016 October Q2
2. A particle of mass 2 kg is moving with velocity \(3 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives an impulse \(( \lambda \mathbf { i } - 2 \lambda \mathbf { j } )\) Ns, where \(\lambda\) is a constant. Immediately after the impulse is received, the speed of the particle is \(6 \mathrm {~ms} ^ { - 1 }\). Find the possible values of \(\lambda\).
Edexcel M2 2016 October Q3
3. A particle \(P\) of mass 4 kg is projected with speed \(6 \mathrm {~ms} ^ { - 1 }\) up a line of greatest slope of a fixed rough inclined plane. The plane is inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 7 }\). The particle is projected from the point \(A\) on the plane and comes to instantaneous rest at the point \(B\) on the plane, where \(A B = 10 \mathrm {~m}\).
  1. Show that the work done against friction as \(P\) moves from \(A\) to \(B\) is 16 joules. After coming to instantaneous rest at \(B\), the particle slides back down the plane.
  2. Use the work-energy principle to find the speed of \(P\) at the instant it returns to \(A\).