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Edexcel FM1 AS 2020 June Q3
12 marks Standard +0.8
  1. Three particles \(A , B\) and \(C\) are at rest on a smooth horizontal plane. The particles lie along a straight line with \(B\) between \(A\) and \(C\).
Particle \(B\) has mass \(4 m\) and particle \(C\) has mass \(k m\), where \(k\) is a positive constant. Particle \(B\) is projected with speed \(u\) along the plane towards \(C\) and they collide directly. The coefficient of restitution between \(B\) and \(C\) is \(\frac { 1 } { 4 }\)
  1. Find the range of values of \(k\) for which there would be no further collisions. The magnitude of the impulse on \(B\) in the collision between \(B\) and \(C\) is \(3 m u\)
  2. Find the value of \(k\).
Edexcel FM1 AS 2020 June Q4
11 marks Standard +0.3
  1. A small ball, of mass \(m\), is thrown vertically upwards with speed \(\sqrt { 8 g H }\) from a point \(O\) on a smooth horizontal floor. The ball moves towards a smooth horizontal ceiling that is a vertical distance \(H\) above \(O\). The coefficient of restitution between the ball and the ceiling is \(\frac { 1 } { 2 }\)
    In a model of the motion of the ball, it is assumed that the ball, as it moves up or down, is subject to air resistance of constant magnitude \(\frac { 1 } { 2 } \mathrm { mg }\).
    Using this model,
    1. use the work-energy principle to find, in terms of \(g\) and \(H\), the speed of the ball immediately before it strikes the ceiling,
    2. find, in terms of \(g\) and \(H\), the speed of the ball immediately before it strikes the floor at \(O\) for the first time.
    In a simplified model of the motion of the ball, it is assumed that the ball, as it moves up or down, is subject to no air resistance. Using this simplified model,
  2. explain, without any detailed calculation, why the speed of the ball, immediately before it strikes the floor at \(O\) for the first time, would still be less than \(\sqrt { 8 g H }\)
Edexcel FM1 AS 2021 June Q1
7 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05f6f654-05e5-41d5-a6e4-11cd91a6df83-02_826_700_244_550} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A small book of mass \(m\) is held on a rough straight desk lid which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The book is released from rest at a distance of 0.5 m from the edge of the desk lid, as shown in Figure 1. The book slides down the desk lid and then hits the floor that is 0.8 m below the edge of the desk lid. The coefficient of friction between the book and the desk lid is 0.4 The book is modelled as a particle which, after leaving the desk lid, is assumed to move freely under gravity.
  1. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction on the book as it slides down the desk lid.
  2. Use the work-energy principle to find the speed of the book as it hits the floor.
Edexcel FM1 AS 2021 June Q2
9 marks Standard +0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05f6f654-05e5-41d5-a6e4-11cd91a6df83-06_458_278_248_986} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle of mass em is at rest on a smooth horizontal plane between two smooth fixed parallel vertical walls, as shown in the plan view in Figure 2. The particle is projected along the plane with speed \(u\) towards one of the walls and strikes the wall at right angles. The coefficient of restitution between the particle and each wall is \(e\) and air resistance is modelled as being negligible. Using the model,
  1. find, in terms of \(m , u\) and \(e\), an expression for the total loss in the kinetic energy of the particle as a result of the first two impacts. Given that \(e\) can vary such that \(0 < e < 1\) and using the model,
  2. find the value of \(e\) for which the total loss in the kinetic energy of the particle as a result of the first two impacts is a maximum,
  3. describe the subsequent motion of the particle.
Edexcel FM1 AS 2021 June Q3
11 marks Standard +0.3
  1. The total mass of a cyclist and his bicycle is 100 kg .
In all circumstances, the magnitude of the resistance to the motion of the cyclist from non-gravitational forces is modelled as being \(k v ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the cyclist. The cyclist can freewheel, without pedalling, down a slope that is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 35 }\), at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) When he is pedalling up a slope that is inclined to the horizontal at an angle \(\beta\), where \(\sin \beta = \frac { 1 } { 70 }\), and he is moving at the same constant speed \(V \mathrm {~ms} ^ { - 1 }\), he is working at a constant rate of \(P\) watts.
  1. Find \(P\) in terms of \(V\). If he pedals and works at a rate of 35 V watts on a horizontal road, he moves at a constant speed of \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  2. Find \(U\) in terms of \(V\).
Edexcel FM1 AS 2021 June Q4
13 marks Standard +0.8
  1. Two particles, \(P\) and \(Q\), have masses \(m\) and \(e m\) respectively. The particles are moving on a smooth horizontal plane in the same direction along the same straight line when they collide directly. The coefficient of restitution between \(P\) and \(Q\) is \(e\), where \(0 < e < 1\)
Immediately before the collision the speed of \(P\) is \(u\) and the speed of \(Q\) is \(e u\).
  1. Show that the speed of \(Q\) immediately after the collision is \(u\).
  2. Show that the direction of motion of \(P\) is unchanged by the collision. The magnitude of the impulse on \(Q\) in the collision is \(\frac { 2 } { 9 } m u\)
  3. Find the possible values of \(e\).
Edexcel FM1 AS 2022 June Q1
5 marks Standard +0.3
  1. A car of mass 1200 kg moves up a straight road that is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 15 }\)
The total resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons. At the instant when the engine of the car is working at a rate of 32 kW and the speed of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the acceleration of the car is \(0.5 \mathrm {~ms} ^ { - 2 }\) Find the value of \(R\)
Edexcel FM1 AS 2022 June Q2
8 marks Standard +0.3
  1. Two particles, \(A\) and \(B\), have masses \(m\) and \(3 m\) respectively. The particles are moving in opposite directions along the same straight line on a smooth horizontal plane when they collide directly.
Immediately before they collide, \(A\) is moving with speed \(2 u\) and \(B\) is moving with speed \(u\). The direction of motion of each particle is reversed by the collision.
In the collision, the magnitude of the impulse exerted on \(A\) by \(B\) is \(\frac { 9 m u } { 2 }\)
  1. Find the value of the coefficient of restitution between \(A\) and \(B\).
  2. Hence, write down the total loss in kinetic energy due to the collision, giving a reason for your answer.
Edexcel FM1 AS 2022 June Q3
12 marks Standard +0.3
  1. A plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\)
A particle \(P\) is held at rest at a point \(A\) on the plane.
The particle \(P\) is then projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\), up a line of greatest slope of the plane. In an initial model, the plane is modelled as being smooth and air resistance is modelled as being negligible. Using this model and the principle of conservation of mechanical energy,
  1. find the speed of \(P\) at the instant when it has travelled a distance \(\frac { 25 } { 6 } \mathrm {~m}\) up the plane from \(A\). In a refined model, the plane is now modelled as being rough, with the coefficient of friction between \(P\) and the plane being \(\frac { 3 } { 5 }\) Air resistance is still modelled as being negligible.
    Using this refined model and the work-energy principle,
  2. find the speed of \(P\) at the instant when it has travelled a distance \(\frac { 25 } { 6 } \mathrm {~m}\) up the plane from \(A\).
Edexcel FM1 AS 2022 June Q4
15 marks Challenging +1.2
  1. A particle \(P\) of mass \(2 m \mathrm {~kg}\) is moving with speed \(2 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal plane. Particle \(P\) collides with a particle \(Q\) of mass \(3 m \mathrm {~kg}\) which is at rest on the plane. The coefficient of restitution between \(P\) and \(Q\) is \(e\). Immediately after the collision the speed of \(Q\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
    1. Show that \(v = \frac { 4 u ( 1 + e ) } { 5 }\)
    2. Show that \(\frac { 4 u } { 5 } \leqslant v \leqslant \frac { 8 u } { 5 }\)
    Given that the direction of motion of \(P\) is reversed by the collision,
  2. find, in terms of \(u\) and \(e\), the speed of \(P\) immediately after the collision. After the collision, \(Q\) hits a wall, that is fixed at right angles to the direction of motion of \(Q\), and rebounds. The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 6 }\)
    Given that \(P\) and \(Q\) collide again,
  3. find the full range of possible values of \(e\).
Edexcel FM1 AS 2023 June Q1
8 marks Standard +0.3
  1. Two particles, \(P\) and \(Q\), of masses \(3 m\) and \(2 m\) respectively, are moving on a smooth horizontal plane. They are moving in opposite directions along the same straight line when they collide directly.
Immediately before the collision, \(P\) is moving with speed \(2 u\).
The magnitude of the impulse exerted on \(P\) by \(Q\) in the collision is \(\frac { 9 m u } { 2 }\)
  1. Find the speed of \(P\) immediately after the collision. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
    Given that the speed of \(Q\) immediately before the collision is \(u\),
  2. find the value of \(e\).
Edexcel FM1 AS 2023 June Q2
8 marks Standard +0.8
  1. A racing car of mass 750 kg is moving along a straight horizontal road at a constant speed of \(U \mathbf { k m ~ h } ^ { - \mathbf { 1 } }\). The engine of the racing car is working at a constant rate of 60 kW .
The resistance to the motion of the racing car is modelled as a force of magnitude \(37.5 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the racing car. Using the model,
  1. find the value of \(U\) Later on, the racing car is accelerating up a straight road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 5 } { 49 }\). The engine of the racing car is working at a constant rate of 60 kW . The total resistance to the motion of the racing car from non-gravitational forces is modelled as a force of magnitude \(37.5 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the racing car. At the instant when the acceleration of the racing car is \(2 \mathrm {~ms} ^ { - 2 }\), the speed of the racing car is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Using the model,
  2. find the value of \(V\)
Edexcel FM1 AS 2023 June Q3
10 marks Standard +0.3
  1. A stone of mass 0.5 kg is projected vertically upwards with a speed \(U \mathrm {~ms} ^ { - 1 }\) from a point \(A\). The point \(A\) is 2.5 m above horizontal ground.
The speed of the stone as it hits the ground is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
The motion of the stone from the instant it is projected from \(A\) until the instant it hits the ground is modelled as that of a particle moving freely under gravity.
  1. Use the model and the principle of conservation of mechanical energy to find the value of \(U\). In reality, the stone will be subject to air resistance as it moves from \(A\) to the ground.
  2. State how this would affect your answer to part (a). The ground is soft and the stone sinks a vertical distance \(d \mathrm {~cm}\) into the ground. The resistive force exerted on the stone by the ground is modelled as a constant force of magnitude 2000 N and the stone is modelled as a particle.
  3. Use the model and the work-energy principle to find the value of \(d\), giving your answer to 3 significant figures.
Edexcel FM1 AS 2023 June Q4
14 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0cec16c3-23a0-4620-a80f-b5d4e014e2fc-12_81_1383_255_342} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Three particles, \(P , Q\) and \(R\), lie at rest on a smooth horizontal plane. The particles are in a straight line with \(Q\) between \(P\) and \(R\), as shown in Figure 1 . Particle \(P\) is projected towards \(Q\) with speed \(u\). At the same time, \(R\) is projected with speed \(\frac { 1 } { 2 } u\) away from \(Q\), in the direction \(Q R\). Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(2 m\).
The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  1. Show that the speed of \(Q\) immediately after the collision between \(P\) and \(Q\) is $$\frac { u ( 1 + e ) } { 3 }$$ It is given that \(e > \frac { 1 } { 2 }\)
  2. Determine whether there is a collision between \(Q\) and \(R\).
  3. Determine the direction of motion of \(P\) immediately after the collision between \(P\) and \(Q\).
  4. Find, in terms of \(m , u\) and \(e\), the total kinetic energy lost in the collision between \(P\) and \(Q\), simplifying your answer.
  5. Explain how using \(e = 1\) could be used to check your answer to part (d).
Edexcel FM1 AS 2024 June Q1
9 marks Standard +0.3
  1. A particle \(A\) has mass \(2 m\) and a particle \(B\) has mass \(3 m\). The particles are moving in opposite directions along the same straight line and collide directly.
Immediately before the collision, the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\). Immediately after the collision, the speed of \(A\) is \(0.5 u\) and the speed of \(B\) is \(w\). Given that the direction of motion of each particle is reversed by the collision,
  1. find \(w\) in terms of \(u\)
  2. find the coefficient of restitution between the particles,
  3. find, in terms of \(m\) and \(u\), the magnitude of the impulse received by \(A\) in the collision.
Edexcel FM1 AS 2024 June Q2
8 marks Standard +0.3
  1. A lorry has mass 5000 kg .
In all circumstances, when the speed of the lorry is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to motion of the lorry from non-gravitational forces is modelled as having magnitude \(490 v\) newtons. The lorry moves along a straight horizontal road at \(12 \mathrm {~ms} ^ { - 1 }\), with its engine working at a constant rate of 84 kW . Using the model,
  1. find the acceleration of the lorry. Another straight road is inclined to the horizontal at an angle \(\alpha\) where \(\sin \alpha = \frac { 1 } { 14 }\)
    With its engine again working at a constant rate of 84 kW , the lorry can maintain a constant speed of \(V \mathrm {~ms} ^ { - 1 }\) up the road. Using the model,
  2. find the value of \(V\).
Edexcel FM1 AS 2024 June Q3
12 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{885dd96e-ecaa-4a7f-acb4-f5cf636f491b-06_458_725_246_671} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the end elevation of a building which sits on horizontal ground. The side of the building is vertical and has height \(h\). A small stone of mass \(m\) is at rest on the roof of the building at the point \(A\). The stone slides from rest down a line of greatest slope of the roof and reaches the edge \(B\) of the roof with speed \(\sqrt { 2 g h }\) The stone then moves under gravity before hitting the ground with speed \(W\).
In a model of the motion of the stone from \(\boldsymbol { B }\) to the ground
  • the stone is modelled as a particle
  • air resistance is ignored
Using the principle of conservation of mechanical energy and the model,
  1. find \(W\) in terms of \(g\) and \(h\). In a model of the motion of the stone from \(\boldsymbol { A }\) to \(\boldsymbol { B }\)
    • the stone is modelled as a particle of mass \(m\)
    • air resistance is ignored
    • the roof of the building is modelled as a rough plane inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 3 } { 4 }\)
    • the coefficient of friction between the stone and the roof is \(\frac { 1 } { 3 }\)
    • \(A B = d\)
    Using this model,
  2. find, in terms of \(m\) and \(g\), the magnitude of the frictional force acting on the stone as it slides down the roof,
  3. use the work-energy principle to find \(d\) in terms of \(h\).
Edexcel FM1 AS 2024 June Q4
11 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{885dd96e-ecaa-4a7f-acb4-f5cf636f491b-10_232_887_246_589} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of mass \(m\) and a particle \(Q\) of mass \(4 m\) are at rest on a smooth horizontal plane, as shown in Figure 2. Particle \(P\) is projected with speed \(u\) along the plane towards \(Q\) and the particles collide.
The coefficient of restitution between the particles is \(e\), where \(e > \frac { 1 } { 4 }\) As a result of the collision, the direction of motion of \(P\) is reversed and \(P\) has speed \(\frac { u } { 5 } ( 4 e - 1 )\).
  1. Find, in terms of \(u\) and \(e\), the speed of \(Q\) after the collision. After the collision, \(P\) goes on to hit a vertical wall which is fixed at right angles to the direction of motion of \(P\). The coefficient of restitution between \(P\) and the wall is \(f\), where \(f > 0\)
    Given that \(e = \frac { 3 } { 4 }\)
  2. find, in terms of \(m , u\) and \(f\), the kinetic energy lost by \(P\) as a result of its impact with the wall. Give your answer in its simplest form. After its impact with the wall, \(P\) goes on to collide with \(Q\) again.
  3. Find the complete range of possible values of \(f\).
Edexcel FM1 AS Specimen Q1
10 marks Moderate -0.8
  1. A small ball of mass 0.1 kg is dropped from a point which is 2.4 m above a horizontal floor. The ball falls freely under gravity, strikes the floor and bounces to a height of 0.6 m above the floor. The ball is modelled as a particle.
    1. Show that the coefficient of restitution between the ball and the floor is 0.5
    2. Find the height reached by the ball above the floor after it bounces on the floor for the second time.
    3. By considering your answer to (b), describe the subsequent motion of the ball.
Edexcel FM1 AS Specimen Q2
6 marks Standard +0.3
  1. A small stone of mass 0.5 kg is thrown vertically upwards from a point A with an initial speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The stone first comes to instantaneous rest at the point B which is 20 m vertically above the point A . As the stone moves it is subject to air resistance. The stone is modelled as a particle.
    1. Find the energy lost due to air resistance by the stone, as it moves from A to B
    The air resistance is modelled as a constant force of magnitude \(R\) newtons.
  2. Find the value of R .
  3. State how the model for air resistance could be refined to make it more realistic.
Edexcel FM1 AS Specimen Q3
8 marks Moderate -0.3
  1. \hspace{0pt} [In this question use \(\mathrm { g } = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) ]
A jogger of mass 60 kg runs along a straight horizontal road at a constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The total resistance to the motion of the jogger is modelled as a constant force of magnitude 30 N .
  1. Find the rate at which the jogger is working. The jogger now comes to a hill which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 15 }\). Because of the hill, the jogger reduces her speed to \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and maintains this constant speed as she runs up the hill. The total resistance to the motion of the jogger from non-gravitational forces continues to be modelled as a constant force of magnitude 30 N .
  2. Find the rate at which she has to work in order to run up the hill at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Edexcel FM1 AS Specimen Q4
16 marks Standard +0.8
  1. A particle P of mass 3 m is moving in a straight line on a smooth horizontal table. A particle \(Q\) of mass \(m\) is moving in the opposite direction to \(P\) along the same straight line. The particles collide directly. Immediately before the collision the speed of P is u and the speed of Q is 2 u . The velocities of P and Q immediately after the collision, measured in the direction of motion of P before the collision, are V and W respectively. The coefficient of restitution between P and Q is e .
    1. Find an expression for v in terms of u and e .
    Given that the direction of motion of P is changed by the collision,
  2. find the range of possible values of e.
  3. Show that \(\mathrm { w } = \frac { \mathrm { u } } { 4 } ( 1 + 9 \mathrm { e } )\). Following the collision with P , the particle Q then collides with and rebounds from a fixed vertical wall which is perpendicular to the direction of motion of \(Q\). The coefficient of restitution between \(Q\) and the wall is \(f\).
    Given that \(\mathrm { e } = \frac { 5 } { 9 }\), and that P and Q collide again in the subsequent motion,
  4. find the range of possible values of f .
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Edexcel FM2 AS 2018 June Q1
7 marks Challenging +1.2
1. Figure 1 A thin uniform rod, of total length \(30 a\) and mass \(M\), is bent to form a frame. The frame is in the shape of a triangle \(A B C\), where \(A B = 12 a , B C = 5 a\) and \(C A = 13 a\), as shown in Figure 1.
  1. Show that the centre of mass of the frame is \(\frac { 3 } { 2 } a\) from \(A B\). The frame is freely suspended from \(A\). A horizontal force of magnitude \(k M g\), where \(k\) is a constant, is applied to the frame at \(B\). The line of action of the force lies in the vertical plane containing the frame. The frame hangs in equilibrium with \(A B\) vertical.
  2. Find the value of \(k\).
Edexcel FM2 AS 2018 June Q2
9 marks Standard +0.3
  1. A car moves round a bend which is banked at a constant angle of \(\theta ^ { \circ }\) to the horizontal.
When the car is travelling at a constant speed of \(80 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) there is no sideways frictional force on the car. The car is modelled as a particle moving in a horizontal circle of radius 500 m .
  1. Find the value of \(\theta\).
  2. Identify one limitation of this model. The speed of the car is increased so that it is now travelling at a constant speed of \(90 \mathrm { kmh } ^ { - 1 }\) The car is still modelled as a particle moving in a horizontal circle of radius 500 m .
  3. Describe the extra force that will now be acting on the car, stating the direction of this force.
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Edexcel FM2 AS 2018 June Q3
11 marks Standard +0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{66c0f4c8-938e-4c05-93a7-99ea26ea0348-08_694_710_382_780} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The lamina \(L\), shown in Figure 2, consists of a uniform square lamina \(A B D F\) and two uniform triangular laminas \(B D C\) and \(F D E\). The square has sides of length \(2 a\). The two triangles are identical. The straight lines \(B D E\) and \(F D C\) are perpendicular with \(B D = D F = 2 a\) and \(D C = D E = a\).
The mass per unit of area of the square is \(M\).
The mass per unit area of each triangle is \(3 M\).
The centre of mass of \(L\) is at the point \(G\).
  1. Without doing any calculations, explain why \(G\) lies on \(A D\).
  2. Show that the distance of \(G\) from \(D\) is \(\frac { \sqrt { 2 } } { 2 } a\) The lamina \(L\) is freely suspended from \(B\) and hangs in equilibrium.
  3. Find the size of the angle between \(B E\) and the downward vertical.
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