Questions — Edexcel M1 (663 questions)

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Edexcel M1 2024 January Q4
8 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e59a66b8-c2ad-41fd-9959-9d21e9455c37-08_399_889_246_587} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows two horizontal forces \(\mathbf { P }\) and \(\mathbf { Q }\) acting on a particle.
The angle between the direction of \(\mathbf { P }\) and the direction of \(\mathbf { Q }\) is \(150 ^ { \circ }\) Force \(\mathbf { P }\) has magnitude \(X\) newtons.
Force \(\mathbf { Q }\) has magnitude \(5 \sqrt { 3 } \mathrm {~N}\).
The resultant of \(\mathbf { P }\) and \(\mathbf { Q }\) has magnitude \(\sqrt { 129 } \mathrm {~N}\).
Find
  1. the value of \(X\).
  2. the angle between \(\mathbf { Q }\) and the resultant, giving your answer to the nearest degree.
Edexcel M1 2024 January Q5
10 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e59a66b8-c2ad-41fd-9959-9d21e9455c37-12_412_1529_242_267} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A beam \(A B\) has mass 30 kg and length 3 m .
The beam rests on supports at \(C\) and \(D\) where \(A C = 0.4 \mathrm {~m}\) and \(D B = 0.4 \mathrm {~m}\), as shown in Figure 4. A person of mass 55 kg stands on the beam between \(C\) and \(D\).
The person is modelled as a particle at the point \(P\), where \(C P = x\) metres and \(0 < x < 2.2\) The beam is modelled as a uniform rod resting in equilibrium in a horizontal position.
Using the model,
  1. show that the magnitude of the reaction at \(C\) is \(( 686 - 245 x ) \mathrm { N }\). The magnitude of the reaction at \(C\) is four times the magnitude of the reaction at \(D\).
    Using the model,
  2. find the value of \(x\) The person steps off the beam and places a package of mass \(M \mathrm {~kg}\) at \(A\).
    The package is modelled as a particle at the point \(A\).
    The beam is now on the point of tilting about \(C\).
    Using the model,
  3. find the value of \(M\)
Edexcel M1 2024 January Q6
12 marks Moderate -0.8
  1. A particle is projected vertically upwards from a point \(A\) with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
The point \(A\) is 2.5 m vertically above the point \(B\).
Point \(B\) lies on horizontal ground.
The particle moves freely under gravity until it hits the ground at \(B\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) After hitting the ground the particle does not rebound.
  1. Find the value of \(V\).
  2. Find the time taken for the particle to reach \(B\). The point \(C\) is 10 m vertically above \(A\).
  3. Find the length of time for which the particle is above \(C\).
  4. Sketch a speed-time graph for the motion of the particle from projection to the instant that it reaches \(B\). (No further calculations are required.)
Edexcel M1 2024 January Q7
11 marks Standard +0.3
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin \(O\).]
At midnight, a ship \(S\) is at the point with position vector ( \(19 \mathbf { i } + 22 \mathbf { j }\) )km
The ship travels with constant velocity \(( 12 \mathbf { i } - 16 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\)
  1. Find the speed of \(S\). At time \(t\) hours after midnight, the position vector of \(S\) is \(\mathbf { s } \mathrm { km }\).
  2. Find an expression for \(\mathbf { s }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(t\). A lighthouse stands on a small rocky island. The lighthouse is modelled as being at the point with position vector \(( 26 \mathbf { i } + 15 \mathbf { j } ) \mathrm { km }\). It is not safe for ships to be within 1.3 km of the lighthouse.
    1. Find the value of \(t\) when \(S\) is closest to the lighthouse.
    2. Hence determine whether it is safe for \(S\) to continue its course.
Edexcel M1 2024 January Q8
12 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e59a66b8-c2ad-41fd-9959-9d21e9455c37-24_346_961_246_543} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A fixed rough plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) A small smooth pulley is fixed at the top of the plane.
One end of a light inextensible string is attached to a particle \(P\) which is at rest on the plane. The string passes over the pulley and the other end of the string is attached to a particle \(Q\) which hangs vertically below the pulley, as shown in Figure 5. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(0.5 m\) The string from \(P\) to the pulley lies along a line of greatest slope of the plane.
The coefficient of friction between \(P\) and the plane is \(\mu\).
The system is in limiting equilibrium with the string taut and \(P\) is on the point of slipping up the plane.
  1. Find the value of \(\mu\). The string breaks and \(P\) begins to move down the plane.
    When particle \(P\) has travelled a distance of 0.8 m down the plane, the speed of \(P\) is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  2. Find the value of \(V\).
Edexcel M1 2014 June Q1
5 marks Moderate -0.8
  1. Two small smooth balls \(A\) and \(B\) have mass 0.6 kg and 0.9 kg respectively. They are moving in a straight line towards each other in opposite directions on a smooth horizontal floor and collide directly. Immediately before the collision the speed of \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The speed of \(A\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) immediately after the collision and \(B\) is brought to rest by the collision.
Find
  1. the value of \(v\),
  2. the magnitude of the impulse exerted on \(A\) by \(B\) in the collision.
Edexcel M1 2014 June Q2
8 marks Standard +0.3
2. A ball is thrown vertically upwards with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\), which is \(h\) metres above the ground. The ball moves freely under gravity until it hits the ground 5 s later.
  1. Find the value of \(h\). A second ball is thrown vertically downwards with speed \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) and moves freely under gravity until it hits the ground. The first ball hits the ground with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the second ball hits the ground with speed \(\frac { 3 } { 4 } V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the value of \(w\).
Edexcel M1 2014 June Q3
12 marks Standard +0.3
3. A particle \(P\) of mass 1.5 kg is placed at a point \(A\) on a rough plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.6
  1. Show that \(P\) rests in equilibrium at \(A\). A horizontal force of magnitude \(X\) newtons is now applied to \(P\), as shown in Figure 1. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{edcc4603-f006-4c4f-a4e5-063cab41da98-04_236_584_667_680} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The particle is on the point of moving up the plane.
  2. Find
    1. the magnitude of the normal reaction of the plane on \(P\),
    2. the value of \(X\).
Edexcel M1 2014 June Q4
10 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{edcc4603-f006-4c4f-a4e5-063cab41da98-06_262_1132_223_415} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A plank \(A B\), of length 6 m and mass 4 kg , rests in equilibrium horizontally on two supports at \(C\) and \(D\), where \(A C = 2 \mathrm {~m}\) and \(D B = 1 \mathrm {~m}\). A brick of mass 2 kg rests on the plank at \(A\) and a brick of mass 3 kg rests on the plank at \(B\), as shown in Figure 2. The plank is modelled as a uniform rod and all bricks are modelled as particles.
  1. Find the magnitude of the reaction exerted on the plank
    1. by the support at \(C\),
    2. by the support at \(D\). The 3 kg brick is now removed and replaced with a brick of mass \(x \mathrm {~kg}\) at \(B\). The plank remains horizontal and in equilibrium but the reactions on the plank at \(C\) and at \(D\) now have equal magnitude.
  2. Find the value of \(x\).
Edexcel M1 2014 June Q5
11 marks Moderate -0.5
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively. Position vectors are given relative to a fixed origin \(O\).]
A boy \(B\) is running in a field with constant velocity ( \(3 \mathbf { i } - 2 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\). At time \(t = 0 , B\) is at the point with position vector 10j m . Find
  1. the speed of \(B\),
  2. the direction in which \(B\) is running, giving your answer as a bearing. At time \(t = 0\), a girl \(G\) is at the point with position vector \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\). The girl is running with constant velocity \(\left( \frac { 5 } { 3 } \mathbf { i } + 2 \mathbf { j } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\) and meets \(B\) at the point \(P\).
  3. Find
    1. the value of \(t\) when they meet,
    2. the position vector of \(P\).
Edexcel M1 2014 June Q6
13 marks Moderate -0.3
  1. A car starts from rest at a point \(A\) and moves along a straight horizontal road. The car moves with constant acceleration \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for the first 8 s . The car then moves with constant acceleration \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for the next 20 s . It then moves with constant speed for \(T\) seconds before slowing down with constant deceleration \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it stops at a point \(B\).
    1. Find the speed of the car 28 s after leaving \(A\).
    2. Sketch, in the space provided, a speed-time graph to illustrate the motion of the car as it travels from \(A\) to \(B\).
    3. Find the distance travelled by the car during the first 28 s of its journey from \(A\).
    The distance from \(A\) to \(B\) is 2 km .
  2. Find the value of \(T\).
Edexcel M1 2014 June Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{edcc4603-f006-4c4f-a4e5-063cab41da98-12_486_1257_230_347} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 3 kg respectively, are connected by a light inextensible string. Initially \(P\) is held at rest on a fixed smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The string passes over a small smooth fixed pulley at the top of the plane. The particle \(Q\) hangs freely below the pulley and 0.6 m above the ground, as shown in Figure 3. The part of the string from \(P\) to the pulley is parallel to a line of greatest slope of the plane. The system is released from rest with the string taut. For the motion before \(Q\) hits the ground,
    1. show that the acceleration of \(Q\) is \(\frac { 2 g } { 5 }\),
    2. find the tension in the string. On hitting the ground \(Q\) is immediately brought to rest by the impact.
  2. Find the speed of \(P\) at the instant when \(Q\) hits the ground. In its subsequent motion \(P\) does not reach the pulley.
  3. Find the total distance moved up the plane by \(P\) before it comes to instantaneous rest.
  4. Find the length of time between \(Q\) hitting the ground and \(P\) first coming to instantaneous rest.
Edexcel M1 2015 June Q1
6 marks Moderate -0.8
  1. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a particle \(P\).
$$\mathbf { F } _ { 1 } = ( 2 \mathbf { i } + 3 a \mathbf { j } ) \mathrm { N } ; \quad \mathbf { F } _ { 2 } = ( 2 a \mathbf { i } + b \mathbf { j } ) \mathrm { N } ; \quad \mathbf { F } _ { 3 } = ( b \mathbf { i } + 4 \mathbf { j } ) \mathrm { N } .$$ The particle \(P\) is in equilibrium under the action of these forces.
Find the value of \(a\) and the value of \(b\).
Edexcel M1 2015 June Q2
9 marks Standard +0.3
2. Particle \(A\) of mass \(2 m\) and particle \(B\) of mass \(k m\), where \(k\) is a positive constant, are moving towards each other in opposite directions along the same straight line on a smooth horizontal plane. The particles collide directly. Immediately before the collision the speed of \(A\) is \(u\) and the speed of \(B\) is \(3 u\). The direction of motion of each particle is reversed by the collision. Immediately after the collision the speed of \(A\) is \(\frac { 1 } { 2 } u\).
  1. Show that \(k < 1\)
  2. Find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) by \(A\) in the collision.
Edexcel M1 2015 June Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3280fdf1-d81a-4729-b065-e84dece6a220-05_325_947_267_493} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle \(P\) of mass 2 kg is pushed by a constant horizontal force of magnitude 30 N up a line of greatest slope of a rough plane. 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 the line of greatest slope of the plane. The particle \(P\) starts from rest. The coefficient of friction between \(P\) and the plane is \(\mu\). After 2 seconds, \(P\) has travelled a distance of 5.5 m up the plane.
  1. Find the acceleration of \(P\) up the plane.
  2. Find the value of \(\mu\).
Edexcel M1 2015 June Q4
7 marks Standard +0.3
  1. A small stone is released from rest from a point \(A\) which is at height \(h\) metres above horizontal ground. Exactly one second later another small stone is projected with speed \(19.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards from a point \(B\), which is also at height \(h\) metres above the horizontal ground. The motion of each stone is modelled as that of a particle moving freely under gravity. The two stones hit the ground at the same time.
Find the value of \(h\).
Edexcel M1 2015 June Q5
10 marks Standard +0.3
5. A car travelling along a straight horizontal road takes 170 s to travel between two sets of traffic lights at \(A\) and \(B\) which are 2125 m apart. The car starts from rest at \(A\) and moves with constant acceleration until it reaches a speed of \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then maintains this speed before moving with constant deceleration, coming to rest at \(B\). The magnitude of the deceleration is twice the magnitude of the acceleration.
  1. Sketch, in the space below, a speed-time graph for the motion of the car between \(A\) and \(B\).
  2. Find the deceleration of the car.
Edexcel M1 2015 June Q6
12 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3280fdf1-d81a-4729-b065-e84dece6a220-10_238_1258_267_342} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A plank \(A B\) has length 4 m and mass 6 kg . The plank rests in a horizontal position on two supports, one at \(B\) and one at \(C\), where \(A C = 1.5 \mathrm {~m}\). A load of mass 15 kg is placed on the plank at the point \(X\), as shown in Figure 2, and the plank remains horizontal and in equilibrium. The plank is modelled as a uniform rod and the load is modelled as a particle. The magnitude of the reaction on the plank at \(C\) is twice the magnitude of the reaction on the plank at \(B\).
  1. Find the magnitude of the reaction on the plank at \(C\).
  2. Find the distance \(A X\). The load is now moved along the plank to a point \(Y\), between \(A\) and \(C\). Given that the plank is on the point of tipping about \(C\),
  3. find the distance \(A Y\).
Edexcel M1 2015 June Q7
5 marks Moderate -0.3
  1. A particle \(P\) moves from point \(A\) to point \(B\) with constant acceleration \(( c \mathbf { i } + d \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\), where \(c\) and \(d\) are positive constants. The velocity of \(P\) at \(A\) is \(( - 3 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(P\) at \(B\) is \(( 2 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The magnitude of the acceleration of \(P\) is \(2.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
Find the value of \(c\) and the value of \(d\).
Edexcel M1 2015 June Q8
16 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3280fdf1-d81a-4729-b065-e84dece6a220-13_648_1280_271_331} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Two particles \(P\) and \(Q\) have masses \(m\) and \(4 m\) 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 light pulley which is fixed at the edge of the table. Particle \(Q\) hangs at rest vertically below the pulley, at a height \(h\) above a horizontal plane, as shown in Figure 3. The coefficient of friction between \(P\) and the table is 0.5 . Particle \(P\) is released from rest with the string taut and slides along the table.
  1. Find, in terms of \(m g\), the tension in the string while both particles are moving. The particle \(P\) does not reach the pulley before \(Q\) hits the plane.
  2. Show that the speed of \(Q\) immediately before it hits the plane is \(\sqrt { 1.4 g h }\) When \(Q\) hits the plane, \(Q\) does not rebound and \(P\) continues to slide along the table. Given that \(P\) comes to rest before it reaches the pulley,
  3. show that the total length of the string must be greater than 2.4 h
Edexcel M1 2017 June Q1
7 marks Moderate -0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c3869c7-008f-4131-b68d-8ecdd4da3377-02_346_499_251_721} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle \(P\) of weight 5 N is attached to one end of a light string. The other end of the string is attached to a fixed point \(O\). A force of magnitude \(F\) newtons is applied to \(P\). The line of action of the force is inclined to the horizontal at \(30 ^ { \circ }\) and lies in the same vertical plane as the string. The particle \(P\) is in equilibrium with the string making an angle of \(40 ^ { \circ }\) with the downward vertical, as shown in Figure 1. Find
  1. the tension in the string,
  2. the value of \(F\).
Edexcel M1 2017 June Q2
9 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5c3869c7-008f-4131-b68d-8ecdd4da3377-04_429_1298_255_324} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A wooden beam \(A B\) has weight 140 N and length \(2 a\) metres. The beam rests horizontally in equilibrium on two supports at \(C\) and \(D\), where \(A C = 2 \mathrm {~m}\) and \(A D = 6 \mathrm {~m}\). A block of weight 30 N is placed on the beam at \(B\) and the beam remains horizontal and in equilibrium, as shown in Figure 2. The reaction on the beam at \(D\) has magnitude 120 N . The block is modelled as a particle and the beam is modelled as a uniform rod.
  1. Find the value of \(a\). The support at \(D\) is now moved to a point \(E\) on the beam and the beam remains horizontal and in equilibrium with the block at \(B\). The magnitude of the reaction on the beam at \(C\) is now equal to the magnitude of the reaction on the beam at \(E\).
  2. Find the distance \(A E\).
Edexcel M1 2017 June Q3
7 marks Standard +0.3
3. Two particles, \(P\) and \(Q\), have masses 0.5 kg and \(m \mathrm {~kg}\) respectively. They are moving in opposite directions towards each other along the same straight line on a smooth horizontal plane and collide directly. Immediately before the collision the speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\) and the speed of \(Q\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The magnitude of the impulse exerted on \(P\) by \(Q\) in the collision is 4.2 N s . As a result of the collision the direction of motion of \(P\) is reversed.
  1. Find the speed of \(P\) immediately after the collision. The speed of \(Q\) immediately after the collision is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the two possible values of \(m\).
Edexcel M1 2017 June Q4
8 marks Moderate -0.8
  1. A small ball of mass 0.2 kg is moving vertically downwards when it hits a horizontal floor. Immediately before hitting the floor the ball has speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after hitting the floor the ball rebounds vertically with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Find the magnitude of the impulse exerted by the floor on the ball.
    By modelling the motion of the ball as that of a particle moving freely under gravity,
  2. find the maximum height above the floor reached by the ball after it has rebounded from the floor,
  3. find the time between the instant when the ball first hits the floor and the instant when the ball is first 1 m above the floor and moving upwards.
Edexcel M1 2017 June Q5
13 marks Standard +0.3
  1. Two trains, \(P\) and \(Q\), move on horizontal parallel straight tracks. Initially both are at rest in a station and level with each other. At time \(t = 0 , P\) starts off and moves with constant acceleration for 10 s up to a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then moves at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t = 20\), where \(t\) is measured in seconds, train \(Q\) starts to move in the same direction as \(P\). Train \(Q\) accelerates with the same initial constant acceleration as \(P\), up to a speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then moves at a constant speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Train \(Q\) overtakes \(P\) at time \(t = T\), after both trains have reached their constant speeds.
    1. Sketch, on the same axes, the speed-time graphs of both trains for \(0 \leqslant t \leqslant T\).
    2. Find the value of \(t\) at the instant when both trains are moving at the same speed.
    3. Find the value of \(T\).