Questions — Edexcel M2 (623 questions)

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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.
Edexcel M2 2024 June Q4
10 marks Standard +0.3
  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
11 marks Standard +0.3
  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
10 marks Standard +0.3
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
11 marks Moderate -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 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
6 marks Moderate -0.8
  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
8 marks Standard +0.3
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
7 marks Standard +0.3
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 \(A\).
Edexcel M2 2016 October Q4
10 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
$$\mathbf { r } = \left( t ^ { 3 } - \frac { 9 } { 2 } t ^ { 2 } - 24 t \right) \mathbf { i } + \left( - t ^ { 3 } + 3 t ^ { 2 } + 12 t \right) \mathbf { j }$$ At time \(T\) seconds, \(P\) is moving in a direction parallel to the vector \(\mathbf { - i } - \mathbf { j }\) Find
  1. the value of \(T\),
  2. the magnitude of the acceleration of \(P\) at the instant when \(t = T\).
Edexcel M2 2016 October Q5
10 marks Standard +0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be16c17a-c4db-4f0c-9f32-8d5614b4f2f3-12_440_1047_246_447} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform rod \(A B\) of length 8 m and weight \(W\) newtons rests in equilibrium against a rough horizontal peg \(P\). The end \(A\) is on rough horizontal ground. The friction is limiting at both \(A\) and \(P\). The distance \(A P\) is 5 m , as shown in Figure 1. The rod rests at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 4 } { 3 }\). The rod is in a vertical plane which is perpendicular to \(P\). The coefficient of friction between the rod and \(P\) is \(\frac { 1 } { 4 }\) and the coefficient of friction between the rod and the ground is \(\mu\).
  1. Show that the magnitude of the normal reaction between the rod and \(P\) is \(0.48 W\) newtons.
  2. Find the value of \(\mu\).
Edexcel M2 2016 October Q6
10 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be16c17a-c4db-4f0c-9f32-8d5614b4f2f3-16_1031_915_116_513} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The uniform lamina \(L\) shown shaded in Figure 2 is formed by removing two circular discs, \(C _ { 1 }\) and \(C _ { 2 }\), from a circular disc with centre \(O\) and radius \(8 a\). Disc \(C _ { 1 }\) has centre \(A\) and radius \(a\). Disc \(C _ { 2 }\) has centre \(B\) and radius \(2 a\). The diameters \(P R\) and \(Q S\) are perpendicular. The midpoint of \(P O\) is \(A\) and the midpoint of \(O R\) is \(B\).
  1. Show that the centre of mass of \(L\) is \(\frac { 484 } { 59 } a\) from \(R\). The mass of \(L\) is \(M\). A particle of mass \(k M\) is attached to \(L\) at \(S\). The lamina with the attached particle is suspended from \(R\) and hangs freely in equilibrium with the diameter \(P R\) at an angle of arctan \(\left( \frac { 1 } { 4 } \right)\) to the downward vertical through \(R\).
  2. Find the value of \(k\).
Edexcel M2 2016 October Q7
10 marks Standard +0.8
7. [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. Position vectors are given relative to a fixed origin O.] At time \(t = 0\) seconds, the particle \(P\) is projected from \(O\) with velocity ( \(3 \mathbf { i } + \lambda \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\), where \(\lambda\) is a positive constant. The particle moves freely under gravity. As \(P\) passes through the fixed point \(A\) it has velocity \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The kinetic energy of \(P\) at the instant it passes through \(A\) is half the initial kinetic energy of \(P\). Find the position vector of \(A\), giving the components to 2 significant figures.
(10)
Edexcel M2 2016 October Q8
14 marks Standard +0.3
8. Particles \(A , B\) and \(C\), of masses \(4 m , k m\) and \(2 m\) respectively, lie at rest in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). Particle \(A\) is projected towards particle \(B\) with speed \(3 u\) and collides directly with \(B\). The coefficient of restitution between each pair of particles is \(\frac { 2 } { 3 }\) Find
  1. the speed of \(A\) immediately after the collision with \(B\), giving your answer in terms of \(u\) and \(k\),
  2. the range of values of \(k\) for which \(A\) and \(B\) will both be moving in the same direction immediately after they collide. After the collision between \(A\) and \(B\), particle \(B\) collides directly with \(C\). Given that \(k = 4\),
  3. show that there will not be a second collision between \(A\) and \(B\).
    DO NOT WRITEIN THIS AREA
Edexcel M2 2017 October Q1
4 marks Moderate -0.8
  1. A small ball \(B\) of mass 0.2 kg is hit by a bat. Immediately before being hit, \(B\) has velocity \(( 10 \mathbf { i } - 17 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after being hit, \(B\) has velocity \(( 5 \mathbf { i } + 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Find the magnitude of the impulse exerted on \(B\) by the bat.
    (4)
Edexcel M2 2017 October Q2
8 marks Moderate -0.3
2. A van of mass 1200 kg is travelling along a straight horizontal road. The resistance to the motion of the van has a constant magnitude of 650 N and the van's engine is working at a rate of 30 kW .
  1. Find the acceleration of the van when its speed is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The van now travels up a straight road which is inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\). The resistance to the motion of the van from non-gravitational forces has a constant magnitude of 650 N . The van moves up the road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  2. Find, in kW , the rate at which the van's engine is now working.
    "
Edexcel M2 2017 October Q3
7 marks Standard +0.3
3. A particle \(P\) of mass 4 kg moves from point \(A\) to point \(B\) down a line of greatest slope of a fixed rough plane. The plane is inclined at \(40 ^ { \circ }\) to the horizontal and \(A B = 12 \mathrm {~m}\). The coefficient of friction between \(P\) and the plane is 0.5
  1. Find the work done against friction as \(P\) moves from \(A\) to \(B\). Given that the speed of \(P\) at \(B\) is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  2. use the work-energy principle to find the speed of \(P\) at \(A\).
Edexcel M2 2017 October Q4
12 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5ef8231d-5b95-4bbb-a8e2-788c708fa078-12_518_696_319_625} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform \(\operatorname { rod } A B\) has mass \(m\) and length \(6 a\). The end \(A\) rests against a rough vertical wall. One end of a light inextensible string is attached to the rod at the point \(C\), where \(A C = 2 a\). The other end of the string is attached to the wall at the point \(D\), where \(D\) is vertically above \(A\), with the string perpendicular to the rod. A particle of mass \(m\) is attached to the rod at the end \(B\). The rod is in equilibrium in a vertical plane which is perpendicular to the wall. The rod is inclined at \(60 ^ { \circ }\) to the wall, as shown in Figure 1. Find, in terms of \(m\) and \(g\),
  1. the tension in the string,
  2. the magnitude of the horizontal component of the force exerted by the wall on the rod. The coefficient of friction between the wall and the rod is \(\mu\). Given that the rod is in limiting equilibrium,
  3. find the value of \(\mu\).
Edexcel M2 2017 October Q5
8 marks Standard +0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5ef8231d-5b95-4bbb-a8e2-788c708fa078-16_632_734_248_605} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The uniform lamina \(A B C\) is in the shape of an equilateral triangle with sides of length \(4 a\). The midpoint of \(B C\) is \(D\). The point \(E\) lies on \(A D\) with \(D E = \frac { 3 a } { 2 }\). A circular hole, with centre \(E\) and radius \(a\), is made in the lamina \(A B C\) to form the lamina \(L\), shown shaded in Figure 2.
  1. Find the distance of the centre of mass of \(L\) from \(D\). The lamina \(L\) is freely suspended from the point \(B\) and hangs in equilibrium.
  2. Find, to the nearest degree, the size of the acute angle between \(A D\) and the downward vertical.
Edexcel M2 2017 October Q6
10 marks Standard +0.3
  1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the acceleration of \(P\) is \(( 2 t - 3 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the positive \(x\) direction. At time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction. When \(t = 3 , v = 2\)
    1. Find \(v\) in terms of \(t\).
      (4)
    The particle first comes to instantaneous rest at the point \(A\) and then comes to instantaneous rest again at the point \(B\).
  2. Find the distance \(A B\).
Edexcel M2 2017 October Q7
14 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5ef8231d-5b95-4bbb-a8e2-788c708fa078-24_711_1009_251_479} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A small ball \(P\) is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) which is 47.5 m above a horizontal beach. The ball moves freely under gravity and hits the beach at the point \(B\), as shown in Figure 3.
  1. By considering energy, find the speed of \(P\) immediately before it hits the beach. The ball was projected from \(A\) at an angle \(\theta\) above the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\)
  2. Find the greatest height above the beach of \(P\) as it moved from \(A\) to \(B\).
  3. Find the least speed of \(P\) as it moved between \(A\) and \(B\).
  4. Find the horizontal distance from \(A\) to \(B\).