Questions M1 (1912 questions)

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Edexcel M1 2012 January Q3
8 marks Moderate -0.8
3. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) acting on a particle \(P\) are given by $$\begin{aligned} & \mathbf { F } _ { 1 } = ( 7 \mathbf { i } - 9 \mathbf { j } ) \mathrm { N } \\ & \mathbf { F } _ { 2 } = ( 5 \mathbf { i } + 6 \mathbf { j } ) \mathrm { N } \\ & \mathbf { F } _ { 3 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N } \end{aligned}$$ where \(p\) and \(q\) are constants.
Given that \(P\) is in equilibrium,
  1. find the value of \(p\) and the value of \(q\). The force \(\mathbf { F } _ { 3 }\) is now removed. The resultant of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) is \(\mathbf { R }\). Find
  2. the magnitude of \(\mathbf { R }\),
  3. the angle, to the nearest degree, that the direction of \(\mathbf { R }\) makes with \(\mathbf { j }\).
Edexcel M1 2012 January Q4
9 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{724254f3-3a6a-4820-b3a1-979458e24437-05_241_794_219_575} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A non-uniform \(\operatorname { rod } A B\), of mass \(m\) and length \(5 d\), rests horizontally in equilibrium on two supports at \(C\) and \(D\), where \(A C = D B = d\), as shown in Figure 1. The centre of mass of the rod is at the point \(G\). A particle of mass \(\frac { 5 } { 2 } m\) is placed on the rod at \(B\) and the rod is on the point of tipping about \(D\).
  1. Show that \(G D = \frac { 5 } { 2 } d\). The particle is moved from \(B\) to the mid-point of the rod and the rod remains in equilibrium.
  2. Find the magnitude of the normal reaction between the support at \(D\) and the rod.
Edexcel M1 2012 January Q5
11 marks Moderate -0.8
  1. A stone is projected vertically upwards from a point \(A\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After projection the stone moves freely under gravity until it returns to \(A\). The time between the instant that the stone is projected and the instant that it returns to \(A\) is \(3 \frac { 4 } { 7 }\) seconds.
Modelling the stone as a particle,
  1. show that \(u = 17 \frac { 1 } { 2 }\),
  2. find the greatest height above \(A\) reached by the stone,
  3. find the length of time for which the stone is at least \(6 \frac { 3 } { 5 } \mathrm {~m}\) above \(A\).
Edexcel M1 2012 January Q6
13 marks Moderate -0.3
  1. A car moves along a straight horizontal road from a point \(A\) to a point \(B\), where \(A B = 885 \mathrm {~m}\). The car accelerates from rest at \(A\) to a speed of \(15 \mathrm {~ms} ^ { - 1 }\) at a constant rate \(a \mathrm {~ms} ^ { - 2 }\). The time for which the car accelerates is \(\frac { 1 } { 3 } T\) seconds. The car maintains the speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(T\) seconds. The car then decelerates at a constant rate of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) stopping at \(B\).
    1. Find the time for which the car decelerates.
    2. Sketch a speed-time graph for the motion of the car.
    3. Find the value of \(T\).
    4. Find the value of \(a\).
    5. Sketch an acceleration-time graph for the motion of the car.
Edexcel M1 2012 January Q7
9 marks Moderate -0.8
7. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively. Position vectors are relative to a fixed origin \(O\).] A boat \(P\) is moving with constant velocity \(( - 4 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
  1. Calculate the speed of \(P\). When \(t = 0\), the boat \(P\) has position vector \(( 2 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km }\). At time \(t\) hours, the position vector of \(P\) is \(\mathbf { p ~ k m }\).
  2. Write down \(\mathbf { p }\) in terms of \(t\). A second boat \(Q\) is also moving with constant velocity. At time \(t\) hours, the position vector of \(Q\) is \(\mathbf { q } \mathrm { km }\), where $$\mathbf { q } = 18 \mathbf { i } + 12 \mathbf { j } - t ( 6 \mathbf { i } + 8 \mathbf { j } )$$ Find
  3. the value of \(t\) when \(P\) is due west of \(Q\),
  4. the distance between \(P\) and \(Q\) when \(P\) is due west of \(Q\).
Edexcel M1 2012 January Q8
14 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{724254f3-3a6a-4820-b3a1-979458e24437-13_334_538_219_703} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of mass 4 kg is moving up a fixed rough plane at a constant speed of \(16 \mathrm {~ms} ^ { - 1 }\) under the action of a force of magnitude 36 N . The plane is inclined at \(30 ^ { \circ }\) to the horizontal. The force acts in the vertical plane containing the line of greatest slope of the plane through \(P\), and acts at \(30 ^ { \circ }\) to the inclined plane, as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\mu\). Find
  1. the magnitude of the normal reaction between \(P\) and the plane,
  2. the value of \(\mu\). The force of magnitude 36 N is removed.
  3. Find the distance that \(P\) travels between the instant when the force is removed and the instant when it comes to rest.
Edexcel M1 2001 June Q1
6 marks Moderate -0.8
  1. Two small balls \(A\) and \(B\) have masses 0.5 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a smooth horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The speed of \(A\) immediately after the collision is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The direction of the motion of \(A\) is unchanged as a result of the collision.
By modelling the balls as particles, find
  1. the speed of \(B\) immediately after the collision,
  2. the magnitude of the impulse exerted on \(B\) in the collision.
Edexcel M1 2001 June Q2
8 marks Moderate -0.8
2. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{218383c1-0875-46f2-9416-8e827065a7a6-2_272_592_1239_648}
\end{figure} Two forces \(\mathbf { P }\) and \(\mathbf { Q }\), act on a particle. The force \(\mathbf { P }\) has magnitude 5 N and the force \(\mathbf { Q }\) has magnitude 3 N . The angle between the directions of \(\mathbf { P }\) and \(\mathbf { Q }\) is \(40 ^ { \circ }\), as shown in Fig. 1. The resultant of \(\mathbf { P }\) and \(\mathbf { Q }\) is \(\mathbf { F }\).
  1. Find, to 3 significant figures, the magnitude of \(\mathbf { F }\).
  2. Find, in degrees to 1 decimal place, the angle between the directions of \(\mathbf { F }\) and \(\mathbf { P }\).
Edexcel M1 2001 June Q3
9 marks Moderate -0.8
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{218383c1-0875-46f2-9416-8e827065a7a6-3_540_1223_348_455}
\end{figure} A car of mass 1200 kg moves along a straight horizontal road. In order to obey a speed restriction, the brakes of the car are applied for 3 s , reducing the car's speed from \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The brakes are then released and the car continues at a constant speed of \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for a further 4 s . Figure 2 shows a sketch of the speed-time graph of the car during the 7 s interval. The graph consists of two straight line segments.
  1. Find the total distance moved by the car during this 7 s interval.
  2. Explain briefly how the speed-time graph shows that, when the brakes are applied, the car experiences a constant retarding force.
  3. Find the magnitude of this retarding force.
Edexcel M1 2001 June Q4
11 marks Standard +0.3
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{218383c1-0875-46f2-9416-8e827065a7a6-4_347_854_356_640}
\end{figure} A small parcel of mass 3 kg is held in equilibrium on a rough plane by the action of a horizontal force of magnitude 30 N acting in a vertical plane through a line of greatest slope. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Fig. 3. The parcel is modelled as a particle. The parcel is on the point of moving up the slope.
  1. Draw a diagram showing all the forces acting on the parcel.
  2. Find the normal reaction on the parcel.
  3. Find the coefficient of friction between the parcel and the plane.
Edexcel M1 2001 June Q5
13 marks Standard +0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{218383c1-0875-46f2-9416-8e827065a7a6-5_328_993_491_483}
\end{figure} A large \(\log A B\) is 6 m long. It rests in a horizontal position on two smooth supports \(C\) and \(D\), where \(A C = 1 \mathrm {~m}\) and \(B D = 1 \mathrm {~m}\), as shown in Figure 4. David needs an estimate of the weight of the log, but the log is too heavy to lift off both supports. When David applies a force of magnitude 1500 N vertically upwards to the \(\log\) at \(A\), the \(\log\) is about to tilt about \(D\).
  1. State the value of the reaction on the \(\log\) at \(C\) for this case. David initially models the log as uniform rod. Using this model,
  2. estimate the weight of the log The shape of the log convinces David that his initial modelling assumption is too simple. He removes the force at \(A\) and applies a force acting vertically upwards at \(B\). He finds that the log is about to tilt about \(C\) when this force has magnitude 1000 N. David now models the log as a non-uniform rod, with the distance of the centre of mass of the \(\log\) from \(C\) as \(x\) metres. Using this model, find
  3. a new estimate for the weight of the log,
  4. the value of \(x\).
  5. State how you have used the modeling assumption that the log is a rod.
Edexcel M1 2001 June Q6
13 marks Moderate -0.3
6. A breakdown van of mass 2000 kg is towing a car of mass 1200 kg along a straight horizontal road. The two vehicles are joined by a tow bar which remains parallel to the road. The van and the car experience constant resistances to motion of magnitudes 800 N and 240 N respectively. There is a constant driving force acting on the van of 2320 N . Find
  1. the magnitude of the acceleration of the van and the car,
  2. the tension in the tow bar. The two vehicles come to a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\). The driving force and the resistances to the motion are unchanged.
  3. Find the magnitude of the acceleration of the van and the car as they move up the hill and state whether their speed increases or decreases.
Edexcel M1 2001 June Q7
15 marks Standard +0.3
7. [In this question, the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and north respectively] A mountain rescue post \(O\) receives a distress call via a mobile phone from a walker who has broken a leg and cannot move. The walker says he is by a pipeline and he can also see a radio mast which he believes to be south-west of him. The pipeline is known to run north-south for a long distance through the point with position vector \(6 \mathbf { i } \mathrm {~km}\), relative to \(O\). The radio mast is known to be at the point with position vector \(2 \mathbf { j } \mathrm {~km}\), relative to \(O\).
  1. Using the information supplied by the walker, write down his position vector in the form \(( a \mathbf { i } + b \mathbf { j } )\). The rescue party moves at a horizontal speed of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). The leader of the party wants to give the walker and idea of how long it will take to for the rescue party to arrive.
  2. Calculate how long it will take for the rescue party to reach the walker's estimated position. The rescue party sets out and walks straight towards the walker's estimated position at a constant horizontal speed of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). After the party has travelled for one hour, the walker rings again. He is very apologetic and says that he now realises that the radio mask is in fact north-west of his position
  3. Find the position vector of the walker.
  4. Find in degrees to one decimal place, the bearing on which the rescue party should now travel in order to reach the walker directly. \section*{END}
Edexcel M1 2003 June Q1
6 marks Moderate -0.3
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d91990b5-b7ea-485c-aa4e-fe42b61ca7f8-2_302_807_379_603}
\end{figure} A uniform plank \(A B\) has mass 40 kg and length 4 m . It is supported in a horizontal position by two smooth pivots, one at the end \(A\), the other at the point \(C\) of the plank where \(A C = 3 \mathrm {~m}\), as shown in Fig. 1. A man of mass 80 kg stands on the plank which remains in equilibrium. The magnitudes of the reactions at the two pivots are each equal to \(R\) newtons. By modelling the plank as a rod and the man as a particle, find
  1. the value of \(R\),
  2. the distance of the man from \(A\).
    (4)
Edexcel M1 2003 June Q2
7 marks Easy -1.3
2. Two particles \(A\) and \(B\) have mass 0.12 kg and 0.08 kg respectively. They are initially at rest on a smooth horizontal table. Particle \(A\) is then given an impulse in the direction \(A B\) so that it moves with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) directly towards \(B\).
  1. Find the magnitude of this impulse, stating clearly the units in which your answer is given.
    (2) Immediately after the particles collide, the speed of \(A\) is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its direction of motion being unchanged.
  2. Find the speed of \(B\) immediately after the collision.
  3. Find the magnitude of the impulse exerted on \(A\) in the collision.
Edexcel M1 2003 June Q3
8 marks Moderate -0.8
3. A competitor makes a dive from a high springboard into a diving pool. She leaves the springboard vertically with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) upwards. When she leaves the springboard, she is 5 m above the surface of the pool. The diver is modelled as a particle moving vertically under gravity alone and it is assumed that she does not hit the springboard as she descends. Find
  1. her speed when she reaches the surface of the pool,
  2. the time taken to reach the surface of the pool.
  3. State two physical factors which have been ignored in the model.
Edexcel M1 2003 June Q4
8 marks Standard +0.3
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d91990b5-b7ea-485c-aa4e-fe42b61ca7f8-3_355_759_1087_605}
\end{figure} A parcel of mass 5 kg lies on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The parcel is held in equilibrium by the action of a horizontal force of magnitude 20 N , as shown in Fig. 2. The force acts in a vertical plane through a line of greatest slope of the plane. The parcel is on the point of sliding down the plane. Find the coefficient of friction between the parcel and the plane.
(8)
Edexcel M1 2003 June Q5
10 marks Moderate -0.3
5. A particle \(P\) moves with constant acceleration \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). At time \(t\) seconds, its velocity is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , \mathbf { v } = - 2 \mathbf { i } + 7 \mathbf { j }\).
  1. Find the value of \(t\) when \(P\) is moving parallel to the vector \(\mathbf { i }\).
  2. Find the speed of \(P\) when \(t = 3\).
  3. Find the angle between the vector \(\mathbf { j }\) and the direction of motion of \(P\) when \(t = 3\).
Edexcel M1 2003 June Q6
11 marks Moderate -0.3
6. A particle \(P\) of mass 3 kg is projected up a line of greatest slope of a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4 . The initial speed of P is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the frictional force acting on \(P\) as it moves up the plane,
  2. the distance moved by \(P\) up the plane before \(P\) comes to instantaneous rest.
Edexcel M1 2003 June Q7
12 marks Standard +0.3
7. Two trains \(A\) and \(B\) run on parallel straight tracks. Initially both are at rest in a station and level with each other. At time \(t = 0 , A\) starts to move. It moves with constant acceleration for 12 s up to a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and then moves at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Train \(B\) starts to move in the same direction as \(A\) when \(t = 40\), where \(t\) is measured in seconds. It accelerates with the same initial acceleration as \(A\), up to a speed of \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then moves at a constant speed of \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Train \(B\) overtakes \(A\) after both trains have reached their maximum speed. Train \(B\) overtakes \(A\) when \(t = T\).
  1. Sketch, on the same diagram, the speed-time graphs of both trains for \(0 \leq t \leq T\).
  2. Find the value of \(T\).
Edexcel M1 2003 June Q8
13 marks Moderate -0.3
8. A car which has run out of petrol is being towed by a breakdown truck along a straight horizontal road. The truck has mass 1200 kg and the car has mass 800 kg . The truck is connected to the car by a horizontal rope which is modelled as light and inextensible. The truck's engine provides a constant driving force of 2400 N . The resistances to motion of the truck and the car are modelled as constant and of magnitude 600 N and 400 N respectively. Find
  1. the acceleration of the truck and the ear,
  2. the tension in the rope. When the truck and car are moving at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rope breaks. The engine of the truck provides the same driving force as before. The magnitude of the resistance to the motion of the truck remains 600 N .
  3. Show that the truck reaches a speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) approximately 6 s earlier than it would have done if the rope had not broken. \section*{END}
Edexcel M1 2004 June Q1
6 marks Moderate -0.8
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{57a51cfd-7206-4f34-9744-44255789188d-2_467_1094_352_511}
\end{figure} A particle of weight \(W\) newtons is attached at \(C\) to the ends of two light inextensible strings \(A C\) and \(B C\). The other ends of the strings are attached to two fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with \(A C\) and \(B C\) inclined to the horizontal at \(30 ^ { \circ }\) and \(60 ^ { \circ }\) respectively, as shown in Fig.1. Given the tension in \(A C\) is 50 N , calculate
  1. the tension in \(B C\), to 3 significant figures,
  2. the value of \(W\).
Edexcel M1 2004 June Q2
7 marks Moderate -0.8
2. A particle \(P\) is moving with constant acceleration along a straight horizontal line \(A B C\), where \(A C = 24 \mathrm {~m}\). Initially \(P\) is at \(A\) and is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(A B\). After 1.5 s , the direction of motion of \(P\) is unchanged and \(P\) is at \(B\) with speed \(9.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that the speed of \(P\) at \(C\) is \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The mass of \(P\) is 2 kg . When \(P\) reaches \(C\), an impulse of magnitude 30 Ns is applied to \(P\) in the direction \(C B\).
  2. Find the velocity of \(P\) immediately after the impulse has been applied, stating clearly the direction of motion of \(P\) at this instant.
    (3)
Edexcel M1 2004 June Q3
9 marks Standard +0.3
3. A particle \(P\) of mass 2 kg is moving with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on a smooth horizontal plane. The particle \(P\) collides directly with a particle \(Q\) of mass 4 kg which is at rest on the same horizontal plane. Immediately after the collision, \(P\) and \(Q\) are moving in opposite directions and the speed of \(P\) is one-third the speed of \(Q\).
  1. Show that the speed of \(P\) immediately after the collision is \(\frac { 1 } { 5 } u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision \(P\) continues to move in the same straight line and is brought to rest by a constant resistive force of magnitude 10 N . The distance between the point of collision and the point where \(P\) comes to rest is 1.6 m .
  2. Calculate the value of \(u\).
    (5)
Edexcel M1 2004 June Q4
11 marks Moderate -0.3
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{57a51cfd-7206-4f34-9744-44255789188d-3_360_1305_1151_416}
\end{figure} A plank \(A E\), of length 6 m and mass 10 kg , rests in a horizontal position on supports at \(B\) and \(D\), where \(A B = 1 \mathrm {~m}\) and \(D E = 2 \mathrm {~m}\). A child of mass 20 kg stands at \(C\), the mid-point of \(B D\), as shown in Fig. 2. The child is modelled as a particle and the plank as a uniform rod. The child and the plank are in equilibrium. Calculate
  1. the magnitude of the force exerted by the support on the plank at \(B\),
  2. the magnitude of the force exerted by the support on the plank at \(D\). The child now stands at a point \(F\) on the plank. The plank is in equilibrium and on the point of tilting about \(D\).
  3. Calculate the distance \(D F\). \section*{5.} \section*{Figure 3}
    \includegraphics[max width=\textwidth, alt={}]{57a51cfd-7206-4f34-9744-44255789188d-4_422_1142_382_455}
    Figure 3 shows a boat \(B\) of mass 400 kg held at rest on a slipway by a rope. The boat is modelled as a particle and the slipway as a rough plane inclined at \(15 ^ { \circ }\) to the horizontal. The coefficient of friction between \(B\) and the slipway is 0.2 . The rope is modelled as a light, inextensible string, parallel to a line of greatest slope of the plane. The boat is in equilibrium and on the point of sliding down the slipway.
  4. Calculate the tension in the rope.
    (6) The boat is 50 m from the bottom of the slipway. The rope is detached from the boat and the boat slides down the slipway.
  5. Calculate the time taken for the boat to slide to the bottom of the slipway.
    (6)