Questions — Edexcel FM1 (51 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 FM1 2022 June Q5
  1. Two particles, \(P\) and \(Q\), are moving in opposite directions along the same straight line on a smooth horizontal surface when they collide directly.
    The mass of \(P\) is \(3 m\) and the mass of \(Q\) is \(4 m\).
    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 \(\frac { u } { 7 } ( 9 e + 2 )\)
    After the collision with \(P\), particle \(Q\) collides directly with a fixed vertical wall and rebounds. The wall is perpendicular to the direction of motion of \(Q\).
    The coefficient of restitution between \(Q\) and the wall is \(\frac { 1 } { 2 }\)
  2. Find the complete range of possible values of \(e\) for which there is a second collision between \(P\) and \(Q\).
Edexcel FM1 2022 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-20_497_1337_246_365} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Two blocks, \(A\) and \(B\), of masses 2 kg and 4 kg respectively are attached to the ends of a light inextensible string. Initially \(A\) is held on a fixed rough plane. The plane is inclined to horizontal ground at an angle \(\theta\), where \(\tan \theta = \frac { 3 } { 4 }\)
The string passes over a small smooth light pulley \(P\) that is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. Block \(A\) is held on the plane with the distance \(A P\) greater than 3 m .
Block \(B\) hangs freely below \(P\) at a distance of 3 m above the ground, as shown in Figure 4. The coefficient of friction between \(A\) and the plane is \(\mu\)
Block \(A\) is released from rest with the string taut.
By modelling the blocks as particles,
  1. find the potential energy lost by the whole system as a result of \(B\) falling 3 m . Given that the speed of \(B\) at the instant it hits the ground is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and ignoring air resistance,
  2. use the work-energy principle to find the value of \(\mu\) After \(B\) hits the ground, \(A\) continues to move up the plane but does not reach the pulley in the subsequent motion.
    Block \(A\) comes to instantaneous rest after moving a total distance of ( \(3 + d\) ) m from its point of release. Ignoring air resistance,
  3. use the work-energy principle to find the value of \(d\)
    \includegraphics[max width=\textwidth, alt={}, center]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-20_2255_50_309_1981}
Edexcel FM1 2022 June Q7
  1. A spring of natural length \(a\) has one end attached to a fixed point \(A\). The other end of the spring is attached to a package \(P\) of mass \(m\).
    The package \(P\) is held at rest at the point \(B\), which is vertically below \(A\) such that \(A B = 3 a\).
    After being released from rest at \(B\), the package \(P\) first comes to instantaneous rest at \(A\). Air resistance is modelled as being negligible.
By modelling the spring as being light and modelling \(P\) as a particle,
  1. show that the modulus of elasticity of the spring is \(2 m g\)
    1. Show that \(P\) attains its maximum speed when the extension of the spring is \(\frac { 1 } { 2 } a\)
    2. Use the principle of conservation of mechanical energy to find the maximum speed, giving your answer in terms of \(a\) and \(g\). In reality, the spring is not light.
  3. State one way in which this would affect your energy equation in part (b).
Edexcel FM1 2022 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{86a37170-046f-46e5-9c8c-06d5f98ca4fe-28_567_1406_244_333} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 represents the plan view of part of a smooth horizontal floor, where \(R S\) and \(S T\) are smooth fixed vertical walls. The vector \(\overrightarrow { R S }\) is in the direction of \(\mathbf { i }\) and the vector \(\overrightarrow { S T }\) is in the direction of \(( 2 \mathbf { i } + \mathbf { j } )\). A small ball \(B\) is projected across the floor towards \(R S\). Immediately before the impact with \(R S\), the velocity of \(B\) is \(( 6 \mathbf { i } - 8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The ball bounces off \(R S\) and then hits \(S T\). The ball is modelled as a particle.
Given that the coefficient of restitution between \(B\) and \(R S\) is \(e\),
  1. find the full range of possible values of \(e\). It is now given that \(e = \frac { 1 } { 4 }\) and that the coefficient of restitution between \(B\) and \(S T\) is \(\frac { 1 } { 2 }\)
  2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) immediately after its impact with \(S T\).
Edexcel FM1 2023 June Q1
  1. A particle \(P\) of mass 2 kg is moving with velocity \(( - 4 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(( - 6 \mathbf { i } + 42 \mathbf { j } )\) N s.
    1. Find the speed of \(P\) immediately after receiving the impulse.
    The angle through which the direction of motion of \(P\) has been deflected by the impulse is \(\alpha ^ { \circ }\)
  2. Find the value of \(\alpha\)
Edexcel FM1 2023 June Q2
  1. A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the motion of the car is a constant force of magnitude 400 N.
The engine of the car is working at a constant rate of 16 kW .
  1. Find the value of \(U\). The car now pulls a trailer of mass 600 kg in a straight line along the road using a tow rope which is parallel to the direction of motion. The resistance to the motion of the car is again a constant force of magnitude 400 N . The resistance to the motion of the trailer is a constant force of magnitude 300 N . The engine of the car is working at a constant rate of 16 kW .
    The tow rope is modelled as being light and inextensible.
    Using the model,
  2. find the tension in the tow rope at the instant when the speed of the car is \(\frac { 20 } { 3 } \mathrm {~ms} ^ { - 1 }\)
Edexcel FM1 2023 June Q3
  1. A particle \(P\) of mass \(2 m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal plane. It collides directly with a particle \(Q\) of mass \(m\) that is moving on the plane with speed \(2 u\) in the opposite direction to \(P\).
    The coefficient of restitution between \(P\) and \(Q\) is \(e\), where \(e > \frac { 4 } { 5 }\)
    1. Show that the speed of \(Q\) immediately after the collision is \(\frac { ( 4 + 10 e ) u } { 3 }\)
    After the collision \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). The coefficient of restitution between \(Q\) and the wall is \(f\).
  2. Find, in terms of \(\boldsymbol { e }\), the set of values of \(f\) for which there will be a second collision between \(P\) and \(Q\).
Edexcel FM1 2023 June Q4
  1. A light elastic string has natural length \(2 a\) and modulus of elasticity \(4 m g\). One end of the elastic string is attached to a fixed point \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the elastic string.
    The particle \(P\) hangs freely in equilibrium at the point \(E\), which is vertically below \(O\)
    1. Find the length \(O E\).
    Particle \(P\) is now pulled vertically downwards to the point \(A\), where \(O A = 4 a\), and released from rest. The resistance to the motion of \(P\) is a constant force of magnitude \(\frac { 1 } { 4 } m g\).
  2. Find, in terms of \(a\) and \(g\), the speed of \(P\) after it has moved a distance \(a\). Particle \(P\) is now held at \(O\) Particle \(P\) is released from rest and reaches its maximum speed at the point \(B\). The resistance to the motion of \(P\) is again a constant force of magnitude \(\frac { 1 } { 4 } m g\).
  3. Find the distance \(O B\).
Edexcel FM1 2023 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0da9cd5b-6f6f-4607-bd4f-c8ae164466ae-16_758_1399_280_333} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A smooth uniform sphere \(S\) of mass \(m\) is moving with speed \(U\) on a smooth horizontal plane. The sphere \(S\) collides obliquely with another uniform sphere of mass \(M\) which is at rest on the plane. The two spheres have the same radius. Immediately before the collision the direction of motion of \(S\) makes an angle \(\alpha\), where \(0 < \alpha < 90 ^ { \circ }\), with the line joining the centres of the spheres. Immediately after the collision the direction of motion of \(S\) makes an angle \(\beta\) with the line joining the centres of the spheres, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\).
  1. Show that \(\tan \beta = \frac { ( m + M ) \tan \alpha } { ( m - e M ) }\) Given that \(m = e M\),
  2. show that the directions of motion of the two spheres immediately after the collision are perpendicular.
Edexcel FM1 2023 June Q6
  1. A particle \(P\) of mass \(m\) is falling vertically when it strikes a fixed smooth inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(0 < \alpha \leqslant 45 ^ { \circ }\)
At the instant immediately before the impact, the speed of \(P\) is \(u\).
At the instant immediately after the impact, \(P\) is moving horizontally with speed \(v\).
  1. Show that the magnitude of the impulse exerted on the plane by \(P\) is \(m u \sec \alpha\) The coefficient of restitution between \(P\) and the plane is \(e\), where \(e > 0\)
  2. Show that \(v ^ { 2 } = u ^ { 2 } \left( \sin ^ { 2 } \alpha + e ^ { 2 } \cos ^ { 2 } \alpha \right)\)
  3. Show that the kinetic energy lost by \(P\) in the impact is $$\frac { 1 } { 2 } m u ^ { 2 } \left( 1 - e ^ { 2 } \right) \cos ^ { 2 } \alpha$$
  4. Hence find, in terms of \(m\), \(u\) and \(e\) only, the kinetic energy lost by \(P\) in the impact.
Edexcel FM1 2023 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0da9cd5b-6f6f-4607-bd4f-c8ae164466ae-24_721_1367_280_349} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A small smooth snooker ball is projected from the corner \(A\) of a horizontal rectangular snooker table \(A B C D\). The ball is projected so it first hits the side \(D C\) at the point \(P\), then hits the side \(C B\) at the point \(Q\) and then returns to \(A\). Angle \(A P D = \alpha\), Angle \(Q P C = \beta\), Angle \(A Q B = \gamma\)
The ball moves along \(A P\) with speed \(U\), along \(P Q\) with speed \(V\) and along \(Q A\) with speed \(W\), as shown in Figure 2. The coefficient of restitution between the ball and side \(D C\) is \(e _ { 1 }\)
The coefficient of restitution between the ball and side \(C B\) is \(e _ { 2 }\)
The ball is modelled as a particle. \section*{Use the model to answer all parts of this question.}
  1. Show that \(\tan \beta = e _ { 1 } \tan \alpha\)
  2. Hence show that \(e _ { 1 } \tan \alpha = e _ { 2 } \cot \gamma\)
  3. By considering (angle \(A P Q\) + angle \(A Q P\) ) or otherwise, show that it would be possible for the ball to return to \(A\) only if \(e _ { 2 } > e _ { 1 }\) If instead \(e _ { 1 } = e _ { 2 }\), the ball would not return to \(A\).
    Given that \(e _ { 1 } = e _ { 2 }\)
  4. use the result from part (b) to describe the path of the ball after it hits \(C B\) at \(Q\), explaining your answer.
Edexcel FM1 2024 June Q1
  1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.]
A particle \(A\) has mass 3 kg and a particle \(B\) has mass 2 kg .
The particles are moving on a smooth horizontal plane when they collide directly.
Immediately before the collision, the velocity of \(A\) is \(( 3 \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the velocity of \(B\) is \(( - 6 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) Immediately after the collision the velocity of \(A\) is \(\left( - 2 \mathbf { i } + \frac { 2 } { 3 } \mathbf { j } \right) \mathrm { ms } ^ { - 1 }\)
  1. Find the total kinetic energy of the two particles before the collision.
  2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the impulse exerted on \(A\) by \(B\) in the collision.
  3. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) immediately after the collision.
Edexcel FM1 2024 June Q2
  1. A rough plane is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 3 } { 4 }\)
A particle \(P\) of mass \(m\) is at rest at a point on the plane. The particle is projected up the plane with speed \(\sqrt { 2 a g }\)
The particle moves up a line of greatest slope of the plane and comes to instantaneous rest after moving a distance \(d\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 7 }\)
  1. Show that the magnitude of the frictional force acting on \(P\) as it moves up the plane is \(\frac { 4 m g } { 35 }\) Air resistance is assumed to be negligible.
    Using the work-energy principle,
  2. find \(d\) in terms of \(a\).
Edexcel FM1 2024 June Q3
  1. A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed of \(72 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)
  • The resistance to the motion of the car is modelled as a constant force of magnitude 900 N
The engine of the car is working at a constant rate of \(P \mathrm {~kW}\).
Using the model,
  1. find the value of \(P\). The car now travels in a straight line up a road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 2 } { 49 }\)
    • In a refined model, the resistance to the motion of the car from non-gravitational forces is now modelled as a force of magnitude \(20 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car
    At the instant when the engine of the car is working at a constant rate of 30 kW and the car is moving up the road at \(10 \mathrm {~ms} ^ { - 1 }\), the acceleration of the car is \(a \mathrm {~ms} ^ { - 2 }\) Using the refined model,
  2. find the value of \(a\). Later on, when the engine of the car is again working at a constant rate of 30 kW , the car is moving up the road at a constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Using the refined model,
  3. find the value of \(U\).
Edexcel FM1 2024 June Q4
  1. A particle \(A\) of mass \(2 m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal plane. Particle \(A\) collides directly with a particle \(B\) of mass \(m\) which is at rest on the plane.
The coefficient of restitution between \(A\) and \(B\) is \(e\), where \(e > 0\)
  1. Show that the speed of \(B\) immediately after the collision is \(2 u ( 1 + e )\). After the collision, \(B\) hits a smooth fixed vertical wall which is perpendicular to the direction of motion of \(B\).
  2. Show that there will be a second collision between \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\)
    Find, in simplified form, in terms of \(m\), \(u\) and \(e\),
  3. the magnitude of the impulse received by \(B\) in its collision with the wall,
  4. the loss in kinetic energy of \(B\) due to its collision with the wall.
Edexcel FM1 2024 June Q5
  1. A light elastic string has natural length \(2 a\) and modulus of elasticity \(2 m g\). One end of the string is attached to a fixed point \(A\) on a horizontal ceiling. The other end is attached to a particle \(P\) of mass \(m\).
The particle \(P\) hangs in equilibrium at the point \(E\), where \(A E = 3 a\).
The particle \(P\) is then projected vertically downwards from \(E\) with speed \(\frac { 3 } { 2 } \sqrt { a g }\)
Air resistance is assumed to be negligible.
Find the elastic energy stored in the string, when \(P\) first comes to instantaneous rest. Give your answer in the form kmga, where \(k\) is a constant to be found.
Edexcel FM1 2024 June Q6
  1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.]
A particle \(P\) is moving with velocity ( \(4 \mathbf { i } - \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) on a smooth horizontal plane. The particle collides with a smooth vertical wall and rebounds with velocity \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) The coefficient of restitution between \(P\) and the wall is \(e\).
  1. Find the value of \(e\). After the collision, \(P\) goes on to hit a second smooth vertical wall, which is parallel to \(\mathbf { i }\).
    The coefficient of restitution between \(P\) and this second wall is \(\frac { 1 } { 3 }\)
    The angle through which the direction of motion of \(P\) has been deflected by its collision with this second wall is \(\alpha ^ { \circ }\).
  2. Find the value of \(\alpha\), giving your answer to the nearest whole number.
Edexcel FM1 2024 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{58a33c19-77c6-4b08-ac09-ce6aa1e641df-20_501_703_251_680} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A smooth uniform sphere \(A\) of mass \(m\) is moving with speed \(U\) on a smooth horizontal plane. The sphere \(A\) collides obliquely with a smooth uniform sphere \(B\) of mass \(3 m\) which is at rest on the plane. The two spheres have the same radius. Immediately before the collision, the direction of motion of \(A\) makes an angle \(\alpha\), where \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), with the line joining the centres of the spheres. Immediately after the collision, the direction of motion of \(A\) is perpendicular to its original direction, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\).
  1. Show that the speed of \(B\) immediately after the collision is $$\frac { 1 } { 4 } ( 1 + e ) U \cos \alpha$$
  2. Show that \(e > \frac { 1 } { 3 }\)
  3. Show that \(0 < \tan \alpha \leqslant \frac { 1 } { \sqrt { 2 } }\)
Edexcel FM1 Specimen Q1
  1. A particle \(P\) of mass 0.5 kg is moving with velocity \(( 4 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(( 2 \mathbf { i } - \mathbf { j } )\) Ns.
Show that the kinetic energy gained by \(P\) as a result of the impulse is 12 J .
Edexcel FM1 Specimen Q2
  1. A parcel of mass 5 kg is projected with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a fixed rough inclined ramp.
    The ramp is inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 7 }\)
    The parcel is projected from the point \(A\) on the ramp and comes to instantaneous rest at the point \(B\) on the ramp, where \(A B = 14 \mathrm {~m}\).
The coefficient of friction between the parcel and the ramp is \(\mu\).
In a model of the parcel's motion, the parcel is treated as a particle.
  1. Use the work-energy principle to find the value of \(\mu\).
  2. Suggest one way in which the model could be refined to make it more realistic.
Edexcel FM1 Specimen Q3
  1. A particle of mass \(m \mathrm {~kg}\) lies on a smooth horizontal surface.
Initially the particle is at rest at a point \(O\) between two fixed parallel vertical walls.
The point \(O\) is equidistant from the two walls and the walls are 4 m apart.
At time \(t = 0\) the particle is projected from \(O\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the walls.
The coefficient of restitution between the particle and each wall is \(\frac { 3 } { 4 }\)
The magnitude of the impulse on the particle due to the first impact with a wall is \(\lambda m u\) Ns.
  1. Find the value of \(\lambda\). The particle returns to \(O\), having bounced off each wall once, at time \(t = 7\) seconds.
  2. Find the value of \(u\).
Edexcel FM1 Specimen Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{681833ac-b266-4ac8-881e-46ede398ce58-08_513_807_303_630} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 represents the plan view of part of a horizontal floor, where \(A B\) and \(B C\) are perpendicular vertical walls. The floor and the walls are modelled as smooth.
A ball is projected along the floor towards \(A B\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a path at an angle of \(60 ^ { \circ }\) to \(A B\). The ball hits \(A B\) and then hits \(B C\). The ball is modelled as a particle.
The coefficient of restitution between the ball and wall \(A B\) is \(\frac { 1 } { \sqrt { 3 } }\)
The coefficient of restitution between the ball and wall \(B C\) is \(\sqrt { \frac { 2 } { 5 } }\)
  1. Show that, using this model, the final kinetic energy of the ball is \(35 \%\) of the initial kinetic energy of the ball.
  2. In reality the floor and the walls may not be smooth. What effect will the model have had on the calculation of the percentage of kinetic energy remaining?
Edexcel FM1 Specimen Q5
  1. A car of mass 600 kg is moving along a straight horizontal road.
At the instant 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 \(( 200 + 2 v ) \mathrm { N }\). The engine of the car is working at a constant rate of 12 kW .
  1. Find the acceleration of the car at the instant when \(v = 20\) Later on the car is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\) At the instant when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude ( \(200 + 2 v ) \mathrm { N }\). The engine is again working at a constant rate of 12 kW .
    At the instant when the car has speed \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the car is decelerating at \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the value of \(w\).
Edexcel FM1 Specimen Q6
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]
A smooth uniform sphere \(A\) has mass \(2 m \mathrm {~kg}\) and another smooth uniform sphere \(B\), with the same radius as \(A\), has mass \(3 m \mathrm {~kg}\). The spheres are moving on a smooth horizontal plane when they collide obliquely.
Immediately before the collision the velocity of \(A\) is \(( 3 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - 5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At the instant of collision, the line joining the centres of the spheres is parallel to \(\mathbf { i }\).
The coefficient of restitution between the spheres is \(\frac { 1 } { 4 }\)
  1. Find the velocity of \(B\) immediately after the collision.
  2. Find, to the nearest degree, the size of the angle through which the direction of motion of \(B\) is deflected as a result of the collision.
Edexcel FM1 Specimen Q7
  1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(3 m g\).
The other end of the string is attached to a fixed point \(O\) on a ceiling.
The particle hangs freely in equilibrium at a distance \(d\) vertically below \(O\).
  1. Show that \(d = \frac { 4 } { 3 } a\). The point \(A\) is vertically below \(O\) such that \(O A = 2 a\).
    The particle is held at rest at \(A\), then released and first comes to instantaneous rest at the point \(B\).
  2. Find, in terms of \(g\), the acceleration of \(P\) immediately after it is released from rest.
  3. Find, in terms of \(g\) and \(a\), the maximum speed attained by \(P\) as it moves from \(A\) to \(B\).
  4. Find, in terms of \(a\), the distance \(O B\).