Questions — Edexcel M2 (551 questions)

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Edexcel M2 2014 June Q1
  1. A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\), in the positive \(x\) direction at time \(t\) seconds, is \(( 2 t - 3 ) \mathrm { m } \mathrm { s } ^ { - 2 }\). The velocity of \(P\), in the positive \(x\) direction at time \(t\) seconds, is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 0 , v = 2\)
    1. Find \(v\) in terms of \(t\).
    The particle is instantaneously at rest at times \(t _ { 1 }\) seconds and \(t _ { 2 }\) seconds, where \(t _ { 1 } < t _ { 2 }\).
  2. Find the values \(t _ { 1 }\) and \(t _ { 2 }\).
  3. Find the distance travelled by \(P\) between \(t = t _ { 1 }\) and \(t = t _ { 2 }\).
Edexcel M2 2014 June Q2
2. A trailer of mass 250 kg is towed by a car of mass 1000 kg . The car and the trailer are travelling down a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\) The resistance to motion of the car is modelled as a single force of magnitude 300 N acting parallel to the road. The resistance to motion of the trailer is modelled as a single force of magnitude 100 N acting parallel to the road. The towbar joining the car to the trailer is modelled as a light rod which is parallel to the direction of motion. At a given instant the car and the trailer are moving down the road with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and acceleration \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the power being developed by the car's engine at this instant.
  2. Find the tension in the towbar at this instant.
Edexcel M2 2014 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d5f79f83-ccfb-47a5-8100-88db81fd0862-05_1102_732_118_651} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform \(\operatorname { rod } A B\) of weight \(W\) is freely hinged at end \(A\) to a vertical wall. The rod is supported in equilibrium at an angle of \(60 ^ { \circ }\) to the wall by a light rigid strut \(C D\). The strut is freely hinged to the rod at the point \(D\) and to the wall at the point \(C\), which is vertically below \(A\), as shown in Figure 1. The rod and the strut lie in the same vertical plane, which is perpendicular to the wall. The length of the rod is \(4 a\) and \(A C = A D = 2.5 a\).
  1. Show that the magnitude of the thrust in the strut is \(\frac { 4 \sqrt { 3 } } { 5 } W\).
  2. Find the magnitude of the force acting on the \(\operatorname { rod }\) at \(A\).
Edexcel M2 2014 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d5f79f83-ccfb-47a5-8100-88db81fd0862-07_728_748_214_639} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The uniform square lamina \(A B C D\) shown in Figure 2 has sides of length 4a. The points \(E\) and \(F\), on \(D A\) and \(D C\) respectively, are both at a distance \(3 a\) from \(D\). The portion \(D E F\) of the lamina is folded through \(180 ^ { \circ }\) about \(E F\) to form the folded lamina \(A B C F E\) shown in Figure 3 below. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d5f79f83-ccfb-47a5-8100-88db81fd0862-07_709_730_1395_639} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}
  1. Show that the distance from \(A B\) of the centre of mass of the folded lamina is \(\frac { 55 } { 32 } a\).
    (6) The folded lamina is freely suspended from \(E\) and hangs in equilibrium.
  2. Find the size of the angle between \(E D\) and the downward vertical.
Edexcel M2 2014 June Q5
5. A particle of mass 0.5 kg is moving on a smooth horizontal surface with velocity \(12 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse \(K ( \mathbf { i } + \mathbf { j } ) \mathrm { N } \mathrm { s }\), where \(K\) is a positive constant. Immediately after receiving the impulse the particle is moving with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction which makes an acute angle \(\theta\) with the vector \(\mathbf { i }\). Find
  1. the value of \(K\),
  2. the size of angle \(\theta\).
Edexcel M2 2014 June Q6
6. Three particles \(P , Q\) and \(R\) have masses \(3 m , k m\) and 7.5m respectively. The three particles lie at rest in a straight line on a smooth horizontal table with \(Q\) between \(P\) and \(R\). Particle \(P\) is projected towards \(Q\) with speed \(u\) and collides directly with \(Q\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 9 }\).
  1. Show that the speed of \(Q\) immediately after the collision is \(\frac { 10 u } { 3 ( 3 + k ) }\).
  2. Find the range of values of \(k\) for which the direction of motion of \(P\) is reversed as a result of the collision. Following the collision between \(P\) and \(Q\) there is a collision between \(Q\) and \(R\). Given that \(k = 7\) and that \(Q\) is brought to rest by the collision with \(R\),
  3. find the total kinetic energy lost in the collision between \(Q\) and \(R\).
Edexcel M2 2014 June Q7
7. A particle \(P\) is projected from a fixed point \(A\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal and moves freely under gravity. When \(P\) passes through the point \(B\) on its path, it has speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. By considering energy, find the vertical distance between \(A\) and \(B\). The minimum speed of \(P\) on its path from \(A\) to \(B\) is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the size of angle \(\alpha\).
  3. Find the horizontal distance between \(A\) and \(B\).
Edexcel M2 2015 June Q1
  1. A particle of mass 0.3 kg is moving with velocity \(( 5 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(( - 3 \mathbf { i } + 3 \mathbf { j } ) \mathrm { N } \mathrm { s }\). Find the change in the kinetic energy of the particle due to the impulse.
    (6)
  2. At time \(t\) seconds, \(t \geqslant 0\), a particle \(P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where
$$\mathbf { v } = \left( 27 - 3 t ^ { 2 } \right) \mathbf { i } + \left( 8 - t ^ { 3 } \right) \mathbf { j }$$ When \(t = 1\), the particle \(P\) is at the point with position vector \(\mathbf { r } \mathrm { m }\) relative to a fixed origin \(O\), where \(\mathbf { r } = - 5 \mathbf { i } + 2 \mathbf { j }\) Find
  1. the magnitude of the acceleration of \(P\) at the instant when it is moving in the direction of the vector \(\mathbf { i }\),
  2. the position vector of \(P\) at the instant when \(t = 3\)
Edexcel M2 2015 June Q3
  1. A thin uniform wire of mass \(12 m\) is bent to form a right-angled triangle \(A B C\). The lengths of the sides \(A B , B C\) and \(A C\) are \(3 a , 4 a\) and \(5 a\) respectively. A particle of mass \(2 m\) is attached to the triangle at \(B\) and a particle of mass \(3 m\) is attached to the triangle at \(C\). The bent wire and the two particles form the system \(S\).
The system \(S\) is freely suspended from \(A\) and hangs in equilibrium.
Find the size of the angle between \(A B\) and the downward vertical.
Edexcel M2 2015 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3f2bf524-ee27-4eef-8c54-48be61c11677-07_531_1194_118_374} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle \(P\) of mass 6.5 kg is projected up a fixed rough plane with initial speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(X\) on the plane, as shown in Figure 1. The particle moves up the plane along the line of greatest slope through \(X\) and comes to instantaneous rest at the point \(Y\), where \(X Y = d\) metres. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 3 } \cdot\)
  1. Use the work-energy principle to show that, to 2 significant figures, \(d = 2.7\) After coming to rest at \(Y\), the particle \(P\) slides back down the plane.
  2. Find the speed of \(P\) as it passes through \(X\).
Edexcel M2 2015 June Q5
  1. Three particles \(A , B\) and \(C\) lie at rest in a straight line on a smooth horizontal table with \(B\) between \(A\) and \(C\). The masses of \(A , B\) and \(C\) are \(3 m\), 4m, and 5m respectively. Particle \(A\) is projected with speed \(u\) towards particle \(B\) and collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 3 }\).
    1. Show that the impulse exerted by \(A\) on \(B\) in this collision has magnitude \(\frac { 16 } { 7 } m u\)
    After the collision between \(A\) and \(B\) there is a direct collision between \(B\) and \(C\).
    After this collision between \(B\) and \(C\), the kinetic energy of \(C\) is \(\frac { 72 } { 245 } m u ^ { 2 }\)
  2. Find the coefficient of restitution between \(B\) and \(C\).
Edexcel M2 2015 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3f2bf524-ee27-4eef-8c54-48be61c11677-11_684_1022_114_479} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform \(\operatorname { rod } A B\) has length \(4 a\) and weight \(W\). A particle of weight \(k W , k < 1\), is attached to the rod at \(B\). The rod rests in equilibrium against a fixed smooth horizontal peg. The end \(A\) of the rod is on rough horizontal ground, as shown in Figure 2. The rod rests on the peg at \(C\), where \(A C = 3 a\), and makes an angle \(\alpha\) with the ground, where \(\tan \alpha = \frac { 1 } { 3 }\). The peg is perpendicular to the vertical plane containing \(A B\).
  1. Give a reason why the force acting on the rod at \(C\) is perpendicular to the rod.
  2. Show that the magnitude of the force acting on the rod at \(C\) is $$\frac { \sqrt { 10 } } { 5 } W ( 1 + 2 k )$$ The coefficient of friction between the rod and the ground is \(\frac { 3 } { 4 }\).
  3. Show that for the rod to remain in equilibrium \(k \leqslant \frac { 2 } { 11 }\).
Edexcel M2 2015 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 \(\mathbf { j }\) being vertically upwards.]
At time \(t = 0\), a particle \(P\) is projected with velocity ( \(4 \mathbf { i } + 9 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) from a fixed point \(O\) on horizontal ground. The particle moves freely under gravity. When \(P\) is at the point \(H\) on its path, \(P\) is at its greatest height above the ground.
  1. Find the time taken by \(P\) to reach \(H\). At the point \(A\) on its path, the position vector of \(P\) relative to \(O\) is \(( k \mathbf { i } + k \mathbf { j } ) \mathrm { m }\), where \(k\) is a positive constant.
  2. Find the value of \(k\).
    (4)
  3. Find, in terms of \(k\), the position vector of the other point on the path of \(P\) which is at the same vertical height above the ground as the point \(A\).
    (3) At time \(T\) seconds the particle is at the point \(B\) and is moving perpendicular to \(( 4 \mathbf { i } + 9 \mathbf { j } )\)
  4. Find the value of \(T\).
Edexcel M2 2016 June Q1
  1. A particle of mass 3 kg is moving with velocity \(( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(( - 4 \mathbf { i } + 3 \mathbf { j } ) \mathrm { N }\) s.
Find
  1. the speed of the particle immediately after receiving the impulse,
  2. the kinetic energy gained by the particle as a result of the impulse.
Edexcel M2 2016 June Q2
2. A truck of mass 1800 kg is moving along a straight horizontal road. The engine of the truck is working at a constant rate of 10 kW . The non-gravitational resistance to motion is modelled as a constant force of magnitude \(R\) newtons. At the instant when the truck is moving with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the truck is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the value of \(R\).
    (4) The truck now moves up a straight road at a constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The road is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\). The non-gravitational resistance to motion is now modelled as a constant force of magnitude 30 V newtons. The engine of the truck is now working at a constant rate of 12 kW .
  2. Find the value of \(V\).
    DO NOT WIRITE IN THIS AREA
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{790546bf-38a4-4eb7-876e-941fe58f4a48-05_529_1040_118_450} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure}
Edexcel M2 2016 June Q3
3. Two particles \(P\) and \(Q\), of mass \(2 m\) and \(3 m\) respectively, are connected by a light inextensible string. Initially \(P\) is held at rest on a fixed rough plane inclined at \(\theta\) to the horizontal ground, where \(\sin \theta = \frac { 2 } { 5 }\). The string passes over a small smooth pulley fixed at the top of the plane. The particle \(Q\) hangs freely below the pulley, as shown in Figure 1. The part of the string from \(P\) to the pulley lies along a line of greatest slope of the plane. At time \(t = 0\) the system is released from rest with the string taut. When \(P\) moves the friction between \(P\) and the plane is modelled as a constant force of magnitude \(\frac { 3 } { 5 } m g\). At the instant when each particle has moved a distance \(d\), they are both moving with speed \(v\), particle \(P\) has not reached the pulley and \(Q\) has not reached the ground.
  1. Show that the total potential energy lost by the system when each particle has moved a distance \(d\) is \(\frac { 11 } { 5 } m g d\).
  2. Use the work-energy principle to find \(v ^ { 2 }\) in terms of \(g\) and \(d\). When \(t = T\) seconds, \(d = 1.5 \mathrm {~m}\).
  3. Find the value of \(T\).
    DO NOT WIRITE IN THIS AREA
Edexcel M2 2016 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{790546bf-38a4-4eb7-876e-941fe58f4a48-07_671_661_239_635} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The uniform lamina \(A B C D E\) is made by joining a rectangular lamina \(A B D E\) to a triangular lamina \(B C D\) along the edge \(B D\). The rectangle has length \(6 a\) and width \(3 a\). The triangle is isosceles, with \(B C = C D\), and the distance from \(C\) to \(B D\) is \(3 a\), as shown in Figure 2.
  1. Find the distance of the centre of mass of the lamina, \(A B C D E\), from \(A E\). The lamina \(A B C D E\) is freely suspended from \(A\). A horizontal force of magnitude \(F\) newtons is applied to the lamina at \(D\). The line of action of the force lies in the vertical plane containing the lamina. The lamina is in equilibrium with \(A E\) vertical. The mass of the lamina is 4 kg .
  2. Find the magnitude of the force exerted on the lamina at \(A\).
    DO NOT WIRITE IN THIS AREA
Edexcel M2 2016 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{790546bf-38a4-4eb7-876e-941fe58f4a48-09_952_664_246_712} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform rod \(A B\) has mass 6 kg and length 2 m . The end \(A\) of the rod rests against a rough vertical wall. One end of a light string is attached to the rod at \(B\). The other end of the string is attached to the wall at \(C\), which is vertically above \(A\). The angle between the rod and the string is \(30 ^ { \circ }\) and the angle between the rod and the wall is \(70 ^ { \circ }\), as shown in Figure 3. The rod is in a vertical plane perpendicular to the wall and rests in limiting equilibrium. Find
  1. the tension in the string,
  2. the coefficient of friction between the rod and the wall,
  3. the direction of the force exerted on the rod by the wall at \(A\).
    DO NOT WIRITE IN THIS AREA
Edexcel M2 2016 June Q6
6. [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.] At \(t = 0\) a particle \(P\) is projected from a fixed point \(O\) with velocity ( \(7 \mathbf { i } + 7 \sqrt { 3 } \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves freely under gravity. The position vector of a point on the path of \(P\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { m }\) relative to \(O\).
  1. Show that $$y = \sqrt { 3 } x - \frac { g } { 98 } x ^ { 2 }$$
  2. Find the direction of motion of \(P\) when it passes through the point on the path where \(x = 20\) At time \(T\) seconds \(P\) passes through the point with position vector \(( 2 \lambda \mathbf { i } + \lambda \mathbf { j } ) \mathrm { m }\) where \(\lambda\) is a positive constant.
  3. Find the value of \(T\).
    DO NOT WIRITE IN THIS AREA
Edexcel M2 2016 June Q7
7. Two particles \(A\) and \(B\), of mass \(m\) and \(2 m\) respectively, are moving in the same direction along the same straight line on a smooth horizontal surface, with \(B\) in front of \(A\). Particle \(A\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and particle \(B\) has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Particle \(A\) collides directly with particle \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\). The direction of motion of both particles is not changed by the collision. Immediately after the collision, \(A\) has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) has speed \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Show that \(w = \frac { 23 } { 9 }\).
    2. Find the value of \(v\). When \(A\) and \(B\) collide they are 3 m from a smooth vertical wall which is perpendicular to their direction of motion. After the collision with \(A\), particle \(B\) hits the wall and rebounds. The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\). There is a second collision between \(A\) and \(B\) at a point \(d \mathrm {~m}\) from the wall.
  2. Find the value of \(d\).
Edexcel M2 2017 June Q1
  1. A particle of mass 4 kg is moving with velocity \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) when it receives an impulse of \(( 7 \mathbf { i } - 5 \mathbf { j } )\) Ns.
Find the speed of the particle immediately after receiving the impulse.
Edexcel M2 2017 June Q2
  1. A cyclist and his bicycle have a total mass of 75 kg . The cyclist is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 21 }\). The non-gravitational resistance to motion is modelled as a constant force of magnitude \(R\) newtons. The cyclist is working at a constant rate of 280 W and moving at a constant speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Find the value of \(R\).
    Later the cyclist cycles down the same road on the same bicycle. He is again working at a constant rate of 280 W and the resistance to motion is now modelled as a constant force of magnitude 60 N .
  2. Find the acceleration of the cyclist at the instant when his speed is \(3.5 \mathrm {~ms} ^ { - 1 }\).
Edexcel M2 2017 June Q3
3. A particle \(P\) moves along the \(x\)-axis. At time \(t = 0 , P\) passes through the origin with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction. The acceleration of \(P\) at time \(t\) seconds, where \(t \geqslant 0\), is \(( 4 t - 8 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the positive \(x\) direction.
    1. Show that \(P\) is instantaneously at rest when \(t = 1\)
    2. Find the other value of \(t\) for which \(P\) is instantaneously at rest.
  2. Find the total distance travelled by \(P\) in the interval \(1 \leqslant t \leqslant 4\)
Edexcel M2 2017 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{266c4f52-f35f-459c-9184-836b0f3baf5b-12_609_639_296_657} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform rod \(A B\) has mass 5 kg and length 4 m . The rod is held in a horizontal position by a light inextensible string. The end \(A\) of the rod rests against a rough vertical wall. One end of the string is attached to the rod at \(B\) and the other end is attached to the wall at a point \(D\). The point \(D\) is vertically above \(A\), with \(A D = 3 \mathrm {~m}\). A particle of mass 2 kg is attached to the rod at \(C\), where \(A C = 0.5 \mathrm {~m}\), as shown in Figure 1. The rod is in equilibrium in a vertical plane perpendicular to the wall. The coefficient of friction between the rod and the wall is \(\mu\). Find
  1. the tension in the string,
  2. the magnitude of the force exerted by the wall on the rod at \(A\),
  3. the range of possible values of \(\mu\).
Edexcel M2 2017 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{266c4f52-f35f-459c-9184-836b0f3baf5b-16_255_1242_301_360} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A smooth straight ramp is fixed to horizontal ground. The ramp has length 8 m and is inclined at \(30 ^ { \circ }\) to the ground, as shown in Figure 2. A particle \(P\) of mass 0.7 kg is projected from a point \(A\) at the bottom of the ramp, up a line of greatest slope of the ramp, with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As \(P\) reaches the point \(B\) at the top of the ramp, \(P\) has speed \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. By considering energy, find the value of \(u\). After leaving the ramp at \(B\), the particle \(P\) moves freely under gravity until it hits the ground at a point \(C\). Immediately before hitting the ground at \(C\), particle \(P\) is moving at \(\theta ^ { \circ }\) below the horizontal with speed \(w \mathrm {~ms} ^ { - 1 }\). Find
    1. the value of \(w\),
    2. the value of \(\theta\),
  3. the horizontal distance from \(B\) to \(C\).