Questions M1 (1912 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel M1 2017 June Q4
9 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c809d34e-83db-4a16-a831-001f9f36b1c3-10_291_926_251_516} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle \(P\) of mass 5 kg is held at rest in equilibrium on a rough inclined plane by a horizontal force of magnitude 10 N . The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Figure 1. The line of action of the force lies in the vertical plane containing \(P\) and a line of greatest slope of the plane. The coefficient of friction between \(P\) and the plane is \(\mu\). Given that \(P\) is on the point of sliding down the plane, find the value of \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{c809d34e-83db-4a16-a831-001f9f36b1c3-13_2460_72_311_27}
Edexcel M1 2017 June Q5
6 marks Moderate -0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c809d34e-83db-4a16-a831-001f9f36b1c3-14_346_241_262_845} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A vertical light rod \(P Q\) has a particle of mass 0.5 kg attached to it at \(P\) and a particle of mass 0.75 kg attached to it at \(Q\), to form a system, as shown in Figure 2. The system is accelerated vertically upwards by a vertical force of magnitude 15 N applied to the particle at \(Q\). Find the thrust in the rod.
Edexcel M1 2017 June Q6
9 marks Moderate -0.8
6. A cyclist is moving along a straight horizontal road and passes a point \(A\). Five seconds later, at the instant when she is moving with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), she passes the point \(B\). She moves with constant acceleration from \(A\) to \(B\). Given that \(A B = 40 \mathrm {~m}\), find
  1. the acceleration of the cyclist as she moves from \(A\) to \(B\),
  2. the time it takes her to travel from \(A\) to the midpoint of \(A B\).
Edexcel M1 2017 June Q7
14 marks Standard +0.3
7. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin \(O\).] Two ships, \(P\) and \(Q\), are moving with constant velocities.
The velocity of \(P\) is \(( 9 \mathbf { i } - 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and the velocity of \(Q\) is \(( 4 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\)
  1. Find the direction of motion of \(P\), giving your answer as a bearing to the nearest degree. When \(t = 0\), the position vector of \(P\) is \(( 9 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\) and the position vector of \(Q\) is \(( \mathbf { i } + 4 \mathbf { j } ) \mathrm { km }\). At time \(t\) hours, the position vectors of \(P\) and \(Q\) are \(\mathbf { p } \mathrm { km }\) and \(\mathbf { q } \mathrm { km }\) respectively.
  2. Find an expression for
    1. \(\mathbf { p }\) in terms of \(t\),
    2. \(\mathbf { q }\) in terms of \(t\).
  3. Hence show that, at time \(t\) hours, $$\overrightarrow { Q P } = ( 8 + 5 t ) \mathbf { i } + ( 6 - 10 t ) \mathbf { j }$$
  4. Find the values of \(t\) when the ships are 10 km apart.
Edexcel M1 2017 June Q8
15 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c809d34e-83db-4a16-a831-001f9f36b1c3-24_638_951_242_500} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Two particles, \(A\) and \(B\), have masses \(2 m\) and \(m\) respectively. The particles are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a fixed rough horizontal table at a distance \(d\) from a small smooth light pulley which is fixed at the edge of the table at the point \(P\). The coefficient of friction between \(A\) and the table is \(\mu\), where \(\mu < \frac { 1 } { 2 }\). The string is parallel to the table from \(A\) to \(P\) and passes over the pulley. Particle \(B\) hangs freely at rest vertically below \(P\) with the string taut and at a height \(h\), ( \(h < d\) ), above a horizontal floor, as shown in Figure 3. Particle \(A\) is released from rest with the string taut and slides along the table.
    1. Write down an equation of motion for \(A\).
    2. Write down an equation of motion for \(B\).
  2. Hence show that, until \(B\) hits the floor, the acceleration of \(A\) is \(\frac { g } { 3 } ( 1 - 2 \mu )\).
  3. Find, in terms of \(g , h\) and \(\mu\), the speed of \(A\) at the instant when \(B\) hits the floor. After \(B\) hits the floor, \(A\) continues to slide along the table. Given that \(\mu = \frac { 1 } { 3 }\) and that \(A\) comes to rest at \(P\),
  4. find \(d\) in terms of \(h\).
  5. Describe what would happen if \(\mu = \frac { 1 } { 2 }\)
    (Total 15 marks)
    Leave blank
    Q8
Edexcel M1 2018 June Q1
6 marks Moderate -0.8
  1. Two particles, \(P\) and \(Q\), have masses \(3 m\) and \(m\) respectively. They are moving in opposite directions towards each other along the same straight line on a smooth horizontal plane and collide directly. The speeds of \(P\) and \(Q\) immediately before the collision are \(2 u\) and \(4 u\) respectively. The magnitude of the impulse received by each particle in the collision is \(\frac { 21 m u } { 4 }\).
    1. Find the speed of \(P\) after the collision.
    2. Find the speed of \(Q\) after the collision.
Edexcel M1 2018 June Q2
10 marks Standard +0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4fd21e83-0bdf-4bb1-8a3f-76beada511ae-04_333_976_287_550} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle of mass 2 kg lies on a rough plane. The plane is inclined to the horizontal at \(30 ^ { \circ }\). The coefficient of friction between the particle and the plane is \(\frac { 1 } { 4 }\). The particle is held in equilibrium by a force of magnitude \(P\) newtons. The force makes an angle of \(20 ^ { \circ }\) with the horizontal and acts in a vertical plane containing a line of greatest slope of the plane, as shown in Figure 1. Find the least possible value of \(P\).
Edexcel M1 2018 June Q3
7 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4fd21e83-0bdf-4bb1-8a3f-76beada511ae-08_426_1226_221_360} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A wooden beam \(A B\), of mass 150 kg and length 9 m , rests in a horizontal position supported by two vertical ropes. The ropes are attached to the beam at \(C\) and \(D\), where \(A C = 1.5 \mathrm {~m}\) and \(B D = 3.5 \mathrm {~m}\). A gymnast of mass 60 kg stands on the beam at the point \(P\), where \(A P = 3 \mathrm {~m}\), as shown in Figure 2. The beam remains horizontal and in equilibrium. By modelling the gymnast as a particle, the beam as a uniform rod and the ropes as light inextensible strings,
  1. find the tension in the rope attached to the beam at \(C\). The gymnast at \(P\) remains on the beam at \(P\) and another gymnast, who is also modelled as a particle, stands on the beam at \(B\). The beam remains horizontal and in equilibrium. The mass of the gymnast at \(B\) is the largest possible for which the beam remains horizontal and in equilibrium.
  2. Find the tension in the rope attached to the beam at \(D\).
Edexcel M1 2018 June Q4
13 marks Standard +0.3
4. A ball of mass 0.2 kg is projected vertically downwards with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point A which is 2.5 m above horizontal ground. The ball hits the ground. Immediately after hitting the ground, the ball rebounds vertically with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball receives an impulse of magnitude 7 Ns in its impact with the ground. By modelling the ball as a particle and ignoring air resistance, find
  1. the value of \(U\). After hitting the ground, the ball moves vertically upwards and passes through a point \(B\) which is 1 m above the ground.
  2. Find the time between the instant when the ball hits the ground and the instant when the ball first passes through \(B\).
  3. Sketch a velocity-time graph for the motion of the ball from when it was projected from \(A\) to when it first passes through \(B\). (You need not make any further calculations to draw this sketch.)
Edexcel M1 2018 June Q5
10 marks Moderate -0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4fd21e83-0bdf-4bb1-8a3f-76beada511ae-16_359_298_233_824} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A lift of mass 250 kg is being raised by a vertical cable attached to the top of the lift. A woman of mass 60 kg stands on the horizontal floor inside the lift, as shown in Figure 3. The lift ascends vertically with constant acceleration \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is a constant downwards resistance of magnitude 100 N on the lift. By modelling the woman as a particle,
  1. find the magnitude of the normal reaction exerted by the floor of the lift on the woman. The tension in the cable must not exceed 10000 N for safety reasons, and the maximum upward acceleration of the lift is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). A typical occupant of the lift is modelled as a particle of mass 75 kg and the cable is modelled as a light inextensible string. There is still a constant downwards resistance of magnitude 100 N on the lift.
  2. Find the maximum number of typical occupants that can be safely carried in the lift when it is ascending with an acceleration of \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
Edexcel M1 2018 June Q6
13 marks Moderate -0.3
6. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively] Two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act on a particle \(P\) of mass 0.5 kg .
\(\mathbf { F } _ { 1 } = ( 4 \mathbf { i } - 6 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }\).
Given that the resultant force of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) is in the same direction as \(- 2 \mathbf { i } - \mathbf { j }\),
  1. show that \(p - 2 q = - 16\) Given that \(q = 3\)
  2. find the magnitude of the acceleration of \(P\),
  3. find the direction of the acceleration of \(P\), giving your answer as a bearing to the nearest degree. XXXXXXXXXXIXITEINTIIS AREA XX女X女X女X女X DO NOT WIRIE IN THS AREA．
Edexcel M1 2018 June Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4fd21e83-0bdf-4bb1-8a3f-76beada511ae-24_391_917_251_516} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A particle \(P\) of mass \(4 m\) is held at rest at the point \(X\) on the surface of a rough inclined plane which is fixed to horizontal ground. The point \(X\) is a distance \(h\) from the bottom of the inclined 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 \(\frac { 1 } { 4 }\). The particle \(P\) is attached to one end of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of the plane. The other end of the string is attached to a particle \(Q\) of mass \(m\) which hangs freely at a distance \(d\), where \(d > h\), below the pulley, as shown in Figure 4. The string lies in a vertical plane through a line of greatest slope of the inclined plane. The system is released from rest with the string taut and \(P\) moves down the plane. For the motion of the particles before \(P\) hits the ground,
  1. state which of the information given above implies that 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. When \(P\) hits the ground, it immediately comes to rest. Given that \(Q\) comes to instantaneous rest before reaching the pulley,
  4. show that \(d > \frac { 28 h } { 25 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{4fd21e83-0bdf-4bb1-8a3f-76beada511ae-27_56_20_109_1950}
    END
Edexcel M1 Q1
7 marks Moderate -0.8
  1. An aircraft moves along a straight horizontal runway with constant acceleration. It passes a point \(A\) on the runway with speed \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then passes the point \(B\) on the runway with speed \(34 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance from \(A\) to \(B\) is 150 m .
    1. Find the acceleration of the aircraft.
    2. Find the time taken by the aircraft in moving from \(A\) to \(B\).
    3. Find, to 3 significant figures, the speed of the aircraft when it passes the point mid-way between \(A\) and \(B\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ee3cbd24-55b1-4003-85bb-26d98f79a118-3_419_569_963_717} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} A particle has mass 2 kg . It is attached at \(B\) to the ends of two light inextensible strings \(A B\) and \(B C\). When the particle hangs in equilibrium, \(A B\) makes an angle of \(30 ^ { \circ }\) with the vertical, as shown in Fig. 1. The magnitude of the tension in \(B C\) is twice the magnitude of the tension in \(A B\).
  2. Find, in degrees to one decimal place, the size of the angle that \(B C\) makes with the vertical.
    (4 marks)
  3. Hence find, to 3 significant figures, the magnitude of the tension in \(A B\).
Edexcel M1 Q3
8 marks Moderate -0.3
3. A racing car is travelling on a straight horizontal road. Its initial speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it accelerates for 4 s to reach a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then travels at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for a further 8 s . The total distance travelled by the car during this 12 s period is 600 m .
  1. Sketch a speed-time graph to illustrate the motion of the car during this 12 s period.
  2. Find the value of \(V\).
  3. Find the acceleration of the car during the initial 4 s period.
Edexcel M1 Q4
11 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ee3cbd24-55b1-4003-85bb-26d98f79a118-4_179_729_449_671} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A plank \(A B\) has length 4 m . It lies on a horizontal platform, with the end \(A\) lying on the platform and the end \(B\) projecting over the edge, as shown in Fig. 2. The edge of the platform is at the point \(C\). Jack and Jill are experimenting with the plank. Jack has mass 40 kg and Jill has mass 25 kg . They discover that, if Jack stands at \(B\) and Jill stands at \(A\) and \(B C = 1.6 \mathrm {~m}\), the plank is in equilibrium and on the point of tilting about \(C\). By modelling the plank as a uniform rod, and Jack and Jill as particles,
  1. find the mass of the plank. They now alter the position of the plank in relation to the platform so that, when Jill stands at \(B\) and Jack stands at \(A\), the plank is again in equilibrium and on the point of tilting about \(C\).
  2. Find the distance \(B C\) in this position.
  3. State how you have used the modelling assumptions that
    1. the plank is uniform,
    2. the plank is a rod,
    3. Jack and Jill are particles.
Edexcel M1 Q5
13 marks Moderate -0.8
5. A post is driven into the ground by means of a blow from a pile-driver. The pile-driver falls from rest from a height of 1.6 m above the top of the post.
  1. Show that the speed of the pile-driver just before it hits the post is \(5.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). (2 marks) The post has mass 6 kg and the pile-driver has mass 78 kg . When the pile-driver hits the top of the post, it is assumed that the there is no rebound and that both then move together with the same speed.
  2. Find the speed of the pile-driver and the post immediately after the pile-driver has hit the post. The post is brought to rest by the action of a resistive force from the ground acting for 0.06 s .
    By modelling this force as constant throughout this time,
  3. find the magnitude of the resistive force,
  4. find, to 2 significant figures, the distance travelled by the post and the pile-driver before they come to rest.
    (4 marks)
Edexcel M1 Q6
13 marks Standard +0.3
6. [In this question, the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due East and North respectively.] A coastguard station \(O\) monitors the movements of ships in a channel. At noon, the station's radar records two ships moving with constant speed. Ship \(A\) is at the point with position vector \(( - 5 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\) relative to \(O\) and has velocity \(( 2 \mathbf { i } + 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Ship \(B\) is at the point with position vector \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { km }\) and has velocity \(( - 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
  1. Given that the two ships maintain these velocities, show that they collide. The coast guard radios ship \(A\) and orders it to reduce its speed to move with velocity \(( \mathbf { i } + \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Given that \(A\) obeys this order and maintains this new constant velocity,
  2. find an expression for the vector \(\overrightarrow { A B }\) at time \(t\) hours after noon.
  3. find, to 3 significant figures, the distance between \(A\) and \(B\) at 1400 hours,
  4. find the time at which \(B\) will be due north of \(A\).
Edexcel M1 Q7
15 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ee3cbd24-55b1-4003-85bb-26d98f79a118-6_271_683_367_717} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} A small parcel of mass 2 kg moves on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The parcel is pulled up a line of greatest slope of the plane by means of a light rope which it attached to it. The rope makes an angle of \(30 ^ { \circ }\) with the plane, as shown in Fig. 3. The coefficient of friction between the parcel and the plane is 0.4 Given that the tension in the rope is 24 N ,
  1. find, to 2 significant figures, the acceleration of the parcel. The rope now breaks. The parcel slows down and comes to rest.
  2. Show that, when the parcel comes to this position of rest, it immediately starts to move down the plane again.
  3. Find, to 2 significant figures, the acceleration of the parcel as it moves down the plane after it has come to this position of instantaneous rest.
Edexcel M1 2002 November Q1
6 marks Moderate -0.8
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{14703bfa-abd8-4a8d-bc18-20d66eea409e-2_671_829_294_663}
\end{figure} A particle \(P\) of weight 6 N is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). A horizontal force of magnitude \(F\) newtons is applied to \(P\). The particle \(P\) is in equilibrium under gravity with the string making an angle of \(30 ^ { \circ }\) with the vertical, as shown in Fig. 1. Find, to 3 significant figures,
  1. the tension in the string,
  2. the value of \(F\).
Edexcel M1 2002 November Q2
7 marks Moderate -0.8
2. A particle \(P\) of mass 1.5 kg is moving under the action of a constant force ( \(3 \mathbf { i } - 7.5 \mathbf { j }\) ) N. Initially \(P\) has velocity \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
  1. the magnitude of the acceleration of \(P\),
  2. the velocity of \(P\), in terms of \(\mathbf { i }\) and \(\mathbf { j }\), when \(P\) has been moving for 4 seconds.
Edexcel M1 2002 November Q3
7 marks Moderate -0.3
3. A car accelerates uniformly from rest to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in \(T\) seconds. The car then travels at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(4 T\) seconds and finally decelerates uniformly to rest in a further 50 s .
  1. Sketch a speed-time graph to show the motion of the car. The total distance travelled by the car is 1220 m . Find
  2. the value of \(T\),
  3. the initial acceleration of the car.
Edexcel M1 2002 November Q4
9 marks Standard +0.2
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{14703bfa-abd8-4a8d-bc18-20d66eea409e-3_282_807_1194_648}
\end{figure} A uniform plank \(A B\) has weight 80 N and length \(x\) metres. The plank rests in equilibrium horizontally on two smooth supports at \(A\) and \(C\), where \(A C = 2 \mathrm {~m}\), as shown in Fig. 2. A rock of weight 20 N is placed at \(B\) and the plank remains in equilibrium. The reaction on the plank at \(C\) has magnitude 90 N . The plank is modelled as a rod and the rock as a particle.
  1. Find the value of \(x\).
  2. State how you have used the model of the rock as a particle. The support at \(A\) is now moved to a point \(D\) on the plank and the plank remains in equilibrium with the rock at \(B\). The reaction on the plank at \(C\) is now three times the reaction at \(D\).
  3. Find the distance \(A D\).
Edexcel M1 2002 November Q5
10 marks Standard +0.3
5. \section*{Figure 3}
\includegraphics[max width=\textwidth, alt={}]{14703bfa-abd8-4a8d-bc18-20d66eea409e-4_502_1154_339_552}
A suitcase of mass 10 kg slides down a ramp which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The suitcase is modelled as a particle and the ramp as a rough plane. The top of the plane is \(A\). The bottom of the plane is \(C\) and \(A C\) is a line of greatest slope, as shown in Fig. 3. The point \(B\) is on \(A C\) with \(A B = 5 \mathrm {~m}\). The suitcase leaves \(A\) with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and passes \(B\) with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the decleration of the suitcase,
  2. the coefficient of friction between the suitcase and the ramp. The suitcase reaches the bottom of the ramp.
  3. Find the greatest possible length of \(A C\).
Edexcel M1 2002 November Q6
11 marks Moderate -0.8
6. A railway truck \(P\) of mass 1500 kg is moving on a straight horizontal track. The truck \(P\) collides with a truck \(Q\) of 2500 kg at a point \(A\). Immediately before the collision, \(P\) and \(Q\) are moving in the same direction with speeds \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Immediately after the collision, the direction of motion of \(P\) is unchanged and its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By modelling the trucks as particles,
  1. show that the speed of \(Q\) immediately after the collision is \(8.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision at \(A\), the truck \(P\) is acted upon by a constant braking force of magnitude 500 N . The truck \(P\) comes to rest at the point \(B\).
  2. Find the distance \(A B\). After the collision \(Q\) continues to move with constant speed \(8.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find the distance between \(P\) and \(Q\) at the instant when \(P\) comes to rest.
Edexcel M1 2002 November Q7
11 marks Moderate -0.8
7. Two helicopters \(P\) and \(Q\) are moving in the same horizontal plane. They are modelled as particles moving in straight lines with constant speeds. At noon \(P\) is at the point with position vector \(( 20 \mathbf { i } + 35 \mathbf { j } ) \mathrm { km }\) with respect to a fixed origin \(O\). At time \(t\) hours after noon the position vector of \(P\) is \(\mathbf { p } \mathrm { km }\). When \(t = \frac { 1 } { 2 }\) the position vector of \(P\) is \(( 50 \mathbf { i } - 25 \mathbf { j } ) \mathrm { km }\). Find
  1. the velocity of \(P\) in the form \(( a \mathbf { i } + b \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\),
  2. an expression for \(\mathbf { p }\) in terms of \(t\). At noon \(Q\) is at \(O\) and at time \(t\) hours after noon the position vector of \(Q\) is \(\mathbf { q } \mathrm { km }\). The velocity of \(Q\) has magnitude \(120 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in the direction of \(4 \mathbf { i } - 3 \mathbf { j }\). Find
  3. an expression for \(\mathbf { q }\) in terms of \(t\),
  4. the distance, to the nearest km , between \(P\) and \(Q\) when \(t = 2\). \section*{8.} \section*{Figure 4}
    \includegraphics[max width=\textwidth, alt={}]{14703bfa-abd8-4a8d-bc18-20d66eea409e-6_695_1153_322_562}
    Two particles \(A\) and \(B\), of mass \(m \mathrm {~kg}\) and 3 kg respectively, are connected by a light inextensible string. The particle \(A\) is held resting on a smooth fixed plane inclined at \(30 ^ { \circ }\) to the horizontal. The string passes over a smooth pulley \(P\) fixed at the top of the plane. The portion \(A P\) of the string lies along a line of greatest slope of the plane and \(B\) hangs freely from the pulley, as shown in Fig. 4. The system is released from rest with \(B\) at a height of 0.25 m above horizontal ground. Immediately after release, \(B\) descends with an acceleration of \(\frac { 2 } { 5 } g\). Given that \(A\) does not reach \(P\), calculate
  5. the tension in the string while \(B\) is descending,
  6. the value of \(m\). The particle \(B\) strikes the ground and does not rebound. Find
  7. the magnitude of the impulse exerted by \(B\) on the ground,
  8. the time between the instant when \(B\) strikes the ground and the instant when \(A\) reaches its highest point.