Questions (30179 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 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 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
Sort by: Default | Easiest first | Hardest first
OCR MEI M4 2016 June Q2
12 marks Challenging +1.8
2 A thin rigid rod PQ has length \(2 a\). Its mass per unit length, \(\rho\), is given by \(\rho = k \left( 1 + \frac { x } { 2 a } \right)\) where \(x\) is the distance from P and \(k\) is a positive constant. The mass of the rod is \(M\) and the moment of inertia of the rod about an axis through P perpendicular to PQ is \(I\).
  1. Show that \(I = \frac { 14 } { 9 } M a ^ { 2 }\). The rod is initially at rest with Q vertically below P . It is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through P . The rod is struck a horizontal blow perpendicular to the fixed axis at the point where \(x = \frac { 3 } { 2 } a\). The magnitude of the impulse of this blow is \(J\).
  2. Find, in terms of \(a , J\) and \(M\), the initial angular speed of the rod.
  3. Find, in terms of \(a , g\) and \(M\), the set of values of \(J\) for which the rod makes complete revolutions.
OCR MEI M4 2016 June Q3
24 marks Challenging +1.8
3 Fig. 3 shows a uniform rigid rod AB of length \(2 a\) and mass \(2 m\). The rod is freely hinged at A so that it can rotate in a vertical plane. One end of a light inextensible string of length \(l\) is attached to B . The string passes over a small smooth fixed pulley at C , where C is vertically above A and \(\mathrm { AC } = 6 a\). A particle of mass \(\lambda m\), where \(\lambda\) is a positive constant, is attached to the other end of the string and hangs freely, vertically below C . The rod makes an angle \(\theta\) with the upward vertical, where \(0 \leqslant \theta \leqslant \pi\). You may assume that the particle does not interfere with the rod AB or the section of the string BC . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3fdb2cff-0f74-4c88-b25a-759bfab1675a-3_878_615_667_717} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Find the potential energy, \(V\), of the system relative to a situation in which the rod AB is horizontal, and hence show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 2 m g a \sin \theta \left( \frac { 3 \lambda } { \sqrt { 10 - 6 \cos \theta } } - 1 \right) .$$
  2. Show that \(\theta = 0\) and \(\theta = \pi\) are the only values of \(\theta\) for which the system is in equilibrium whatever the value of \(\lambda\).
  3. Show that, if there is a third value of \(\theta\) for which the system is in equilibrium, then \(\frac { 2 } { 3 } < \lambda < \frac { 4 } { 3 }\).
  4. Given that there are three positions of equilibrium, establish whether each of these positions is stable or unstable. It is given that, for small values of \(\theta\), $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } \approx 2 m g a \left[ \left( \frac { 3 } { 2 } \lambda - 1 \right) \theta - \left( \frac { 13 } { 16 } \lambda - \frac { 1 } { 6 } \right) \theta ^ { 3 } \right] .$$
  5. Investigate the stability of the equilibrium position given by \(\theta = 0\) in the case when \(\lambda = \frac { 2 } { 3 }\).
OCR MEI M4 2016 June Q4
24 marks Challenging +1.8
4 A raindrop falls from rest through a stationary cloud. The raindrop has mass \(m\) and speed \(v\) when it has fallen a distance \(x\). You may assume that resistances to motion are negligible.
  1. Derive the equation of motion $$m v \frac { \mathrm {~d} v } { \mathrm {~d} x } + v ^ { 2 } \frac { \mathrm {~d} m } { \mathrm {~d} x } = m g .$$ Initially the mass of the raindrop is \(m _ { 0 }\). Two different models for the mass of the raindrop are suggested.
    In the first model \(m = m _ { 0 } \mathrm { e } ^ { k _ { 1 } x }\), where \(k _ { 1 }\) is a positive constant.
  2. Show that $$v ^ { 2 } = \frac { g } { k _ { 1 } } \left( 1 - \mathrm { e } ^ { - 2 k _ { 1 } x } \right) ,$$ and hence state, in terms of \(g\) and \(k _ { 1 }\), the terminal velocity of the raindrop according to this first model. In the second model \(m = m _ { 0 } \left( 1 + k _ { 2 } x \right)\), where \(k _ { 2 }\) is a positive constant.
  3. By considering the expression obtained from differentiating \(v ^ { 2 } \left( 1 + k _ { 2 } x \right) ^ { 2 }\) with respect to \(x\), show that, for the second model, the equation of motion in part (i) may be written as $$\frac { d } { d x } \left[ v ^ { 2 } \left( 1 + k _ { 2 } x \right) ^ { 2 } \right] = 2 g \left( 1 + k _ { 2 } x \right) ^ { 2 } .$$ Hence find an expression for \(v ^ { 2 }\) in terms of \(g , k _ { 2 }\) and \(x\).
  4. Suppose that the models give the same value for the speed of the raindrop at the instant when it has doubled its initial mass. Find the exact value of \(\frac { k _ { 1 } } { k _ { 2 } }\), giving your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers. are integers. \section*{END OF QUESTION PAPER}
Edexcel M5 Q1
5 marks Standard +0.8
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors.]
A small smooth ring of mass 0.5 kg moves along a smooth horizontal wire. The only forces acting on the ring are its weight, the normal reaction from the wire, and a constant force \(( 5 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } ) \mathrm { N }\). The ring is initially at rest at the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\), relative to a fixed origin. Find the speed of the ring as it passes through the point with position vector \(( 3 \mathbf { i } + \mathbf { k } ) \mathrm { m }\).
Edexcel M5 Q2
6 marks Standard +0.8
2. Three forces, \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body. \(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( \mathbf { i } + \mathbf { j } - 4 \mathbf { k } )\) N and \(\mathbf { F } _ { 3 } = ( p \mathbf { i } + q \mathbf { j } + r \mathbf { k } ) \mathrm { N }\), where \(p , q\) and \(r\) are constants. All three forces act through the point with position vector \(( 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } ) \mathrm { m }\), relative to a fixed origin. The three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are equivalent to a single force ( \(5 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }\) ) N , acting at the origin, together with a couple \(\mathbf { G }\).
  1. Find the values of \(p , q\) and \(r\).
  2. Find \(\mathbf { G }\).
Edexcel M5 Q3
7 marks Standard +0.3
3. At time \(t\) seconds, the position vector of a particle \(P\) is \(\mathbf { r }\) metres, relative to a fixed origin. The particle moves in such a way that $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t } = \mathbf { 0 }$$ At \(t = 0 , P\) is moving with velocity ( \(8 \mathbf { i } - 6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Find the speed of \(P\) when \(t = \frac { 1 } { 2 } \ln 2\).
Edexcel M5 Q4
9 marks Challenging +1.8
4. A uniform plane lamina of mass \(m\) is in the shape of an equilateral triangle of side \(2 a\). Find, using integration, the moment of inertia of the lamina about one of its edges.
(9)
Edexcel M5 Q5
14 marks Challenging +1.8
5. A rocket is launched vertically upwards from rest. Initially, the total mass of the rocket and its fuel is 1000 kg . The rocket burns fuel at a rate of \(10 \mathrm {~kg} \mathrm {~s} ^ { - 1 }\). The burnt fuel is ejected vertically downwards with a speed of \(2000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the rocket, and burning stops after one minute. At time \(t\) seconds, \(t \leq 60\), after the launch, the speed of the rocket is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Air resistance is assumed to be negligible.
  1. Show that $$- 9.8 ( 100 - t ) = ( 100 - t ) \frac { \mathrm { d } v } { \mathrm {~d} t } - 2000 .$$
  2. Find the speed of the rocket when burning stops.
Edexcel M5 Q6
17 marks Challenging +1.8
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{9e3d76a7-b997-4e46-a5dd-aeeaa5abfa4e-02_211_611_388_1845}
\end{figure} A rough uniform rod, of mass \(m\) and length \(4 a\), is rod is held on a rough horizontal table. The rod is perpendicular to the edge of the table and a length \(3 a\) projects horizontally over the edge, as shown in Fig. 1.
  1. Show that the moment of inertia of the rod about the edge of the table is \(\frac { 7 } { 3 } m a ^ { 2 }\). The rod is released from rest and rotates about the edge of the table. When the rod has turned through an angle \(\theta\), its angular speed is \(\dot { \theta }\). Assuming that the rod has not started to slip,
  2. show that \(\dot { \theta } ^ { 2 } = \frac { 6 g \sin \theta } { 7 a }\),
  3. find the angular acceleration of the rod,
  4. find the normal reaction of the table on the rod. The coefficient of friction between the rod and the edge of the table is \(\mu\).
  5. Show that the rod starts to slip when \(\tan \theta = \frac { 4 } { 13 } \mu\)
Edexcel M5 Q8
17 marks Challenging +1.8
8. A pendulum consists of a uniform rod \(P Q\), of mass \(3 m\) and length \(2 a\), which is rigidly fixed at its end \(Q\) to the centre of a uniform circular disc of mass \(m\) and radius \(a\). The rod is perpendicular to the plane of the disc. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through the end \(P\) of the rod and is perpendicular to the rod.
  1. Show that the moment of inertia of the pendulum about \(L\) is \(\frac { 33 } { 4 } m a ^ { 2 }\). The pendulum is released from rest in the position where \(P Q\) makes an angle \(\alpha\) with the downward vertical. At time \(t , P Q\) makes an angle \(\theta\) with the downward vertical.
  2. Show that the angular speed, \(\dot { \theta }\), of the pendulum satisfies $$\dot { \theta } ^ { 2 } = \frac { 40 g ( \cos \theta - \cos \alpha ) } { 33 a } .$$
  3. Hence, or otherwise, find the angular acceleration of the pendulum. Given that \(\alpha = \frac { \pi } { 20 }\) and that \(P Q\) has length \(\frac { 8 } { 33 } \mathrm {~m}\),
  4. find, to 3 significant figures, an approximate value for the angular speed of the pendulum 0.2 s after it has been released from rest. \section*{Advanced Level} \section*{Monday 25 June 2012 - Afternoon} \section*{Materials required for examination
    Mathematical Formulae (Pink)} Items included with question papers
    Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.
Edexcel M5 2006 January Q1
4 marks Moderate -0.3
  1. A bead is threaded on a straight wire. The vector equation of the wire is
$$\mathbf { r } = \mathbf { i } - 3 \mathbf { j } + \mathbf { k } + t ( 2 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )$$ where the unit of length is the metre. The bead is moved from a point \(A\) on the wire through a distance of 6 m along the wire to a point \(B\) by a force \(\mathbf { F } = ( 7 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } ) \mathrm { N }\). Find the magnitude of the work done by \(\mathbf { F }\) in moving the bead from \(A\) to \(B\).
(Total 4 marks)
Edexcel M5 2006 January Q2
5 marks Standard +0.8
2. A uniform circular disc has radius \(a\) and mass \(m\). Prove, using integration, that the moment of inertia of the disc about an axis through its centre and perpendicular to the plane of the disc is \(\frac { 1 } { 2 } m a ^ { 2 }\).
(Total 5 marks)
Edexcel M5 2006 January Q3
6 marks Standard +0.3
3. The position vector \(\mathbf { r }\) of a particle \(P\) at time \(t\) satisfies the vector differential equation $$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + 2 \mathbf { r } = 4 \mathbf { i }$$ Given that the position vector of \(P\) at time \(t = 0\) is \(2 \mathbf { j }\), find the position vector of \(P\) at time \(t\).
(Total 6 marks)
Edexcel M5 2006 January Q4
6 marks Standard +0.8
4. A uniform \(\operatorname { rod } A B\), of mass \(m\) and length \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\) and perpendicular to the plane. The rod hangs in equilibrium with \(B\) below \(A\). The rod is rotated through a small angle and released from rest at time \(t = 0\).
  1. Show that the motion of the rod is approximately simple harmonic.
  2. Using this approximation, find the time \(t\) when the rod is first vertical after being released.
    (Total 6 marks)
Edexcel M5 2006 January Q5
10 marks Challenging +1.8
5. A uniform circular disc has mass \(m\) and radius \(a\). The disc can rotate freely about an axis that is in the same plane as the disc and tangential to the disc at a point \(A\) on its circumference. The disc hangs at rest in equilibrium with its centre \(O\) vertically below \(A\). A particle \(P\) of mass \(m\) is moving horizontally and perpendicular to the disc with speed \(\sqrt { } ( k g a )\), where \(k\) is a constant. The particle then strikes the disc at \(O\) and adheres to it at \(O\). Given that the disc rotates through an angle of \(90 ^ { \circ }\) before first coming to instantaneous rest, find the value of \(k\).
(Total 10 marks)
Edexcel M5 2006 January Q6
12 marks Challenging +1.3
6. The vertices of a tetrahedron \(P Q R S\) have position vectors \(\mathbf { p } , \mathbf { q } , \mathbf { r }\) and \(\mathbf { s }\) respectively, where $$\mathbf { p } = - 3 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } , \quad \mathbf { q } = 4 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } , \quad \mathbf { r } = \mathbf { i } - 2 \mathbf { j } + \mathbf { k } , \quad \mathbf { s } = 4 \mathbf { i } + \mathbf { k }$$ Forces of magnitude 20 N and \(2 \sqrt { } 13 \mathrm {~N}\) act along \(S Q\) and \(S R\) respectively. A third force acts at \(P\).
Given that the system of three forces reduces to a couple \(\mathbf { G }\), find
  1. the third force,
  2. the magnitude of \(\mathbf { G }\).
    (6)
    (Total 12 marks)
Edexcel M5 2006 January Q7
15 marks Challenging +1.8
7. At time \(t = 0\), a small body is projected vertically upwards. While ascending it picks up small drops of moisture from the atmosphere. The drops of moisture are at rest before they are picked up. At time \(t\), the combined body \(P\) has mass \(m\) and speed \(v\).
  1. Show that, while \(P\) is moving upwards, \(m \frac { \mathrm {~d} v } { \mathrm {~d} t } + v \frac { \mathrm {~d} m } { \mathrm {~d} t } = - m g\). The initial mass of \(P\) is \(M\), and \(m = M \mathrm { e } ^ { k t }\), where \(k\) is a positive constant.
  2. Show that, while \(P\) is moving upwards, \(\frac { \mathrm { d } } { \mathrm { d } t } \left( v \mathrm { e } ^ { k t } \right) = - g \mathrm { e } ^ { k t }\). Given that the initial projection speed of \(P\) is \(\frac { g } { 2 k }\),
  3. find, in terms of \(M\), the mass of \(P\) when it reaches its highest point.
    (Total 15 marks)
Edexcel M5 2006 January Q8
17 marks Challenging +1.2
8. Four uniform rods, each of mass \(m\) and length \(2 a\), are joined together at their ends to form a plane rigid square framework \(A B C D\) of side \(2 a\). The framework is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\). The axis is perpendicular to the plane of the framework.
  1. Show that the moment of inertia of the framework about the axis is \(\frac { 40 m a ^ { 2 } } { 3 }\). The framework is slightly disturbed from rest when \(C\) is vertically above \(A\). Find
  2. the angular acceleration of the framework when \(A C\) is horizontal,
  3. the angular speed of the framework when \(A C\) is horizontal,
  4. the magnitude of the force acting on the framework at \(A\) when \(A C\) is horizontal.
Edexcel M5 2002 June Q1
5 marks Standard +0.8
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors.]
A small smooth ring of mass 0.5 kg moves along a smooth horizontal wire. The only forces acting on the ring are its weight, the normal reaction from the wire, and a constant force ( \(5 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }\) ) N. The ring is initially at rest at the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\), relative to a fixed origin. Find the speed of the ring as it passes through the point with position vector \(( 3 \mathbf { i } + \mathbf { k } ) \mathrm { m }\).
Edexcel M5 2002 June Q2
6 marks Standard +0.8
2. Three forces, \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body. \(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( \mathbf { i } + \mathbf { j } - 4 \mathbf { k } )\) N and \(\mathbf { F } _ { 3 } = ( p \mathbf { i } + q \mathbf { j } + r \mathbf { k } ) \mathrm { N }\), where \(p , q\) and \(r\) are constants. All three forces act through the point with position vector ( \(3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\) ) m, relative to a fixed origin. The three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are equivalent to a single force ( \(5 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }\) ) N, acting at the origin, together with a couple \(\mathbf { G }\).
  1. Find the values of \(p , q\) and \(r\).
  2. Find \(\mathbf { G }\).
Edexcel M5 2002 June Q3
7 marks Standard +0.3
3. At time \(t\) seconds, the position vector of a particle \(P\) is \(\mathbf { r }\) metres, relative to a fixed origin. The particle moves in such a way that $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t } = \mathbf { 0 }$$ At \(t = 0 , P\) is moving with velocity ( \(8 \mathbf { i } - 6 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\).
Find the speed of \(P\) when \(t = \frac { 1 } { 2 } \ln 2\).
Edexcel M5 2002 June Q4
9 marks Challenging +1.2
4. A uniform plane lamina of mass \(m\) is in the shape of an equilateral triangle of side \(2 a\). Find, using integration, the moment of inertia of the lamina about one of its edges.
Edexcel M5 2002 June Q5
14 marks Challenging +1.8
5. A rocket is launched vertically upwards from rest. Initially, the total mass of the rocket and its fuel is 1000 kg . The rocket burns fuel at a rate of \(10 \mathrm {~kg} \mathrm {~s} ^ { - 1 }\). The burnt fuel is ejected vertically downwards with a speed of \(2000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the rocket, and burning stops after one minute. At time \(t\) seconds, \(t \leq 60\), after the launch, the speed of the rocket is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Air resistance is assumed to be negligible.
  1. Show that $$- 9.8 ( 100 - t ) = ( 100 - t ) \frac { \mathrm { d } v } { \mathrm {~d} t } - 2000$$ (8)
  2. Find the speed of the rocket when burning stops.
    (6) \section*{6.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{4c585ec7-7b3e-4ff8-b7c2-83f58ad82ae9-4_316_929_391_573}
    \end{figure} A rough uniform rod, of mass \(m\) and length \(4 a\), is rod is held on a rough horizontal table. The rod is perpendicular to the edge of the table and a length \(3 a\) projects horizontally over the edge, as shown in Fig. 1.
Edexcel M5 2002 June Q7
17 marks Challenging +1.8
7. A uniform plane circular disc, of mass \(m\) and radius \(a\), hangs in equilibrium from a point \(B\) on its circumference. The disc is free to rotate about a fixed smooth horizontal axis which is in the plane of the disc and tangential to the disc at \(B\). A particle \(P\), of mass \(m\), is moving horizontally with speed \(u\) in a direction which is perpendicular to the plane of the disc. At time \(t = 0 , P\) strikes the disc at its centre and adheres to the disc.
  1. Show that the angular speed of the disc immediately after it has been struck by \(P\) is \(\frac { 4 u } { 9 a }\).
    (6) It is given that \(u ^ { 2 } = \frac { 1 } { 10 } a g\), and that air resistance is negligible.
  2. Find the angle through which the disc turns before it first comes to instantaneous rest. The disc first returns to its initial position at time \(t = T\).
    1. Write down an equation of motion for the disc.
    2. Hence find \(T\) in terms of \(a , g\) and \(m\), using a suitable approximation which should be justified.
Edexcel M5 2003 June Q1
6 marks Challenging +1.2
  1. In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane and \(\mathbf { k }\) is a unit vector vertically upwards.
A small smooth ring of mass 0.1 kg is threaded onto a smooth horizontal wire which is parallel to \(( \mathbf { i } + 2 \mathbf { j } )\). The only forces acting on the ring are its weight, the normal reaction from the wire and a constant force \(( \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )\) N. The ring starts from rest at the point \(A\) on the wire, whose position vector relative to a fixed origin is \(( 2 \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k } ) \mathrm { m }\), and passes through the point \(B\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the position vector of \(B\).
(6)