Questions M1 (2067 questions)

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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
    (d) an expression for \(\mathbf { q }\) in terms of \(t\),
    (e) 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
    (a) the tension in the string while \(B\) is descending,
    (b) the value of \(m\). The particle \(B\) strikes the ground and does not rebound. Find
  3. the magnitude of the impulse exerted by \(B\) on the ground,
  4. the time between the instant when \(B\) strikes the ground and the instant when \(A\) reaches its highest point.
Edexcel M1 2014 January Q1
6 marks Moderate -0.8
A truck \(P\) of mass \(2 M\) is moving with speed \(U\) on smooth straight horizontal rails. It collides directly with another truck \(Q\) of mass \(3 M\) which is moving with speed \(4 U\) in the opposite direction on the same rails. The trucks join so that immediately after the collision they move together. By modelling the trucks as particles, find
  1. the speed of the trucks immediately after the collision,
  2. the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision.
Edexcel M1 2014 January Q2
6 marks Moderate -0.8
2. A particle \(P\) is moving with constant velocity ( \(2 \mathbf { i } - 3 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Find the speed of \(P\). The particle \(P\) passes through the point \(A\) and 4 seconds later passes through the point with position vector ( \(\mathbf { i } - 4 \mathbf { j }\) ) m.
  2. Find the position vector of \(A\).
Edexcel M1 2014 January Q3
5 marks Standard +0.3
3. A beam \(A B\) has length 15 m and mass 25 kg . The beam is smoothly supported at the point \(P\), where \(A P = 8 \mathrm {~m}\). A man of mass 100 kg stands on the beam at a distance of 2 m from \(A\) and another man stands on the beam at a distance of 1 m from \(B\). The beam is modelled as a non-uniform rod and the men are modelled as particles. The beam is in equilibrium in a horizontal position with the reaction on the beam at \(P\) having magnitude 2009 N. Find the distance of the centre of mass of the beam from \(A\).
Edexcel M1 2014 January Q4
11 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fade35da-8dca-4d98-a07c-ed3a173fccda-08_396_483_214_735} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A fixed rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\) A small box of mass \(m\) is at rest on the plane. A force of magnitude \(k m g\), where \(k\) is a constant, is applied to the box. The line of action of the force is at angle \(\alpha\) to the line of greatest slope of the plane through the box, as shown in Figure 1, and lies in the same vertical plane as this line of greatest slope. The coefficient of friction between the box and the plane is \(\mu\). The box is on the point of slipping up the plane. By modelling the box as a particle, find \(k\) in terms of \(\mu\).
Edexcel M1 2014 January Q5
7 marks Moderate -0.3
5. A racing car is moving along a straight horizontal track with constant acceleration. There are three checkpoints, \(P , Q\) and \(R\), on the track, where \(P Q = 48 \mathrm {~m}\) and \(Q R = 200 \mathrm {~m}\). The car takes 3 s to travel from \(P\) to \(Q\) and 5 s to travel from \(Q\) to \(R\). Find
  1. the acceleration of the car,
  2. the speed of the car as it passes \(P\).
Edexcel M1 2014 January Q6
11 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fade35da-8dca-4d98-a07c-ed3a173fccda-16_398_860_210_543} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Two particles \(P\) and \(Q\) have masses 0.1 kg and 0.5 kg respectively. The particles are attached to the ends of a light inextensible string. Particle \(P\) is held at rest on a rough horizontal table. The string lies along the table and passes over a small smooth pulley which is fixed to the edge of the table. Particle \(Q\) is at rest on a smooth plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 4 } { 3 }\) The string lies in the vertical plane which contains the pulley and a line of greatest slope of the inclined plane, as shown in Figure 2. Particle \(P\) is released from rest with the string taut. During the first 0.5 s of the motion \(P\) does not reach the pulley and \(Q\) moves 0.75 m down the plane.
  1. Find the tension in the string during the first 0.5 s of the motion.
  2. Find the coefficient of friction between \(P\) and the table. \includegraphics[max width=\textwidth, alt={}, center]{fade35da-8dca-4d98-a07c-ed3a173fccda-19_72_59_2613_1886}
Edexcel M1 2014 January Q7
12 marks Moderate -0.3
7. A force \(\mathbf { F }\) is given by \(\mathbf { F } = ( 9 \mathbf { i } + 13 \mathbf { j } )\) N.
  1. Find the size of the angle between the direction of \(\mathbf { F }\) and the vector \(\mathbf { j }\). The force \(\mathbf { F }\) is the resultant of two forces \(\mathbf { P }\) and \(\mathbf { Q }\). The line of action of \(\mathbf { P }\) is parallel to the vector ( \(2 \mathbf { i } - \mathbf { j }\) ). The line of action of \(\mathbf { Q }\) is parallel to the vector ( \(\mathbf { i } + 3 \mathbf { j }\) ).
  2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\),
    1. the force \(\mathbf { P }\),
    2. the force \(\mathbf { Q }\).
Edexcel M1 2014 January Q8
17 marks Moderate -0.3
8. Two trains, \(A\) and \(B\), start together from rest, at time \(t = 0\), at a station and move along parallel straight horizontal tracks. Both trains come to rest at the next station after 180 s . Train \(A\) moves with constant acceleration \(\frac { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 30 s , then moves at constant speed for 120 s and then moves with constant deceleration for the final 30 s . Train \(B\) moves with constant acceleration for 90 s and then moves with constant deceleration for the final 90 s .
  1. Sketch, on the same axes, the speed-time graphs for the motion of the two trains between the two stations.
  2. Find the acceleration of train \(B\) for the first half of its journey.
  3. Find the times when the two trains are moving at the same speed.
  4. Find the distance between the trains 96 s after they start. \includegraphics[max width=\textwidth, alt={}, center]{fade35da-8dca-4d98-a07c-ed3a173fccda-28_43_58_2457_1893}
Edexcel M1 2017 January Q1
9 marks Easy -1.2
  1. A train moves along a straight horizontal track between two stations \(R\) and \(S\). Initially the train is at rest at \(R\). The train accelerates uniformly at \(\frac { 1 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) from rest at \(R\) until it is moving with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For the next 200 seconds the train maintains a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The train then decelerates uniformly at \(\frac { 1 } { 4 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest at \(S\).
Find
  1. the time taken by the train to travel from \(R\) to \(S\),
  2. the distance from \(R\) to \(S\),
  3. the average speed of the train during the journey from \(R\) to \(S\).
Edexcel M1 2017 January Q2
9 marks Moderate -0.8
A particle \(P\) of mass 0.5 kg moves under the action of a single constant force ( \(2 \mathbf { i } + 3 \mathbf { j }\) )N.
  1. Find the acceleration of \(P\). At time \(t\) seconds, \(P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , \mathbf { v } = 4 \mathbf { i }\)
  2. Find the speed of \(P\) when \(t = 3\) Given that \(P\) is moving parallel to the vector \(2 \mathbf { i } + \mathbf { j }\) at time \(t = T\)
  3. find the value of \(T\).
Edexcel M1 2017 January Q3
8 marks Moderate -0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ba698f74-a51c-409a-a9d9-e9080fc87be2-05_520_730_264_607} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Two forces \(\mathbf { P }\) and \(\mathbf { Q }\) act on a particle at a point \(O\). Force \(\mathbf { P }\) has magnitude 6 N and force \(\mathbf { Q }\) has magnitude 7 N . The angle between the line of action of \(\mathbf { P }\) and the line of action of \(\mathbf { Q }\) is \(120 ^ { \circ }\), as shown in Figure 1. The resultant of \(\mathbf { P }\) and \(\mathbf { Q }\) is \(\mathbf { R }\). Find
  1. the magnitude of \(\mathbf { R }\),
  2. the angle between the line of action of \(\mathbf { R }\) and the line of action of \(\mathbf { P }\).
Edexcel M1 2017 January Q4
13 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ba698f74-a51c-409a-a9d9-e9080fc87be2-06_266_1440_239_251} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A plank \(A B\) of mass 20 kg and length 8 m is resting in a horizontal position on two supports at \(C\) and \(D\), where \(A C = 1.5 \mathrm {~m}\) and \(D B = 2 \mathrm {~m}\). A package of mass 8 kg is placed on the plank at \(C\), as shown in Figure 2. The plank remains horizontal and in equilibrium. The plank is modelled as a uniform rod and the package is modelled as a particle.
  1. Find the magnitude of the normal reaction
    1. between the plank and the support at \(C\),
    2. between the plank and the support at \(D\).
      (6) The package is now moved along the plank to the point \(E\). When the package is at \(E\), the magnitude of the normal reaction between the plank and the support at \(C\) is \(R\) newtons and the magnitude of the normal reaction between the plank and the support at \(D\) is \(2 R\) newtons.
  2. Find the distance \(A E\).
  3. State how you have used the fact that the package is modelled as a particle.
Edexcel M1 2017 January Q5
8 marks Standard +0.3
Two particles \(P\) and \(Q\) have masses \(4 m\) and \(k m\) respectively. They are moving towards each other in opposite directions along the same straight line on a smooth horizontal table when they collide directly. Immediately before the collision the speed of \(P\) is \(3 u\) and the speed of \(Q\) is \(u\). Immediately after the collision both particles have speed \(2 u\) and the direction of motion of \(Q\) has been reversed.
  1. Find, in terms of \(k , m\) and \(u\), the magnitude of the impulse received by \(Q\) in the collision.
  2. Find the two possible values of \(k\).
Edexcel M1 2017 January Q6
14 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ba698f74-a51c-409a-a9d9-e9080fc87be2-10_609_1013_118_456} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A particle \(P\) of mass 4 kg is held at rest at the point \(A\) on a rough plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The point \(B\) lies on the line of greatest slope of the plane that passes through \(A\). The point \(B\) is 5 m down the plane from \(A\), as shown in Figure 3. The coefficient of friction between the plane and \(P\) is 0.3 The particle is released from rest at \(A\) and slides down the plane.
  1. Find the speed of \(P\) at the instant it reaches \(B\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ba698f74-a51c-409a-a9d9-e9080fc87be2-10_478_1011_1343_456} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} The particle is now returned to \(A\) and is held in equilibrium by a horizontal force of magnitude \(H\) newtons, as shown in Figure 4. The line of action of the force lies in the vertical plane containing the line of greatest slope of the plane through \(A\). The particle is on the point of moving up the plane.
  2. Find the value of \(H\).
Edexcel M1 2017 January Q7
14 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ba698f74-a51c-409a-a9d9-e9080fc87be2-12_524_586_274_696} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Two particles \(P\) and \(Q\) have masses 3 kg and \(m \mathrm {~kg}\) respectively ( \(m > 3\) ). The particles are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut and the hanging parts of the string vertical. The particle \(Q\) is at a height of 10.5 m above the horizontal ground, as shown in Figure 5. The system is released from rest and \(Q\) moves downwards. In the subsequent motion \(P\) does not reach the pulley. After the system is released, the tension in the string is 33.6 N .
  1. Show that the magnitude of the acceleration of \(P\) is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the value of \(m\). The system is released from rest at time \(t = 0\). At time \(T _ { 1 }\) seconds after release, \(Q\) strikes the ground and does not rebound. The string goes slack and \(P\) continues to move upwards.
  3. Find the value of \(T _ { 1 }\) At time \(T _ { 2 }\) seconds after release, \(P\) comes to instantaneous rest.
  4. Find the value of \(T _ { 2 }\) At time \(T _ { 3 }\) seconds after release ( \(T _ { 3 } > T _ { 1 }\) ) the string becomes taut again.
  5. Sketch a velocity-time graph for the motion of \(P\) in the interval \(0 \leqslant t \leqslant T _ { 3 }\)
Edexcel M1 2018 January Q1
7 marks Moderate -0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{04b73f81-3316-4f26-ad98-a7be3a4b738f-02_297_812_240_567} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle of weight \(W\) is attached at \(C\) to two light inextensible strings \(A C\) and \(B C\). The other ends of the strings are attached to fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with the strings in a vertical plane and with \(A C\) and \(B C\) inclined to the horizontal at \(30 ^ { \circ }\) and \(45 ^ { \circ }\) respectively, as shown in Figure 1. Find, in terms of \(W\),
  1. the tension in \(A C\),
  2. the tension in \(B C\).
    VILLI SIHI NITIIIUM ION OC
    VILV SIHI NI JAHM ION OC
    VI4V SIHI NI JIIIM ION OC
Edexcel M1 2018 January Q2
6 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{04b73f81-3316-4f26-ad98-a7be3a4b738f-06_241_768_214_589} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of weight 40 N lies at rest in equilibrium on a fixed rough horizontal surface. A force of magnitude 20 N is applied to \(P\). The force acts at angle \(\theta\) to the horizontal, as shown in Figure 2. The coefficient of friction between \(P\) and the surface is \(\mu\). Given that the particle remains at rest, show that $$\mu \geqslant \frac { \cos \theta } { 2 + \sin \theta }$$ \includegraphics[max width=\textwidth, alt={}, center]{04b73f81-3316-4f26-ad98-a7be3a4b738f-07_119_167_2615_1777}
Edexcel M1 2018 January Q3
7 marks Standard +0.3
3. Two particles \(A\) and \(B\) have mass \(2 m\) and \(k m\) respectively. The particles are moving in opposite directions along the same straight smooth horizontal line so that the particles collide directly. Immediately before the collision \(A\) has speed \(2 u\) and \(B\) has speed \(u\). The direction of motion of each particle is reversed by the collision. Immediately after the collision the speed of \(A\) is \(\frac { u } { 2 }\).
  1. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted by \(B\) on \(A\) in the collision.
  2. Show that \(k < 5\)
Edexcel M1 2018 January Q4
8 marks Moderate -0.3
A package of mass 6 kg is held at rest at a fixed point \(A\) on a rough plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. The package is released from rest and slides down a line of greatest slope of the plane. The coefficient of friction between the package and the plane is \(\frac { 1 } { 4 }\). The package is modelled as a particle.
  1. Find the magnitude of the acceleration of the package. As it slides down the slope the package passes through the point \(B\), where \(A B = 10 \mathrm {~m}\).
  2. Find the speed of the package as it passes through \(B\).
Edexcel M1 2018 January Q5
12 marks Standard +0.3
5. A cyclist is travelling along a straight horizontal road. The cyclist starts from rest at point \(A\) on the road and accelerates uniformly at \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 20 seconds. He then moves at constant speed for \(4 T\) seconds, where \(T < 20\). He then decelerates uniformly at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and after \(T\) seconds passes through point \(B\) on the road. The distance from \(A\) to \(B\) is 705 m .
  1. Sketch a speed-time graph for the motion of the cyclist between points \(A\) and \(B\).
  2. Find the value of \(T\). The cyclist continues his journey, still decelerating uniformly at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), until he comes to rest at point \(C\) on the road.
  3. Find the total time taken by the cyclist to travel from \(A\) to \(C\).
Edexcel M1 2018 January Q6
9 marks Moderate -0.3
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.]
A particle \(P\) of mass 2 kg moves under the action of two forces, \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { N }\) and \(( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }\).
  1. Find the magnitude of the acceleration of \(P\). At time \(t = 0 , P\) has velocity ( \(- u \mathbf { i } + u \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\), where \(u\) is a positive constant. At time \(t = T\) seconds, \(P\) has velocity \(( 10 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  2. Find
    1. the value of \(T\),
    2. the value of \(u\).
Edexcel M1 2018 January Q7
12 marks Standard +0.3
7. A non-uniform rod \(A B\) has length 6 m and mass 8 kg . The rod rests in equilibrium, in a horizontal position, on two smooth supports at \(C\) and at \(D\), where \(A C = 1 \mathrm {~m}\) and \(D B = 1 \mathrm {~m}\), as shown in Figure 3. The magnitude of the reaction between the rod and the support at \(D\) is twice the magnitude of the reaction between the rod and the support at \(C\). The centre of mass of the rod is at \(G\), where \(A G = x \mathrm {~m}\).
  1. Show that \(x = \frac { 11 } { 3 }\). The support at \(C\) is moved to the point \(F\) on the rod, where \(A F = 2 \mathrm {~m}\). A particle of mass 3 kg is placed on the rod at \(A\). The rod remains horizontal and in equilibrium. The magnitude of the reaction between the rod and the support at \(D\) is \(k\) times the magnitude of the reaction between the rod and the support at \(F\).
  2. Find the value of \(k\).
    7. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{04b73f81-3316-4f26-ad98-a7be3a4b738f-20_223_1262_127_338}
    \end{figure}