Questions — Edexcel M2 (623 questions)

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Edexcel M2 2013 January Q4
10 marks Moderate -0.3
4. At time \(t\) seconds the velocity of a particle \(P\) is \([ ( 4 t - 5 ) \mathbf { i } + 3 \mathbf { j } ] \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the position vector of \(P\) is \(( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\), relative to a fixed origin \(O\).
  1. Find the value of \(t\) when the velocity of \(P\) is parallel to the vector \(\mathbf { j }\).
  2. Find an expression for the position vector of \(P\) at time \(t\) seconds. A second particle \(Q\) moves with constant velocity \(( - 2 \mathbf { i } + c \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the position vector of \(Q\) is \(( 11 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m }\). The particles \(P\) and \(Q\) collide at the point with position vector ( \(d \mathbf { i } + 14 \mathbf { j }\) ) m.
  3. Find
    1. the value of \(c\),
    2. the value of \(d\).
Edexcel M2 2013 January Q5
11 marks Standard +0.3
5. The point \(A\) lies on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 24 } { 25 }\). A particle \(P\) is projected from \(A\), up a line of greatest slope of the plane, with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The mass of \(P\) is 2 kg and the coefficient of friction between \(P\) and the plane is \(\frac { 5 } { 12 }\). The particle comes to instantaneous rest at the point \(B\) on the plane, where \(A B = 1.5 \mathrm {~m}\). It then moves back down the plane to \(A\).
  1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
  2. Use the work-energy principle to find the value of \(U\).
  3. Find the speed of \(P\) when it returns to \(A\).
Edexcel M2 2013 January Q6
15 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ad18c22c-2fc5-4844-99b8-492f758bb24e-11_531_931_230_520} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A ball is thrown from a point \(O\), which is 6 m above horizontal ground. The ball is projected with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. There is a thin vertical post which is 4 m high and 8 m horizontally away from the vertical through \(O\), as shown in Figure 2. The ball passes just above the top of the post 2 s after projection. The ball is modelled as a particle.
  1. Show that \(\tan \theta = 2.2\)
  2. Find the value of \(u\). The ball hits the ground \(T\) seconds after projection.
  3. Find the value of \(T\). Immediately before the ball hits the ground the direction of motion of the ball makes an angle \(\alpha\) with the horizontal.
  4. Find \(\alpha\).
Edexcel M2 2013 January Q7
16 marks Challenging +1.2
7. A particle \(A\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal floor when it collides directly with another particle \(B\), of mass \(3 m\), which is at rest on the floor. The coefficient of restitution between the particles is \(e\). The direction of motion of \(A\) is reversed by the collision.
  1. Find, in terms of \(e\) and \(u\),
    1. the speed of \(A\) immediately after the collision,
    2. the speed of \(B\) immediately after the collision. After being struck by \(A\) the particle \(B\) collides directly with another particle \(C\), of mass \(4 m\), which is at rest on the floor. The coefficient of restitution between \(B\) and \(C\) is \(2 e\). Given that the direction of motion of \(B\) is reversed by this collision,
  2. find the range of possible values of \(e\),
  3. determine whether there will be a second collision between \(A\) and \(B\).
Edexcel M2 2004 June Q1
7 marks Moderate -0.3
  1. A lorry of mass 1500 kg moves along a straight horizontal road. The resistance to the motion of the lorry has magnitude 750 N and the lorry's engine is working at a rate of 36 kW .
    1. Find the acceleration of the lorry when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    The lorry comes to a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 10 }\). The magnitude of the resistance to motion from non-gravitational forces remains 750 N . The lorry moves up the hill at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the rate at which the lorry's engine is now working.
    (3)
Edexcel M2 2004 June Q2
9 marks Moderate -0.8
2. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.] A ball has mass 0.2 kg . It is moving with velocity ( 30 i ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it is struck by a bat. The bat exerts an impulse of \(( - 4 \mathbf { i } + 4 \mathbf { j } )\) Ns on the ball. Find
  1. the velocity of the ball immediately after the impact,
  2. the angle through which the ball is deflected as a result of the impact,
  3. the kinetic energy lost by the ball in the impact.
Edexcel M2 2004 June Q3
9 marks Standard +0.3
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{8e694174-b9a9-4018-8896-31a3b4f0d344-3_860_565_269_740}
\end{figure} Figure 1 shows a decoration which is made by cutting the shape of a simple tree from a sheet of uniform card. The decoration consists of a triangle \(A B C\) and a rectangle \(P Q R S\). The points \(P\) and \(S\) lie on \(B C\) and \(M\) is the mid-point of both \(B C\) and \(P S\). The triangle \(A B C\) is isosceles with \(A B = A C , B C = 4 \mathrm {~cm} , A M = 6 \mathrm {~cm} , P S = 2 \mathrm {~cm}\) and \(P Q = 3 \mathrm {~cm}\).
  1. Find the distance of the centre of mass of the decoration from \(B C\). The decoration is suspended from \(Q\) and hangs freely.
  2. Find, in degrees to one decimal place, the angle between \(P Q\) and the vertical.
Edexcel M2 2004 June Q4
10 marks Standard +0.3
4. At time \(t\) seconds, the velocity of a particle \(P\) is \([ ( 4 t - 7 ) \mathbf { i } - 5 \mathbf { j } ] \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , P\) is at the point with position vector \(( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\) relative to a fixed origin \(O\).
  1. Find an expression for the position vector of \(P\) after \(t\) seconds, giving your answer in the form \(( a \mathbf { i } + b \mathbf { j } ) \mathrm { m }\). A second particle \(Q\) moves with constant velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the position vector of \(Q\) is \(( - 7 \mathrm { i } ) \mathrm { m }\).
  2. Prove that \(P\) and \(Q\) collide.
Edexcel M2 2004 June Q5
11 marks Standard +0.8
5. Two small smooth spheres, \(P\) and \(Q\), of equal radius, have masses \(2 m\) and \(3 m\) respectively. The sphere \(P\) is moving with speed \(5 u\) on a smooth horizontal table when it collides directly with \(Q\), which is at rest on the table. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  1. Show that the speed of \(Q\) immediately after the collision is \(2 ( 1 + e ) u\). After the collision, \(Q\) hits a smooth vertical wall which is at the edge of the table and perpendicular to the direction of motion of \(Q\). The coefficient of restitution between \(Q\) and the wall is \(f , 0 < f \leqslant 1\).
  2. Show that, when \(e = 0.4\), there is a second collision between \(P\) and \(Q\). Given that \(e = 0.8\) and there is a second collision between \(P\) and \(Q\),
  3. find the range of possible values of \(f\).
Edexcel M2 2004 June Q6
12 marks Challenging +1.2
6. A uniform ladder \(A B\), of mass \(m\) and length \(2 a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is 0.6 . The other end \(B\) of the ladder rests against a smooth vertical wall. A builder of mass 10 m stands at the top of the ladder. To prevent the ladder from slipping, the builder's friend pushes the bottom of the ladder horizontally towards the wall with a force of magnitude \(P\). This force acts in a direction perpendicular to the wall. The ladder rests in equilibrium in a vertical plane perpendicular to the wall and makes an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac { 3 } { 2 }\).
  1. Show that the reaction of the wall on the ladder has magnitude 7 mg .
  2. Find, in terms of \(m\) and \(g\), the range of values of \(P\) for which the ladder remains in equilibrium.
Edexcel M2 2004 June Q7
17 marks Standard +0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{8e694174-b9a9-4018-8896-31a3b4f0d344-5_424_1324_264_383}
\end{figure} In a ski-jump competition, a skier of mass 80 kg moves from rest at a point \(A\) on a ski-slope. The skier's path is an arc \(A B\). The starting point \(A\) of the slope is 32.5 m above horizontal ground. The end \(B\) of the slope is 8.1 m above the ground. When the skier reaches \(B\), she is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and moving upwards at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Fig. 2. The distance along the slope from \(A\) to \(B\) is 60 m . The resistance to motion while she is on the slope is modelled as a force of constant magnitude \(R\) newtons. By using the work-energy principle,
  1. find the value of \(R\). On reaching \(B\), the skier then moves through the air and reaches the ground at the point \(C\). The motion of the skier in moving from \(B\) to \(C\) is modelled as that of a particle moving freely under gravity.
  2. Find the time for the skier to move from \(B\) to \(C\).
  3. Find the horizontal distance from \(B\) to \(C\).
  4. Find the speed of the skier immediately before she reaches \(C\). END
Edexcel M2 2007 June Q1
4 marks Moderate -0.3
  1. A cyclist and his bicycle have a combined mass of 90 kg . He rides on a straight road up a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 21 }\). He works at a constant rate of 444 W and cycles up the hill at a constant speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find the magnitude of the resistance to motion from non-gravitational forces as he cycles up the hill.
Edexcel M2 2007 June Q2
6 marks Moderate -0.8
2. A particle \(P\) of mass 0.5 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of \(P\) is given by $$\mathbf { v } = 3 t ^ { 2 } \mathbf { i } + ( 1 - 4 t ) \mathbf { j }$$ Find
  1. the acceleration of \(P\) at time \(t\) seconds,
  2. the magnitude of \(\mathbf { F }\) when \(t = 2\).
Edexcel M2 2007 June Q3
8 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{778a0276-6738-40e6-90b2-a536ce5abe6a-04_568_568_205_685} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform lamina \(A B C D E F\) is formed by taking a uniform sheet of card in the form of a square \(A X E F\), of side \(2 a\), and removing the square \(B X D C\) of side \(a\), where \(B\) and \(D\) are the mid-points of \(A X\) and \(X E\) respectively, as shown in Figure 1.
  1. Find the distance of the centre of mass of the lamina from \(A F\). The lamina is freely suspended from \(A\) and hangs in equilibrium.
  2. Find, in degrees to one decimal place, the angle which \(A F\) makes with the vertical.
Edexcel M2 2007 June Q4
7 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{778a0276-6738-40e6-90b2-a536ce5abe6a-06_330_1118_203_411} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Two particles \(A\) and \(B\), of mass \(m\) and \(2 m\) respectively, are attached to the ends of a light inextensible string. The particle \(A\) lies on a rough plane inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). The string passes over a small light smooth pulley \(P\) fixed at the top of the plane. The particle \(B\) hangs freely below \(P\), as shown in Figure 2. The particles are released from rest with the string taut and the section of the string from \(A\) to \(P\) parallel to a line of greatest slope of the plane. The coefficient of friction between \(A\) and the plane is \(\frac { 5 } { 8 }\). When each particle has moved a distance \(h , B\) has not reached the ground and \(A\) has not reached \(P\).
  1. Find an expression for the potential energy lost by the system when each particle has moved a distance \(h\). When each particle has moved a distance \(h\), they are moving with speed \(v\). Using the workenergy principle,
  2. find an expression for \(v ^ { 2 }\), giving your answer in the form \(k g h\), where \(k\) is a number.
Edexcel M2 2007 June Q5
9 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{778a0276-6738-40e6-90b2-a536ce5abe6a-08_376_874_205_525} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform beam \(A B\) of mass 2 kg is freely hinged at one end \(A\) to a vertical wall. The beam is held in equilibrium in a horizontal position by a rope which is attached to a point \(C\) on the beam, where \(A C = 0.14 \mathrm {~m}\). The rope is attached to the point \(D\) on the wall vertically above \(A\), where \(\angle A C D = 30 ^ { \circ }\), as shown in Figure 3. The beam is modelled as a uniform rod and the rope as a light inextensible string. The tension in the rope is 63 N . Find
  1. the length of \(A B\),
  2. the magnitude of the resultant reaction of the hinge on the beam at \(A\).
Edexcel M2 2007 June Q6
12 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{778a0276-6738-40e6-90b2-a536ce5abe6a-10_447_908_205_516} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A golf ball \(P\) is projected with speed \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) on a cliff above horizontal ground. The angle of projection is \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 4 } { 3 }\). The ball moves freely under gravity and hits the ground at the point \(B\), as shown in Figure 4.
  1. Find the greatest height of \(P\) above the level of \(A\). The horizontal distance from \(A\) to \(B\) is 168 m .
  2. Find the height of \(A\) above the ground. By considering energy, or otherwise,
  3. find the speed of \(P\) as it hits the ground at \(B\).
Edexcel M2 2007 June Q7
13 marks Standard +0.8
  1. Two small spheres \(P\) and \(Q\) of equal radius have masses \(m\) and \(5 m\) respectively. They lie on a smooth horizontal table. Sphere \(P\) is moving with speed \(u\) when it collides directly with sphere \(Q\) which is at rest. The coefficient of restitution between the spheres is \(e\), where \(e > \frac { 1 } { 5 }\).
    1. (i) Show that the speed of \(P\) immediately after the collision is \(\frac { u } { 6 } ( 5 e - 1 )\).
      (ii) Find an expression for the speed of \(Q\) immediately after the collision, giving your answer in the form \(\lambda u\), where \(\lambda\) is in terms of \(e\).
      (6)
    Three small spheres \(A , B\) and \(C\) of equal radius lie at rest in a straight line on a smooth horizontal table, with \(B\) between \(A\) and \(C\). The spheres \(A\) and \(C\) each have mass \(5 m\), and the mass of \(B\) is \(m\). Sphere \(B\) is projected towards \(C\) with speed \(u\). The coefficient of restitution between each pair of spheres is \(\frac { 4 } { 5 }\).
  2. Show that, after \(B\) and \(C\) have collided, there is a collision between \(B\) and \(A\).
  3. Determine whether, after \(B\) and \(A\) have collided, there is a further collision between \(B\) and \(C\).
Edexcel M2 2007 June Q8
16 marks Standard +0.3
  1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where \(v\) is given by
$$v = \left\{ \begin{array} { l c } 8 t - \frac { 3 } { 2 } t ^ { 2 } , & 0 \leqslant t \leqslant 4 , \\ 16 - 2 t , & t > 4 . \end{array} \right.$$ When \(t = 0 , P\) is at the origin \(O\).
Find
  1. the greatest speed of \(P\) in the interval \(0 \leqslant t \leqslant 4\),
  2. the distance of \(P\) from \(O\) when \(t = 4\),
  3. the time at which \(P\) is instantaneously at rest for \(t > 4\),
  4. the total distance travelled by \(P\) in the first 10 s of its motion.
Edexcel M2 2008 June Q1
6 marks Moderate -0.8
  1. A lorry of mass 2000 kg is moving down a straight road inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 25 }\). The resistance to motion is modelled as a constant force of magnitude 1600 N . The lorry is moving at a constant speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find, in kW , the rate at which the lorry's engine is working.
Edexcel M2 2008 June Q2
9 marks Standard +0.3
2. A particle \(A\) of mass \(4 m\) is moving with speed \(3 u\) in a straight line on a smooth horizontal table. The particle \(A\) collides directly with a particle \(B\) of mass \(3 m\) moving with speed \(2 u\) in the same direction as \(A\). The coefficient of restitution between \(A\) and \(B\) is \(e\). Immediately after the collision the speed of \(B\) is \(4 e u\).
  1. Show that \(e = \frac { 3 } { 4 }\).
  2. Find the total kinetic energy lost in the collision.
Edexcel M2 2008 June Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2738ce4-4dc5-4cd1-ac3d-0c3fcf21ea71-04_511_922_260_511} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A package of mass 3.5 kg is sliding down a ramp. The package is modelled as a particle and the ramp as a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The package slides down a line of greatest slope of the plane from a point \(A\) to a point \(B\), where \(A B = 14 \mathrm {~m}\). At \(A\) the package has speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at \(B\) the package has speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Figure 1. Find
  1. the total energy lost by the package in travelling from \(A\) to \(B\),
  2. the coefficient of friction between the package and the ramp.
Edexcel M2 2008 June Q4
12 marks Standard +0.3
  1. A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds,
$$\mathbf { F } = ( 6 t - 5 ) \mathbf { i } + \left( t ^ { 2 } - 2 t \right) \mathbf { j }$$ The velocity of \(P\) at time \(t\) seconds is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , \mathbf { v } = \mathbf { i } - 4 \mathbf { j }\).
  1. Find \(\mathbf { v }\) at time \(t\) seconds. When \(t = 3\), the particle \(P\) receives an impulse ( \(- 5 \mathbf { i } + 12 \mathbf { j }\) ) N s.
  2. Find the speed of \(P\) immediately after it receives the impulse.
Edexcel M2 2008 June Q5
11 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2738ce4-4dc5-4cd1-ac3d-0c3fcf21ea71-07_501_918_274_502} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A plank rests in equilibrium against a fixed horizontal pole. The plank is modelled as a uniform rod \(A B\) and the pole as a smooth horizontal peg perpendicular to the vertical plane containing \(A B\). The rod has length \(3 a\) and weight \(W\) and rests on the peg at \(C\), where \(A C = 2 a\). The end \(A\) of the rod rests on rough horizontal ground and \(A B\) makes an angle \(\alpha\) with the ground, as shown in Figure 2.
  1. Show that the normal reaction on the rod at \(A\) is \(\frac { 1 } { 4 } \left( 4 - 3 \cos ^ { 2 } \alpha \right) W\). Given that the rod is in limiting equilibrium and that \(\cos \alpha = \frac { 2 } { 3 }\),
  2. find the coefficient of friction between the rod and the ground.
Edexcel M2 2008 June Q6
13 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a2738ce4-4dc5-4cd1-ac3d-0c3fcf21ea71-09_600_968_292_486} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a rectangular lamina \(O A B C\). The coordinates of \(O , A , B\) and \(C\) are ( 0,0 ), \(( 8,0 ) , ( 8,5 )\) and \(( 0,5 )\) respectively. Particles of mass \(k m , 5 m\) and \(3 m\) are attached to the lamina at \(A , B\) and \(C\) respectively. The \(x\)-coordinate of the centre of mass of the three particles without the lamina is 6.4.
  1. Show that \(k = 7\). The lamina \(O A B C\) is uniform and has mass \(12 m\).
  2. Find the coordinates of the centre of mass of the combined system consisting of the three particles and the lamina. The combined system is freely suspended from \(O\) and hangs at rest.
  3. Find the angle between \(O C\) and the horizontal.