Questions — Edexcel M1 (663 questions)

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Edexcel M1 Q4
10 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4fe54579-ac77-46f9-85e1-2e95963d6b3e-3_467_348_201_708} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a weight \(A\) of mass 6 kg connected by a light, inextensible string which passes over a smooth, fixed pulley to a box \(B\) of mass 5 kg . There is an object \(C\) of mass 3 kg resting on the horizontal floor of box \(B\). The system is released from rest. Find, giving your answers in terms of \(g\),
  1. the acceleration of the system,
  2. the force on the pulley.
  3. Show that the reaction between \(C\) and the floor of \(B\) is \(\frac { 18 } { 7 } \mathrm {~g}\) newtons.
Edexcel M1 Q5
11 marks Standard +0.3
5. Two flies \(P\) and \(Q\), are crawling vertically up a wall. At time \(t = 0\), the flies are at the same height above the ground, with \(P\) crawling at a steady speed of \(4 \mathrm { cms } ^ { - 1 }\). \(Q\) starts from rest at time \(t = 0\) and accelerates uniformly to a speed of \(6 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\) in 6 seconds. Fly \(Q\) then maintains this speed.
  1. Find the value of \(t\) when the two flies are moving at the same speed.
  2. Sketch on the same diagram, speed-time graphs to illustrate the motion of the two flies. Given that the distance of the two flies from the top of the wall at time \(t = 0\) is \(x \mathrm {~cm}\) and that \(Q\) reaches the top of the wall first,
  3. show that \(x > 36\).
Edexcel M1 Q6
14 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4fe54579-ac77-46f9-85e1-2e95963d6b3e-4_288_1275_201_410} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows a uniform plank \(A B\) of length 8 m and mass 50 kg suspended horizontally by two light vertical inextensible strings attached at either end of the plank. The maximum tension that either string can support is 40 gN . A rock of mass \(M \mathrm {~kg}\) is placed on the plank at \(A\) and rolled along the plank to \(B\) without either string breaking.
  1. Explain, with the aid of a sketch-graph, how the tension in the string at \(A\) varies with \(x\), the distance of the rock from \(A\).
  2. Show that \(M \leq 15\). The first rock is removed and a second rock of mass 20 kg is placed on the plank.
  3. Find the fraction of the plank on which the rock can be placed without one of the strings breaking.
Edexcel M1 Q7
14 marks Standard +0.3
7. At 6 a.m. a cargo ship has position vector \(( 7 \mathbf { i } + 56 \mathbf { j } ) \mathrm { km }\) relative to a fixed origin \(O\) on the coast and moves with constant velocity \(( 9 \mathbf { i } - 6 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }\). A ferry sails from \(O\) at 6 a.m. and moves with constant velocity \(( 12 \mathbf { i } + 18 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and due north respectively.
  1. Show that the position vector of the cargo ship \(t\) hours after 6 a.m. is given by $$[ ( 7 + 9 t ) \mathbf { i } + ( 56 - 6 t ) \mathbf { j } ] \mathrm { km }$$ and find the position vector of the ferry in terms of \(t\).
  2. Show that if both vessels maintain their course and speed, they will collide and find the time and position vector at which this occurs.
    (6 marks)
    At 8 a.m. the captain of the ferry realises that a collision is imminent and changes course so that the ferry now has velocity \(( 21 \mathbf { i } + 6 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }\).
  3. Find the distance between the two ships at the time when they would have collided.
Edexcel M1 Q3
Moderate -0.8
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-005_851_1073_312_456}
\end{figure} A sprinter runs a race of 200 m . Her total time for running the race is 25 s . Figure 2 is a sketch of the speed-time graph for the motion of the sprinter. She starts from rest and accelerates uniformly to a speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 4 s . The speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is maintained for 16 s and she then decelerates uniformly to a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the end of the race. Calculate
  1. the distance covered by the sprinter in the first 20 s of the race,
  2. the value of \(u\),
  3. the deceleration of the sprinter in the last 5 s of the race.
Edexcel M1 Q4
Moderate -0.8
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-007_330_675_287_644}
\end{figure} A particle \(P\) of mass 2.5 kg rests in equilibrium on a rough plane under the action of a force of magnitude \(X\) newtons acting up a line of greatest slope of the plane, as shown in Figure 3. The plane is inclined at \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4 . The particle is in limiting equilibrium and is on the point of moving up the plane. Calculate
  1. the normal reaction of the plane on \(P\),
  2. the value of \(X\). The force of magnitude \(X\) newtons is now removed.
  3. Show that \(P\) remains in equilibrium on the plane.
Edexcel M1 Q5
Standard +0.3
5. Figure 4 \includegraphics[max width=\textwidth, alt={}, center]{94d9432d-1723-4549-ad5e-d4be0f5fd083-009_609_1026_301_516} A block of wood \(A\) of mass 0.5 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. The other end of the string is attached to a ball \(B\) of mass 0.8 kg which hangs freely below the pulley, as shown in Figure 4. The coefficient of friction between \(A\) and the table is \(\mu\). The system is released from rest with the string taut. After release, \(B\) descends a distance of 0.4 m in 0.5 s . Modelling \(A\) and \(B\) as particles, calculate
  1. the acceleration of \(B\),
  2. the tension in the string,
  3. the value of \(\mu\).
  4. State how in your calculations you have used the information that the string is inextensible.
Edexcel M1 Q7
Moderate -0.3
7. Two ships \(P\) and \(Q\) are travelling at night with constant velocities. At midnight, \(P\) is at the point with position vector \(( 20 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\) relative to a fixed origin \(O\). At the same time, \(Q\) is at the point with position vector \(( 14 \mathbf { i } - 6 \mathbf { j } ) \mathrm { km }\). Three hours later, \(P\) is at the point with position vector \(( 29 \mathbf { i } + 34 \mathbf { j } ) \mathrm { km }\). The ship \(Q\) travels with velocity \(12 \mathbf { j } \mathrm {~km} \mathrm {~h} ^ { - 1 }\). At time \(t\) hours after midnight, the position vectors of \(P\) and \(Q\) are \(\mathbf { p } \mathrm { km }\) and \(\mathbf { q } \mathrm { km }\) respectively. Find
  1. the velocity of \(P\), in terms of \(\mathbf { i }\) and \(\mathbf { j }\),
  2. expressions for \(\mathbf { p }\) and \(\mathbf { q }\), in terms of \(t\), i and \(\mathbf { j }\). At time \(t\) hours after midnight, the distance between \(P\) and \(Q\) is \(d \mathrm {~km}\).
  3. By finding an expression for \(\overrightarrow { P Q }\), show that $$d ^ { 2 } = 25 t ^ { 2 } - 92 t + 292$$ Weather conditions are such that an observer on \(P\) can only see the lights on \(Q\) when the distance between \(P\) and \(Q\) is 15 km or less. Given that when \(t = 1\), the lights on \(Q\) move into sight of the observer,
  4. find the time, to the nearest minute, at which the lights on \(Q\) move out of sight of the observer.
    1. In taking off, an aircraft moves on a straight runway \(A B\) of length 1.2 km . The aircraft moves from \(A\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It moves with constant acceleration and 20 s later it leaves the runway at \(C\) with speed \(74 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
    2. the acceleration of the aircraft,
    3. the distance \(B C\).
    4. Two small steel balls \(A\) and \(B\) have mass 0.6 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 \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, the direction of motion of \(A\) is unchanged and the speed of \(B\) is twice the speed of \(A\). Find
    5. the speed of \(A\) immediately after the collision,
    6. the magnitude of the impulse exerted on \(B\) in the collision.
    \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{94d9432d-1723-4549-ad5e-d4be0f5fd083-018_282_707_278_699}
    \end{figure}
Edexcel M1 Q8
Moderate -0.3
  1. \hspace{0pt} [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal vectors due east and north respectively.]
Edexcel M1 2024 October Q1
Moderate -0.3
  1. Particle \(A\) has mass \(4 m\) and particle \(B\) has mass \(3 m\).
The particles are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly. Immediately before the collision, the speed of \(A\) is \(2 x\) and the speed of \(B\) is \(x\).
Immediately after the collision, the speed of \(A\) is \(y\) and the speed of \(B\) is \(5 y\).
The direction of motion of each particle is reversed as a result of the collision.
  1. Show that \(y = \frac { 5 } { 11 } x\).
  2. Find, in terms of \(m\) and \(x\), the magnitude of the impulse received by \(A\) in the collision.
Edexcel M1 2024 October Q2
Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2f2f89a6-cec4-444d-95d9-0112887d87eb-04_282_1075_296_495} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A non-uniform beam \(A B\) has length 6 m and mass 50 kg . The beam rests horizontally on two supports at \(C\) and \(D\), where \(A C = 0.9 \mathrm {~m}\) and \(D B = 1.8 \mathrm {~m}\). A child of mass 25 kg stands on the beam at \(E\), where \(A E = E B = 3 \mathrm {~m}\), as shown in Figure 1. The beam is in equilibrium.
The magnitude of the normal reaction between the beam and the support at \(C\) is \(R _ { C }\) newtons. The magnitude of the normal reaction between the beam and the support at \(D\) is \(R _ { D }\) newtons. The beam is modelled as a rod and the child is modelled as a particle.
The centre of mass of the beam is between \(C\) and \(D\) and is a distance \(x\) metres from \(D\).
Given that \(2 R _ { D } = 3 R _ { C }\)
  1. show that \(x = 1.38\) The child remains at \(E\) and a block of mass \(M \mathrm {~kg}\) is placed on the beam at \(B\).
    The block is modelled as a particle.
    Given that the beam is on the point of tilting,
  2. find the value of \(M\).
Edexcel M1 2024 October Q3
Moderate -0.8
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors and position vectors are given relative to a fixed origin.]
A ship \(A\) is moving with constant velocity.
At 1 pm , the position vector of \(A\) is \(( 25 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\).
At 3 pm , the position vector of \(A\) is \(( 55 \mathbf { i } + 34 \mathbf { j } ) \mathrm { km }\).
At time \(t\) hours after 1 pm , the position vector of \(A\) is \(\mathbf { r } _ { A } \mathrm {~km}\).
  1. Show that \(\mathbf { r } _ { A } = ( 25 + 15 t ) \mathbf { i } + ( 10 + 12 t ) \mathbf { j }\) The speed of \(A\) is \(V \mathrm {~ms} ^ { - 1 }\)
  2. Find the value of \(V\). A ship \(B\) is moving with constant velocity \(( 20 \mathbf { i } - 6 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) At 1 pm , the position vector of \(B\) is \(( 35 \mathbf { i } + 51 \mathbf { j } ) \mathrm { km }\).
    At 2:30 pm, \(B\) passes through the point \(P\).
  3. Show that \(A\) also passes through \(P\).
Edexcel M1 2024 October Q4
Moderate -0.8
  1. The points \(A\) and \(B\) lie on the same straight horizontal road.
Figure 2, on page 11, shows the speed-time graph of a cyclist \(P\), for his journey from \(A\) to \(B\).
At time \(t = 0 , P\) starts from rest at \(A\) and accelerates uniformly for 9 seconds until his speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) He then travels at constant speed \(V \mathrm {~ms} ^ { - 1 }\) When \(t = 42\), cyclist \(P\) passes \(B\).
Given that the distance \(A B\) is 120 m ,
  1. show that \(V = 3.2\)
  2. Find the acceleration of cyclist \(P\) between \(t = 0\) and \(t = 9\) Cyclist \(P\) continues to cycle along the road in the same direction at the same constant speed, \(V \mathrm {~ms} ^ { - 1 }\) When \(t = 6\), a second cyclist \(Q\) sets off from \(A\) and travels in the same direction as \(P\) along the same road. She accelerates for \(T\) seconds until her speed is \(3.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) She then travels at constant speed \(3.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Cyclist \(Q\) catches up with \(P\) when \(t = 54\)
  3. On Figure 2, on page 11, sketch a speed-time graph showing the journeys of both cyclists, for the interval \(0 \leqslant t \leqslant 54\)
  4. Find the value of \(T\) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2f2f89a6-cec4-444d-95d9-0112887d87eb-11_661_1509_292_278} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A copy of Figure 2 is on page 13 if you need to redraw your answer to part (c). Only use this copy of Figure 2 if you need to redraw your answer to part (c). \includegraphics[max width=\textwidth, alt={}, center]{2f2f89a6-cec4-444d-95d9-0112887d87eb-13_666_1509_374_278} \section*{Copy of Figure 2}
Edexcel M1 2024 October Q5
Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2f2f89a6-cec4-444d-95d9-0112887d87eb-14_588_908_292_794} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Two particles, \(P\) and \(Q\), have masses 3 kg and 5 kg respectively. The particles are connected by a light inextensible string which passes over a small smooth fixed pulley. The particles are released from rest with the string taut and the hanging parts of the string vertical, as shown in Figure 3. Immediately after the particles are released from rest, \(P\) moves upwards with acceleration \(a \mathrm {~ms} ^ { - 2 }\) and the tension in the string is \(T\) newtons.
  1. Write down an equation of motion for \(P\).
  2. Find the value of \(T\). The total force acting on the pulley due to the string has magnitude \(F\) newtons.
  3. Find the value of \(F\). Initially, \(Q\) is 10 m above horizontal ground and \(P\) is more than 2 m below the pulley.
    At the instant when \(Q\) has descended a distance of 2 m , the string breaks and \(Q\) falls to the ground.
  4. Find the speed of \(Q\) at the instant it hits the ground.
Edexcel M1 2024 October Q6
Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2f2f89a6-cec4-444d-95d9-0112887d87eb-18_335_682_296_696} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A particle \(P\) of mass 5 kg lies on the surface of a rough plane.
The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\) The particle is held in equilibrium by a horizontal force of magnitude \(H\) newtons, as shown in Figure 4. The horizontal force acts in a vertical plane containing a line of greatest slope of the inclined plane. The coefficient of friction between the particle and the plane is \(\frac { 1 } { 4 }\)
  1. Find the smallest possible value of \(H\). The horizontal force is now removed, and \(P\) starts to slide down the slope.
    In the first \(T\) seconds after \(P\) is released from rest, \(P\) slides 1.5 m down the slope.
  2. Find the value of \(T\).
Edexcel M1 2024 October Q7
Moderate -0.3
7 At time \(t = 0\), a small ball \(A\) is projected vertically upwards with speed \(8 \mathrm {~ms} ^ { - 1 }\) from a fixed point on horizontal ground.
The ball hits the ground again for the first time at time \(t = T _ { 1 }\) seconds.
Ball \(A\) is modelled as a particle moving freely under gravity.
  1. Show that \(T _ { 1 } = 1.63\) to 3 significant figures. After the first impact with the ground, \(A\) rebounds to a height of 2 m above the ground.
    Given that the mass of \(A\) is 0.1 kg ,
  2. find the magnitude of the impulse received by \(A\) as a result of its first impact with the ground. At time \(t = 1\) second, another small ball \(B\) is projected vertically upwards from another point on the ground with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Ball \(B\) is modelled as a particle moving freely under gravity.
    At time \(t = T _ { 2 }\) seconds ( \(T _ { 2 } > 1\) ), \(A\) and \(B\) are at the same height above the ground for the first time.
  3. Find the value of \(T _ { 2 }\)
Edexcel M1 2015 January Q1
7 marks Moderate -0.3
A railway truck \(A\) of mass \(m\) and a second railway truck \(B\) of mass \(4m\) are moving in opposite directions on a smooth straight horizontal track when they collide directly. Immediately before the collision the speed of truck \(A\) is \(3u\) and the speed of truck \(B\) is \(2u\). In the collision the trucks join together. Modelling the trucks as particles, find
  1. the speed of \(A\) immediately after the collision, [3]
  2. the direction of motion of \(A\) immediately after the collision, [1]
  3. the magnitude of the impulse exerted by \(A\) on \(B\) in the collision. [3]
Edexcel M1 2015 January Q2
8 marks Standard +0.3
\includegraphics{figure_1} A block of mass 50 kg lies on a rough plane which is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \frac{7}{24}\). The block is held at rest by a vertical rope, as shown in Figure 1, and is on the point of sliding down the plane. The block is modelled as a particle and the rope is modelled as a light inextensible string. Given that the friction force acting on the block has magnitude 65.8 N, find
  1. the tension in the rope, [4]
  2. the coefficient of friction between the block and the plane. [4]
Edexcel M1 2015 January Q3
7 marks Moderate -0.8
[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors directed due east and due north respectively.] A particle \(P\) is moving with constant velocity \((-6\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\). At time \(t = 0\), \(P\) passes through the point with position vector \((21\mathbf{i} + 5\mathbf{j})\) m, relative to a fixed origin \(O\).
  1. Find the direction of motion of \(P\), giving your answer as a bearing to the nearest degree. [3]
  2. Write down the position vector of \(P\) at time \(t\) seconds. [1]
  3. Find the time at which \(P\) is north-west of \(O\). [3]
Edexcel M1 2015 January Q4
7 marks Standard +0.3
The points \(P\) and \(Q\) are at the same height \(h\) metres above horizontal ground. A small stone is dropped from rest from \(P\). Half a second later a second small stone is thrown vertically downwards from \(Q\) with speed 7.35 m s\(^{-1}\). Given that the stones hit the ground at the same time, find the value of \(h\). [7]
Edexcel M1 2015 January Q5
10 marks Standard +0.8
\includegraphics{figure_2} A particle \(P\) of mass 2 kg is pushed up a line of greatest slope of a rough plane by a horizontal force of magnitude \(X\) newtons, as shown in Figure 2. The force acts in the vertical plane which contains \(P\) and a line of greatest slope of the plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \frac{3}{4}\). The coefficient of friction between \(P\) and the plane is 0.5 Given that the acceleration of \(P\) is 1.45 m s\(^{-2}\), find the value of \(X\). [10]
Edexcel M1 2015 January Q6
10 marks Standard +0.3
A uniform rod \(AC\), of weight \(W\) and length \(3l\), rests horizontally on two supports, one at \(A\) and one at \(B\), where \(AB = 2l\). A particle of weight \(2W\) is placed on the rod at a distance \(x\) from \(A\). The rod remains horizontal and in equilibrium.
  1. Find the greatest possible value of \(x\). [5]
The magnitude of the reaction of the support at \(A\) is \(R\). Due to a weakness in the support at \(A\), the greatest possible value of \(R\) is \(4W\).
  1. find the least possible value of \(x\). [5]
Edexcel M1 2015 January Q7
10 marks Moderate -0.8
A train travels along a straight horizontal track between two stations \(A\) and \(B\). The train starts from rest at \(A\) and moves with constant acceleration until it reaches its maximum speed of 108 km h\(^{-1}\). The train then travels at this speed before it moves with constant deceleration coming to rest at \(B\). The journey from \(A\) to \(B\) takes 8 minutes.
  1. Change 108 km h\(^{-1}\) into m s\(^{-1}\). [2]
  2. Sketch a speed-time graph for the motion of the train between the two stations \(A\) and \(B\). [2]
Given that the distance between the two stations is 12 km and that the time spent decelerating is three times the time spent accelerating,
  1. find the acceleration, in m s\(^{-2}\), of the train. [6]
Edexcel M1 2015 January Q8
16 marks Standard +0.3
\includegraphics{figure_3} A particle \(A\) of mass \(3m\) is held at rest on a rough horizontal table. The particle is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) which is fixed at the edge of the table. The other end of the string is attached to a particle \(B\) of mass \(2m\), which hangs freely, vertically below \(P\). The system is released from rest, with the string taut, when \(A\) is 1.3 m from \(P\) and \(B\) is 1 m above the horizontal floor, as shown in Figure 3. Given that \(B\) hits the floor 2 s after release and does not rebound,
  1. find the acceleration of \(A\) during the first two seconds, [2]
  2. find the coefficient of friction between \(A\) and the table, [8]
  3. determine whether \(A\) reaches the pulley. [6]
Edexcel M1 2016 January Q1
7 marks Moderate -0.3
A truck of mass 2400 kg is pulling a trailer of mass \(M\) kg along a straight horizontal road. The tow bar, connecting the truck to the trailer, is horizontal and parallel to the direction of motion. The tow bar is modelled as being light and inextensible. The resistance forces acting on the truck and the trailer are constant and of magnitude 400 N and 200 N respectively. The acceleration of the truck is 0.5 m s\(^{-2}\) and the tension in the tow bar is 600 N.
  1. Find the magnitude of the driving force of the truck. [3]
  2. Find the value of \(M\). [3]
  3. Explain how you have used the fact that the tow bar is inextensible in your calculations. [1]