Questions — Edexcel M1 (599 questions)

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Edexcel M1 2008 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7ba14d10-1b57-4930-8d65-f21088c5d513-06_305_607_246_701} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle \(P\) of mass 6 kg lies on the surface of a smooth plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The particle is held in equilibrium by a force of magnitude 49 N , acting at an angle \(\theta\) to the plane, as shown in Figure 1. The force acts in a vertical plane through a line of greatest slope of the plane.
  1. Show that \(\cos \theta = \frac { 3 } { 5 }\).
  2. Find the normal reaction between \(P\) and the plane. The direction of the force of magnitude 49 N is now changed. It is now applied horizontally to \(P\) so that \(P\) moves up the plane. The force again acts in a vertical plane through a line of greatest slope of the plane.
  3. Find the initial acceleration of \(P\). \(\_\_\_\_\)}
Edexcel M1 2008 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7ba14d10-1b57-4930-8d65-f21088c5d513-08_315_817_255_587} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A beam \(A B\) has mass 12 kg and length 5 m . It is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. One rope is attached to \(A\), the other to the point \(C\) on the beam, where \(B C = 1 \mathrm {~m}\), as shown in Figure 2. The beam is modelled as a uniform rod, and the ropes as light strings.
  1. Find
    1. the tension in the rope at \(C\),
    2. the tension in the rope at \(A\). A small load of mass 16 kg is attached to the beam at a point which is \(y\) metres from \(A\). The load is modelled as a particle. Given that the beam remains in equilibrium in a horizontal position,
  2. find, in terms of \(y\), an expression for the tension in the rope at \(C\). The rope at \(C\) will break if its tension exceeds 98 N. The rope at \(A\) cannot break.
  3. Find the range of possible positions on the beam where the load can be attached without the rope at \(C\) breaking.
Edexcel M1 2008 January Q6
6. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively.] A particle \(P\) is moving with constant velocity \(( - 5 \mathbf { i } + 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
  1. the speed of \(P\),
  2. the direction of motion of \(P\), giving your answer as a bearing. At time \(t = 0 , P\) is at the point \(A\) with position vector ( \(7 \mathbf { i } - 10 \mathbf { j }\) ) m relative to a fixed origin \(O\). When \(t = 3 \mathrm {~s}\), the velocity of \(P\) changes and it moves with velocity \(( u \mathbf { i } + v \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(u\) and \(v\) are constants. After a further 4 s , it passes through \(O\) and continues to move with velocity ( \(u \mathbf { i } + v \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  3. Find the values of \(u\) and \(v\).
  4. Find the total time taken for \(P\) to move from \(A\) to a position which is due south of A.
Edexcel M1 2008 January Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7ba14d10-1b57-4930-8d65-f21088c5d513-12_292_897_278_415} \captionsetup{labelformat=empty} \caption{Figure 3}
\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 horizontal table. The string passes over a small smooth pulley \(P\) fixed on the edge of the table. The particle \(B\) hangs freely below the pulley, as shown in Figure 3. The coefficient of friction between \(A\) and the table is \(\mu\). The particles are released from rest with the string taut. Immediately after release, the magnitude of the acceleration of \(A\) and \(B\) is \(\frac { 4 } { 9 } g\). By writing down separate equations of motion for \(A\) and \(B\),
  1. find the tension in the string immediately after the particles begin to move,
  2. show that \(\mu = \frac { 2 } { 3 }\). When \(B\) has fallen a distance \(h\), it hits the ground and does not rebound. Particle \(A\) is then a distance \(\frac { 1 } { 3 } h\) from \(P\).
  3. Find the speed of \(A\) as it reaches \(P\).
  4. State how you have used the information that the string is light.
Edexcel M1 2009 January Q1
  1. A particle \(P\) moves with constant acceleration \(( 2 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). At time \(t = 0 , P\) has speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t = 3 \mathrm {~s} , P\) has velocity \(( - 6 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Find the value of \(u\).
(5)
Edexcel M1 2009 January Q2
2. A small ball is projected vertically upwards from ground level with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball takes 4 s to return to ground level.
  1. Draw, in the space below, a velocity-time graph to represent the motion of the ball during the first 4 s .
  2. The maximum height of the ball above the ground during the first 4 s is 19.6 m . Find the value of \(u\).
Edexcel M1 2009 January Q3
3. Two particles \(A\) and \(B\) are moving on a smooth horizontal plane. The mass of \(A\) is \(k m\), where \(2 < k < 3\), and the mass of \(B\) is \(m\). The particles are moving along the same straight line, but in opposite directions, and they collide directly. Immediately before they collide the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(4 u\). As a result of the collision the speed of \(A\) is halved and its direction of motion is reversed.
  1. Find, in terms of \(k\) and \(u\), the speed of \(B\) immediately after the collision.
  2. State whether the direction of motion of \(B\) changes as a result of the collision, explaining your answer. Given that \(k = \frac { 7 } { 3 }\),
  3. find, in terms of \(m\) and \(u\), the magnitude of the impulse that \(A\) exerts on \(B\) in the collision.
Edexcel M1 2009 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{86bb11a4-b409-49b1-bffb-d0e3727d345c-05_349_869_303_532} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A bench consists of a plank which is resting in a horizontal position on two thin vertical legs. The plank is modelled as a uniform rod \(P S\) of length 2.4 m and mass 20 kg . The legs at \(Q\) and \(R\) are 0.4 m from each end of the plank, as shown in Figure 1. Two pupils, Arthur and Beatrice, sit on the plank. Arthur has mass 60 kg and sits at the middle of the plank and Beatrice has mass 40 kg and sits at the end \(P\). The plank remains horizontal and in equilibrium. By modelling the pupils as particles, find
  1. the magnitude of the normal reaction between the plank and the leg at \(Q\) and the magnitude of the normal reaction between the plank and the leg at \(R\). Beatrice stays sitting at \(P\) but Arthur now moves and sits on the plank at the point \(X\). Given that the plank remains horizontal and in equilibrium, and that the magnitude of the normal reaction between the plank and the leg at \(Q\) is now twice the magnitude of the normal reaction between the plank and the leg at \(R\),
  2. find the distance \(Q X\).
Edexcel M1 2009 January Q5
5.
\includegraphics[max width=\textwidth, alt={}]{86bb11a4-b409-49b1-bffb-d0e3727d345c-07_352_834_300_551}
\section*{Figure 2} A small package of mass 1.1 kg is held in equilibrium on a rough plane by a horizontal force. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The force acts in a vertical plane containing a line of greatest slope of the plane and has magnitude \(P\) newtons, as shown in Figure 2. The coefficient of friction between the package and the plane is 0.5 and the package is modelled as a particle. The package is in equilibrium and on the point of slipping down the plane.
  1. Draw, on Figure 2, all the forces acting on the package, showing their directions clearly.
    1. Find the magnitude of the normal reaction between the package and the plane.
    2. Find the value of \(P\).
Edexcel M1 2009 January Q6
6. Two forces, \(( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }\) and \(( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }\), act on a particle \(P\) of mass \(m \mathrm {~kg}\). The resultant of the two forces is \(\mathbf { R }\). Given that \(\mathbf { R }\) acts in a direction which is parallel to the vector ( \(\mathbf { i } - 2 \mathbf { j }\) ),
  1. find the angle between \(\mathbf { R }\) and the vector \(\mathbf { j }\),
  2. show that \(2 p + q + 3 = 0\). Given also that \(q = 1\) and that \(P\) moves with an acceleration of magnitude \(8 \sqrt { } 5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), (c) find the value of \(m\).
Edexcel M1 2009 January Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{86bb11a4-b409-49b1-bffb-d0e3727d345c-11_495_892_301_523} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} One end of a light inextensible string is attached to a block \(P\) of mass 5 kg . The block \(P\) is held at rest on a smooth fixed plane which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 3 } { 5 }\). The string lies along a line of greatest slope of the plane and passes over a smooth light pulley which is fixed at the top of the plane. The other end of the string is attached to a light scale pan which carries two blocks \(Q\) and \(R\), with block \(Q\) on top of block \(R\), as shown in Figure 3. The mass of block \(Q\) is 5 kg and the mass of block \(R\) is 10 kg . The scale pan hangs at rest and the system is released from rest. By modelling the blocks as particles, ignoring air resistance and assuming the motion is uninterrupted, find
    1. the acceleration of the scale pan,
    2. the tension in the string,
  2. the magnitude of the force exerted on block \(Q\) by block \(R\),
  3. the magnitude of the force exerted on the pulley by the string.
Edexcel M1 2013 January Q1
  1. Two particles \(P\) and \(Q\) have masses \(4 m\) and \(m\) respectively. The particles are moving towards each other on a smooth horizontal plane and collide directly. The speeds of \(P\) and \(Q\) immediately before the collision are \(2 u\) and \(5 u\) respectively. Immediately after the collision, the speed of \(P\) is \(\frac { 1 } { 2 } u\) and its direction of motion is reversed.
    1. Find the speed and direction of motion of \(Q\) after the collision.
    2. Find the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision.
    3. A steel girder \(A B\), of mass 200 kg and length 12 m , rests horizontally in equilibrium on two smooth supports at \(C\) and at \(D\), where \(A C = 2 \mathrm {~m}\) and \(D B = 2 \mathrm {~m}\). A man of mass 80 kg stands on the girder at the point \(P\), where \(A P = 4 \mathrm {~m}\), as shown in Figure 1.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6e887c0f-911a-4f18-81a1-2f90025f5410-03_318_1061_420_440} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The man is modelled as a particle and the girder is modelled as a uniform rod.
  2. Find the magnitude of the reaction on the girder at the support at \(C\). The support at \(D\) is now moved to the point \(X\) on the girder, where \(X B = x\) metres. The man remains on the girder at \(P\), as shown in Figure 2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6e887c0f-911a-4f18-81a1-2f90025f5410-03_324_1058_1292_443} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Given that the magnitudes of the reactions at the two supports are now equal and that the girder again rests horizontally in equilibrium, find
  3. the magnitude of the reaction at the support at \(X\),
  4. the value of \(x\).
Edexcel M1 2013 January Q3
3. A particle \(P\) of mass 2 kg is attached to one end of a light string, the other end of which is attached to a fixed point \(O\). The particle is held in equilibrium, with \(O P\) at \(30 ^ { \circ }\) to the downward vertical, by a force of magnitude \(F\) newtons. The force acts in the same vertical plane as the string and acts at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Figure 3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6e887c0f-911a-4f18-81a1-2f90025f5410-05_444_510_477_699} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Find
  1. the value of \(F\),
  2. the tension in the string.
Edexcel M1 2013 January Q4
4. A lifeboat slides down a straight ramp inclined at an angle of \(15 ^ { \circ }\) to the horizontal. The lifeboat has mass 800 kg and the length of the ramp is 50 m . The lifeboat is released from rest at the top of the ramp and is moving with a speed of \(12.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it reaches the end of the ramp. By modelling the lifeboat as a particle and the ramp as a rough inclined plane, find the coefficient of friction between the lifeboat and the ramp.
(9)
Edexcel M1 2013 January Q5
5．\(v \left( \mathrm {~m} \mathrm {~s} ^ { - 1 } \right) ^ { \wedge }\) Figure 4 The velocity－time graph in Figure 4 represents the journey of a train \(P\) travelling along a straight horizontal track between two stations which are 1.5 km apart．The train \(P\) leaves the first station，accelerating uniformly from rest for 300 m until it reaches a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) ．The train then maintains this speed for \(T\) seconds before decelerating uniformly at \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) ，coming to rest at the next station．
（a）Find the acceleration of \(P\) during the first 300 m of its journey．
（b）Find the value of \(T\) ． A second train \(Q\) completes the same journey in the same total time．The train leaves the first station，accelerating uniformly from rest until it reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then immediately decelerates uniformly until it comes to rest at the next station．
（c）Sketch on the diagram above，a velocity－time graph which represents the journey of train \(Q\) ．
（d）Find the value of \(V\) ．
Edexcel M1 2013 January Q6
6. [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship sets sail at 9 am from a port \(P\) and moves with constant velocity. The position vector of \(P\) is \(( 4 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km }\). At 9.30 am the ship is at the point with position vector \(( \mathbf { i } - 4 \mathbf { j } ) \mathrm { km }\).
  1. Find the speed of the ship in \(\mathrm { km } \mathrm { h } ^ { - 1 }\).
  2. Show that the position vector \(\mathbf { r } \mathrm { km }\) of the ship, \(t\) hours after 9 am , is given by \(\mathbf { r } = ( 4 - 6 t ) \mathbf { i } + ( 8 t - 8 ) \mathbf { j }\). At 10 am , a passenger on the ship observes that a lighthouse \(L\) is due west of the ship. At 10.30 am , the passenger observes that \(L\) is now south-west of the ship.
  3. Find the position vector of \(L\).
Edexcel M1 2013 January Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6e887c0f-911a-4f18-81a1-2f90025f5410-12_337_1084_228_429} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows two particles \(A\) and \(B\), of mass \(2 m\) and \(4 m\) respectively, connected by a light inextensible string. Initially \(A\) is held at rest on a rough inclined plane which is fixed to horizontal ground. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 4 }\). The string passes over a small smooth pulley \(P\) which is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane and \(B\) hangs vertically below \(P\). The system is released from rest with the string taut, with \(A\) at the point \(X\) and with \(B\) at a height \(h\) above the ground. For the motion until \(B\) hits the ground,
  1. give a reason why the magnitudes of the accelerations of the two particles are the same,
  2. write down an equation of motion for each particle,
  3. find the acceleration of each particle. Particle \(B\) does not rebound when it hits the ground and \(A\) continues moving up the plane towards \(P\). Given that \(A\) comes to rest at the point \(Y\), without reaching \(P\),
  4. find the distance \(X Y\) in terms of \(h\).
Edexcel M1 Specimen Q1
  1. A particle \(P\) is moving with constant velocity \(( - 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At time \(t = 6 \mathrm {~s} P\) is at the point with position vector \(( - 4 \mathbf { i } - 7 \mathbf { j } ) \mathrm { m }\). Find the distance of \(P\) from the origin at time \(t = 2 \mathrm {~s}\).
Edexcel M1 Specimen Q2
2. Particle \(P\) has mass \(m \mathrm {~kg}\) and particle \(Q\) has mass \(3 m \mathrm {~kg}\). The particles are moving in opposite directions along a smooth horizontal plane when they collide directly. Immediately before the collision \(P\) has speed \(4 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) has speed \(k u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(k\) is a constant. As a result of the collision the direction of motion of each particle is reversed and the speed of each particle is halved.
  1. Find the value of \(k\).
  2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(P\) by \(Q\).
Edexcel M1 Specimen Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ab3c0d61-3cab-4050-8288-6052e8404eb1-06_182_872_310_543} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A small box is pushed along a floor. The floor is modelled as a rough horizontal plane and the box is modelled as a particle. The coefficient of friction between the box and the floor is \(\frac { 1 } { 2 }\). The box is pushed by a force of magnitude 100 N which acts at an angle of \(30 ^ { \circ }\) with the floor, as shown in Figure 1. Given that the box moves with constant speed, find the mass of the box.
Edexcel M1 Specimen Q6
  1. A ball is projected vertically upwards with a speed of \(14.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point which is 49 m above horizontal ground. Modelling the ball as a particle moving freely under gravity, find
    1. the greatest height, above the ground, reached by the ball,
    2. the speed with which the ball first strikes the ground,
    3. the total time from when the ball is projected to when it first strikes the ground.
Edexcel M1 Specimen Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ab3c0d61-3cab-4050-8288-6052e8404eb1-20_264_684_319_639} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle of mass 0.4 kg is held at rest on a fixed rough plane by a horizontal force of magnitude \(P\) newtons. The force acts in the vertical plane containing the line of greatest slope of the inclined plane which passes through the particle. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Figure 2. The coefficient of friction between the particle and the plane is \(\frac { 1 } { 3 }\).
Given that the particle is on the point of sliding up the plane, find
  1. the magnitude of the normal reaction between the particle and the plane,
  2. the value of \(P\).
Edexcel M1 Specimen Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ab3c0d61-3cab-4050-8288-6052e8404eb1-24_862_412_310_774} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Two particles \(A\) and \(B\) have mass 0.4 kg and 0.3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed above a horizontal floor. Both particles are held, with the string taut, at a height of 1 m above the floor, as shown in Figure 3. The particles are released from rest and in the subsequent motion \(B\) does not reach the pulley.
  1. Find the tension in the string immediately after the particles are released.
  2. Find the acceleration of \(A\) immediately after the particles are released. When the particles have been moving for 0.5 s , the string breaks.
  3. Find the further time that elapses until \(B\) hits the floor.
Edexcel M1 Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a9f91ceb-662a-40cd-956b-815052b8f1a0-01_173_520_360_1891} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A uniform \(\operatorname { rod } A B\) has weight 70 N and length 3 m . It rests in a horizontal position on two smooth supports placed at \(P\) and \(Q\), where \(A P = 0.5 \mathrm {~m}\), as shown in Fig. 1 . The reaction on the rod at \(P\) has magnitude 20 N . Find
  1. the magnitude of the reaction on the rod at \(Q\),
  2. the distance \(A Q\).
    . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a9f91ceb-662a-40cd-956b-815052b8f1a0-01_190_476_964_1905} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} A particle \(P\) of mass 2 kg is held in equilibrium under gravity by two light inextensible strings. One string is horizontal and the other is inclined at an angle \(\alpha\) to the horizontal, as shown in Fig. 2. The tension in the horizontal string is 15 N . The tension in the other string is \(T\) newtons.
  3. Find the size of the angle \(\alpha\).
    (6 marks)
  4. Find the value of \(T\). You must ensure that your answers to parts of questions are clearly labelled.
    You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
Edexcel M1 Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a9f91ceb-662a-40cd-956b-815052b8f1a0-02_280_428_340_516} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Two particles \(A\) and \(B\) have masses \(3 m\) and \(k m\) respectively, where \(k > 3\). They are connected by a light inextensible string which passes over a smooth fixed pulley. The system is released from rest with the string taut and the hanging parts of the string vertical, as shown in Fig. 3. While the particles are moving freely, \(A\) has an acceleration of magnitude \(\frac { 2 } { 5 } g\).
  1. Find, in terms of \(m\) and \(g\), the tension in the string.
    (3 marks)
  2. State why \(B\) also has an acceleration of magnitude \(\frac { 2 } { 5 } g\).
  3. Find the value of \(k\).
  4. State how you have used the fact that the string is light.
    (1 mark)