Questions — Edexcel FM1 AS (32 questions)

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Edexcel FM1 AS 2018 June Q1
  1. A small ball of mass 0.3 kg is released from rest from a point 3.6 m above horizontal ground. The ball falls freely under gravity, hits the ground and rebounds vertically upwards.
In the first impact with the ground, the ball receives an impulse of magnitude 4.2 Ns . The ball is modelled as a particle.
  1. Find the speed of the ball immediately after it first hits the ground.
  2. Find the kinetic energy lost by the ball as a result of the impact with the ground.
Edexcel FM1 AS 2018 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cfa9b998-d57d-4980-9316-1bddeac55b90-04_267_891_346_687} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a ramp inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 2 } { 7 }\)
A parcel of mass 4 kg is projected, with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), from a point \(A\) on the ramp.
The parcel moves up a line of greatest slope of the ramp and first comes to instantaneous rest at the point \(B\), where \(A B = 2.5 \mathrm {~m}\).
The parcel is modelled as a particle.
The total resistance to the motion of the parcel from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons.
  1. Use the work-energy principle to show that \(R = 8.8\) After coming to instantaneous rest at \(B\), the parcel slides back down the ramp. The total resistance to the motion of the particle is modelled as a constant force of magnitude 8.8N.
  2. Find the speed of the parcel at the instant it returns to \(A\).
  3. Suggest two improvements that could be made to the model.
    VILU SIHI NI IIIUM ION OCVGHV SIHILNI IMAM ION OOVJYV SIHI NI JIIYM ION OC
Edexcel FM1 AS 2018 June Q3
  1. A van of mass 750 kg is moving along a straight horizontal road. At the instant when the van is moving at \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the van is modelled as a force of magnitude \(\lambda \nu \mathrm { N }\), where \(\lambda\) is a constant.
The engine of the van is working at a constant rate of 18 kW .
At the instant when \(v = 15\), the acceleration of the van is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
  1. Show that \(\lambda = 50\) The van now moves up a straight road inclined at an angle to the horizontal, where \(\sin \alpha = \frac { 1 } { 15 }\)
    At the instant when the van is moving at \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the van from non-gravitational forces is modelled as a force of magnitude 50 v . When the engine of the van is working at a constant rate of 12 kW , the van is moving at a constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  2. Find the value of \(V\).
    V349 SIHI NI IMIMM ION OCVJYV SIHIL NI LIIIM ION OOVJYV SIHIL NI JIIYM ION OC
Edexcel FM1 AS 2018 June Q4
  1. A particle \(P\) of mass \(3 m\) is moving in a straight line on a smooth horizontal floor. A particle \(Q\) of mass \(5 m\) 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 \(2 u\) and the speed of \(Q\) is \(u\). The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  1. Show that the speed of \(Q\) immediately after the collision is \(\frac { u } { 8 } ( 9 e + 1 )\)
  2. Find the range of values of \(e\) for which the direction of motion of \(P\) is not changed as a result of the collision. When \(P\) and \(Q\) collide they are at a distance \(d\) from a smooth fixed vertical wall, which is perpendicular to their direction of motion. After the collision with \(P\), particle \(Q\) collides directly with the wall and rebounds so that there is a second collision between \(P\) and \(Q\). This second collision takes place at a distance \(x\) from the wall. Given that \(e = \frac { 1 } { 18 }\) and the coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 3 }\)
  3. find \(x\) in terms of \(d\).
Edexcel FM1 AS 2019 June Q1
  1. A lorry of mass 16000 kg moves along a straight horizontal road.
The lorry moves at a constant speed of \(25 \mathrm {~ms} ^ { - 1 }\)
In an initial model for the motion of the lorry, the resistance to the motion of the lorry is modelled as having constant magnitude 16000 N .
  1. Show that the engine of the lorry is working at a rate of 400 kW . The model for the motion of the lorry along the same road is now refined so that when the speed of the lorry along the same road is \(V \mathrm {~ms} ^ { - 1 }\), the resistance to the motion of the lorry is modelled as having magnitude 640 V newtons. Assuming that the engine of the lorry is working at the same rate of 400 kW
  2. use the refined model to find the speed of the lorry when it is accelerating at \(2.1 \mathrm {~ms} ^ { - 2 }\)
Edexcel FM1 AS 2019 June Q2
  1. Two particles, \(A\) and \(B\), of masses \(2 m\) and \(3 m\) respectively, are moving on a smooth horizontal plane. The particles are moving in opposite directions towards each other along the same straight line when they collide directly. Immediately before the collision the speed of \(A\) is \(2 u\) and the speed of \(B\) is \(u\). In the collision the impulse of \(A\) on \(B\) has magnitude 5 mu .
    1. Find the coefficient of restitution between \(A\) and \(B\).
    2. Find the total loss in kinetic energy due to the collision.
Edexcel FM1 AS 2019 June Q3
  1. A particle, \(P\), of mass \(m \mathrm {~kg}\) is projected with speed \(5 \mathrm {~ms} ^ { - 1 }\) down a line of greatest slope of a rough plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 3 } { 5 }\) The total resistance to the motion of \(P\) is a force of magnitude \(\frac { 1 } { 5 } m g\)
    Use the work-energy principle to find the speed of \(P\) at the instant when it has moved a distance 8 m down the plane from the point of projection.
Edexcel FM1 AS 2019 June Q4
  1. Three particles, \(P , Q\) and \(R\), are at rest on a smooth horizontal plane. The particles lie along a straight line with \(Q\) between \(P\) and \(R\). The particles \(Q\) and \(R\) have masses \(m\) and \(k m\) respectively, where \(k\) is a constant.
Particle \(Q\) is projected towards \(R\) with speed \(u\) and the particles collide directly.
The coefficient of restitution between each pair of particles is \(e\).
  1. Find, in terms of \(e\), the range of values of \(k\) for which there is a second collision. Given that the mass of \(P\) is \(k m\) and that there is a second collision,
  2. write down, in terms of \(u , k\) and \(e\), the speed of \(Q\) after this second collision.
Edexcel FM1 AS 2020 June Q1
  1. Two particles \(P\) and \(Q\) have masses \(m\) and \(4 m\) respectively. The particles are at rest on a smooth horizontal plane. Particle \(P\) is given a horizontal impulse, of magnitude \(I\), in the direction \(P Q\). Particle \(P\) then collides directly with \(Q\). Immediately after this collision, \(P\) is at rest and \(Q\) has speed \(w\). The coefficient of restitution between the particles is \(e\).
    1. Find \(I\) in terms of \(m\) and \(w\).
    2. Show that \(e = \frac { 1 } { 4 }\)
    3. Find, in terms of \(m\) and \(w\), the total kinetic energy lost in the collision between \(P\) and \(Q\).
Edexcel FM1 AS 2020 June Q2
  1. A car of mass 1000 kg moves along a straight horizontal road.
In all circumstances, when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car is modelled as a force of magnitude \(c v ^ { 2 } \mathrm {~N}\), where \(c\) is a constant. The maximum power that can be developed by the engine of the car is 50 kW .
At the instant when the speed of the car is \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) and the engine is working at its maximum power, the acceleration of the car is \(2.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
  1. Convert \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) into \(\mathrm { m } \mathrm { s } ^ { - 1 }\)
  2. Find the acceleration of the car at the instant when the speed of the car is \(144 \mathrm { kmh } ^ { - 1 }\) and the engine is working at its maximum power. The maximum speed of the car when the engine is working at its maximum power is \(V \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
  3. Find, to the nearest whole number, the value of \(V\).
Edexcel FM1 AS 2020 June Q3
  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
  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
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
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
  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
  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
  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
  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
  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
  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
  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
  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
  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
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
  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.