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
Edexcel M3 Q3
9 marks Standard +0.3
3. At time \(t\) seconds the acceleration, \(a \mathrm {~ms} ^ { - 2 }\), of a particle is given by $$a = \frac { 4 } { ( 1 + t ) ^ { 3 } }$$ When \(t = 0\), the particle has velocity \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and displacement 3 m from a fixed origin \(O\).
  1. Find an expression for the velocity of the particle in terms of \(t\).
  2. Show that when \(t = 3\) the particle is 10.5 m from \(O\).
Edexcel M3 Q4
10 marks Standard +0.3
4. A particle of mass 0.5 kg is moving on a straight line with simple harmonic motion. At time \(t = 0\) the particle is instantaneously at rest at the point \(A\). It next comes instantaneously to rest 3 seconds later at the point \(B\) where \(A B = 4 \mathrm {~m}\).
  1. For the motion of the particle write down
    1. the period,
    2. the amplitude.
  2. Find the maximum kinetic energy of the particle in terms of \(\pi\). The point \(C\) lies on \(A B\) at a distance of 1.2 m from \(B\).
  3. Find the time it takes the particle to travel directly from \(A\) to \(C\), giving your answer in seconds correct to 2 decimal places.
    (4 marks)
Edexcel M3 Q5
11 marks Standard +0.8
5. When a particle of mass \(M\) is at a distance of \(x\) metres from the centre of the moon, the gravitational force, \(F\) N, acting on it and directed towards the centre of the moon is given by $$F = \frac { \left( 4.90 \times 10 ^ { 12 } \right) M } { x ^ { 2 } }$$ A rocket is projected vertically into space from a point on the surface of the moon with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that the radius of the moon is \(\left( 1.74 \times 10 ^ { 6 } \right) \mathrm { m }\),
  1. show that the speed of the rocket, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when it is \(x\) metres from the centre of the moon is given by $$v ^ { 2 } = u ^ { 2 } + \frac { a } { x } - b$$ where \(a\) and \(b\) are constants which should be found correct to 3 significant figures.
  2. Find, correct to 2 significant figures, the minimum value of \(u\) needed for the rocket to escape the moon's gravitational attraction.
Edexcel M3 Q6
13 marks Challenging +1.2
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8b7133ed-3748-46cb-99d2-570ee33c7393-4_526_620_196_598} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a bowl formed by removing from a solid hemisphere of radius \(\frac { 3 } { 2 } r\) a smaller hemisphere of radius \(r\) having the same axis of symmetry and the same plane face.
  1. Show that the centre of mass of the bowl is a distance of \(\frac { 195 } { 304 } r\) from its plane face.
    (7 marks)
    The bowl has mass \(M\) and is placed with its curved surface on a smooth horizontal plane. A stud of mass \(\frac { 1 } { 2 } M\) is attached to the outer rim of the bowl. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8b7133ed-3748-46cb-99d2-570ee33c7393-4_517_729_1318_539} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} When the bowl is in equilibrium its plane surface is inclined at an angle \(\alpha\) to the horizontal as shown in Figure 2.
  2. Find tan \(\alpha\).
    (6 marks)
Edexcel M3 Q7
18 marks Standard +0.8
7. A cyclist is travelling round a circular bend of radius 25 m on a track which is banked at an angle of \(35 ^ { \circ }\) to the horizontal. In a model of the situation, the cyclist and her bicycle are represented by a particle of mass 60 kg and air resistance and friction are ignored. Using this model and assuming that the cyclist is not slipping,
  1. find, correct to 3 significant figures, the speed at which she is travelling. In tests it is found that the cyclist must travel at a minimum speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to prevent the bicycle from slipping down the slope. A more refined model is now used with a coefficient of friction between the bicycle and the track of \(\mu\). Using this model,
  2. show that \(\mu = 0.227\), correct to 3 significant figures,
  3. find, correct to 2 significant figures, the maximum speed at which the cyclist can travel without slipping up the slope. END
Edexcel M3 Q1
7 marks Standard +0.3
  1. The velocity, \(\mathbf { v ~ c m ~ s } { } ^ { - 1 }\), at time \(t\) seconds, of a radio-controlled toy is modelled by the formula
$$\mathbf { v } = \mathrm { e } ^ { 2 t } \mathbf { i } + 2 t \mathbf { j } ,$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.
  1. Find the acceleration of the toy in terms of \(t\).
  2. Find, correct to 2 significant figures, the time at which the acceleration of the toy is parallel to the vector \(( 4 \mathbf { i } + \mathbf { j } )\).
  3. Explain why this model is unlikely to be realistic for large values of \(t\).
Edexcel M3 Q2
8 marks Standard +0.3
2. A particle \(P\) of mass 0.4 kg is moving in a straight line through a fixed point \(O\). At time \(t\) seconds after it passes through \(O\), the distance \(O P\) is \(x\) metres and the resultant force acting on \(P\) is of magnitude ( \(5 + 4 \mathrm { e } ^ { - x }\) ) N in the direction \(O P\). When \(x = 1 , P\) is at the point \(A\).
  1. Find, correct to 3 significant figures, the work done in moving \(P\) from \(O\) to \(A\). Given that \(P\) passes through \(O\) with speed \(2 \mathrm {~ms} ^ { - 1 }\),
  2. find, correct to 3 significant figures, the speed of \(P\) as it passes through \(A\).
Edexcel M3 Q3
8 marks Moderate -0.5
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00776cc0-0214-4029-8ef1-c1cba89f4b87-2_382_796_1640_479} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A popular racket game involves a tennis ball of mass 0.1 kg which is attached to one end of a light inextensible string. The other end of the string is attached to the top of a fixed rigid pole. A boy strikes the ball such that it moves in a horizontal circle with angular speed \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and the string makes an angle of \(60 ^ { \circ }\) with the downward vertical as shown in Figure 1.
  1. Find the tension in the string.
  2. Find the length of the string.
Edexcel M3 Q4
9 marks Standard +0.3
4. A particle moves with simple harmonic motion along a straight line. When the particle is 3 cm from its centre of motion it has a speed of \(8 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\) and an acceleration of magnitude \(12 \mathrm {~cm} \mathrm {~s} ^ { - 2 }\).
  1. Show that the period of the motion is \(\pi\) seconds.
  2. Find the amplitude of the motion.
  3. Hence, find the greatest speed of the particle.
Edexcel M3 Q5
10 marks Standard +0.3
5. A physics student is set the task of finding the mass of an object without using a set of scales. She decides to use a light elastic string of natural length 2 m and modulus of elasticity 280 N attached to two points \(A\) and \(B\) which are on the same horizontal level and 2.4 m apart. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00776cc0-0214-4029-8ef1-c1cba89f4b87-3_307_1072_993_438} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} She attaches the object to the midpoint of the string so that it hangs in equilibrium 0.35 m below \(A B\) as shown in Figure 2.
  1. Explain why it is reasonable to assume that the tensions in each half of the string are equal.
  2. Find the mass of the object.
  3. Find the elastic potential energy of the string when the object is suspended from it.
Edexcel M3 Q6
13 marks Challenging +1.2
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00776cc0-0214-4029-8ef1-c1cba89f4b87-4_455_540_201_660} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows part of the curve \(y = x ^ { 2 } + 1\). The shaded region enclosed by the curve, the coordinate axes and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  1. Find the coordinates of the centre of mass of the solid obtained. The solid is suspended from a point on its larger circular rim and hangs in equilibrium.
  2. Find, correct to the nearest degree, the acute angle which the plane surfaces of the solid make with the vertical.
    (3 marks)
Edexcel M3 Q7
20 marks Standard +0.8
7. A particle of mass 0.5 kg is hanging vertically at one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point. The particle is given an initial horizontal speed of \(u \mathrm {~ms} ^ { - 1 }\).
  1. Show that the particle will perform complete circles if \(u \geq \sqrt { 3 g }\). Given that \(u = 5\),
  2. find, correct to the nearest degree, the angle through which the string turns before it becomes slack,
  3. find, correct to the nearest centimetre, the greatest height the particle reaches above its position when the string becomes slack.
Edexcel M3 Q1
8 marks Moderate -0.3
  1. A particle \(P\) of mass 1.5 kg moves from rest at the origin such that at time \(t\) seconds it is subject to a single force of magnitude \(( 4 t + 3 ) \mathrm { N }\) in the direction of the positive \(x\)-axis.
    1. Find the magnitude of the impulse exerted by the force during the interval \(1 \leq t \leq 4\).
    Given that at time \(T\) seconds, \(P\) has a speed of \(22 \mathrm {~ms} ^ { - 1 }\),
  2. find the value of \(T\) correct to 3 significant figures.
Edexcel M3 Q2
8 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-2_469_465_776_680} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A particle \(P\) of mass 0.5 kg is at rest at the highest point \(A\) of a smooth sphere, centre \(O\), of radius 1.25 m which is fixed to a horizontal surface. When \(P\) is slightly disturbed it slides along the surface of the sphere. Whilst \(P\) is in contact with the sphere it has speed \(v \mathrm {~ms} ^ { - 1 }\) when \(\angle A O P = \theta\) as shown in Figure 1.
  1. Show that \(v ^ { 2 } = 24.5 ( 1 - \cos \theta )\).
  2. Find the value of \(\cos \theta\) when \(P\) leaves the surface of the sphere.
Edexcel M3 Q3
12 marks Standard +0.8
3. A car starts from rest at the point \(O\) and moves along a straight line. The car accelerates to a maximum velocity, \(V \mathrm {~ms} ^ { - 1 }\), before decelerating and coming to rest again at the point \(A\). The acceleration of the car during this journey, \(a \mathrm {~ms} ^ { - 2 }\), is modelled by the formula $$a = \frac { 500 - k x } { 150 }$$ where \(x\) is the distance in metres of the car from \(O\).
Using this model and given that the car is travelling at \(16 \mathrm {~ms} ^ { - 1 }\) when it is 40 m from \(O\),
  1. find \(k\),
  2. show that \(V = 41\), correct to 2 significant figures,
  3. find the distance \(O A\).
Edexcel M3 Q4
12 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-3_316_536_1087_639} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A particle \(P\) of mass 2 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity \(\lambda\). The other end of the string is fixed to a point \(A\) on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 6 } \sqrt { 3 }\). \(P\) is held at rest at \(A\) and then released. It first comes to instantaneous rest at the point \(B , 2.2 \mathrm {~m}\) from \(A\). For the motion of \(P\) from \(A\) to \(B\),
  1. show that the work done against friction is 10.78 J ,
  2. find the change in the gravitational potential energy of \(P\). By using the work-energy principle, or otherwise,
  3. find \(\lambda\).
Edexcel M3 Q5
16 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-4_693_554_196_717} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} A flask is modelled as a uniform solid formed by removing a cylinder of radius \(r\) and height \(h\) from a cylinder of radius \(\frac { 4 } { 3 } r\) and height \(\frac { 3 } { 2 } h\) with the same axis of symmetry and a common plane as shown in Figure 3.
  1. Show that the centre of mass of the flask is a distance of \(\frac { 9 } { 10 } h\) from the open end of the flask. The flask is made from a material of density \(\rho\) and is filled to the level of the open plane face with a liquid of density \(k \rho\). Given that the centre of mass of the flask and liquid together is a distance of \(\frac { 15 } { 22 } h\) from the open end of the flask,
  2. find the value of \(k\).
  3. Explain why it may be advantageous to make the base of the flask from a more dense material.
    (2 marks)
Edexcel M3 Q6
19 marks Standard +0.8
6. A particle \(P\) of mass 2.5 kg is moving with simple harmonic motion in a straight line between two points \(A\) and \(B\) on a smooth horizontal table. When \(P\) is 3 m from \(O\), the centre of the oscillations, its speed is \(6 \mathrm {~ms} ^ { - 1 }\). When \(P\) is 2.25 m from \(O\), its speed is \(8 \mathrm {~ms} ^ { - 1 }\).
  1. Show that \(A B = 7.5 \mathrm {~m}\).
  2. Find the period of the motion.
  3. Find the kinetic energy of \(P\) when it is 2.7 m from \(A\).
  4. Show that the time taken by \(P\) to travel directly from \(A\) to the midpoint of \(O B\) is \(\frac { \pi } { 4 }\) seconds.
Edexcel M4 Q3
8 marks Challenging +1.8
3. A smooth uniform sphere \(P\) of mass \(m\) is falling vertically and strikes a fixed smooth inclined plane with speed \(u\). The plane is inclined at an angle \(\theta , \theta < 45 ^ { \circ }\), to the horizontal. The coefficient of restitution between \(P\) and the inclined plane is \(e\). Immediately after \(P\) strikes the plane, \(P\) moves horizontally.
  1. Show that \(e = \tan ^ { 2 } \theta\).
Edexcel M4 2003 January Q1
6 marks Standard +0.8
  1. A boy enters a large horizontal field and sees a friend 100 m due north. The friend is walking in an easterly direction at a constant speed of \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The boy can walk at a maximum speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find the shortest time for the boy to intercept his friend and the bearing on which he must travel to achieve this.
(6)
Edexcel M4 2003 January Q2
7 marks Challenging +1.2
2. Boat \(A\) is sailing due cast at a constant speed of \(10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). To an observer on \(A\), the wind appears to be blowing from due south. A second boat \(B\) is sailing due north at a constant speed of \(14 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). To an observer on \(B\), the wind appears to be blowing from the south west. The velocity of the wind relative to the earth is constant and is the same for both boats. Find the velocity of the wind relative to the earth, stating its magnitude and direction.
Edexcel M4 2003 January Q3
11 marks Challenging +1.8
3. A small pebble of mass \(m\) is placed in a viscous liquid and sinks vertically from rest through the liquid. When the speed of the pebble is \(v\) the magnitude of the resistance due to the liquid is modelled as \(m k v ^ { 2 }\), where \(k\) is a positive constant. Find the speed of the pebble after it has fallen a distance \(D\) through the liquid. \includegraphics[max width=\textwidth, alt={}, center]{618fdb9c-cc0b-4a80-a148-4311c908c94e-3_813_699_397_674} Figure 1 shows a uniform rod \(A B\), of mass \(m\) and length \(4 a\), resting on a smooth fixed sphere of radius \(a\). A light elastic string, of natural length \(a\) and modulus of elasticity \(\frac { 3 } { 4 } m g\), has one end attached to the lowest point \(C\) of the sphere and the other end attached to \(A\). The points \(A\), \(B\) and \(C\) lie in a vertical plane with \(\angle B A C = 2 \theta\), where \(\theta < \frac { \pi } { 4 }\). Given that \(A C\) is always horizontal,
  1. show that the potential energy of the system is $$\frac { m g a } { 8 } \left( 16 \sin 2 \theta + 3 \cot ^ { 2 } \theta - 6 \cot \theta \right) + \text { constant } ,$$
  2. show that there is a value of \(\theta\) for which the system is in equilibrium such that \(0.535 < \theta < 0.545\).
  3. Determine whether this position of equilibrium is stable or unstable.
Edexcel M4 2003 January Q5
17 marks Standard +0.8
5. A particle \(P\) moves in a straight line. At time \(t\) seconds its displacement from a fixed point \(O\) on the line is \(x\) metres. The motion of \(P\) is modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 x = 12 \cos 2 t - 6 \sin 2 t$$ When \(t = 0 , P\) is at rest at \(O\).
  1. Find, in terms of \(t\), the displacement of \(P\) from \(O\).
  2. Show that \(P\) comes to instantaneous rest when \(t = \frac { \pi } { 4 }\).
  3. Find, in metres to 3 significant figures, the displacement of \(P\) from \(O\) when \(t = \frac { \pi } { 4 }\).
  4. Find the approximate period of the motion for large values of \(t\). \section*{6.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{618fdb9c-cc0b-4a80-a148-4311c908c94e-5_534_923_388_541}
    \end{figure} A small ball \(Q\) of mass \(2 m\) is at rest at the point \(B\) on a smooth horizontal plane. A second small ball \(P\) of mass \(m\) is moving on the plane with speed \(\frac { 13 } { 12 } u\) and collides with \(Q\). Both the balls are smooth, uniform and of the same radius. The point \(C\) is on a smooth vertical wall \(W\) which is at a distance \(d _ { 1 }\) from \(B\), and \(B C\) is perpendicular to \(W\). A second smooth vertical wall is perpendicular to \(W\) and at a distance \(d _ { 2 }\) from \(B\). Immediately before the collision occurs, the direction of motion of \(P\) makes an angle \(\alpha\) with \(B C\), as shown in Fig. 2, where tan \(\alpha = \frac { 5 } { 12 }\). The line of centres of \(P\) and \(Q\) is parallel to \(B C\). After the collision \(Q\) moves towards \(C\) with speed \(\frac { 3 } { 5 } u\).
  5. Show that, after the collision, the velocity components of \(P\) parallel and perpendicular to \(C B\) are \(\frac { 1 } { 5 } u\) and \(\frac { 5 } { 12 } u\) respectively.
  6. Find the coefficient of restitution between \(P\) and \(Q\).
  7. Show that when \(Q\) reaches \(C , P\) is at a distance \(\frac { 4 } { 3 } d _ { 1 }\) from \(W\). For each collision between a ball and a wall the coefficient of restitution is \(\frac { 1 } { 2 }\).
    Given that the balls collide with each other again,
  8. show that the time between the two collisions of the balls is \(\frac { 15 d _ { 1 } } { u }\),
  9. find the ratio \(d _ { 1 } : d _ { 2 }\). \section*{END}
Edexcel M4 2004 January Q1
5 marks Standard +0.3
  1. A particle \(P\) of mass 3 kg moves in a straight line on a smooth horizontal plane. When the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resultant force acting on \(P\) is a resistance to motion of magnitude \(2 v \mathrm {~N}\). Find the distance moved by \(P\) while slowing down from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    (5)
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{08aefb06-102f-4a1b-ae3c-2d0079b86045-2_731_1662_554_227}
\end{figure} Two smooth uniform spheres \(A\) and \(B\) of equal radius have masses 2 kg and 1 kg respectively. They are moving on a smooth horizontal plane when they collide. Immediately before the collision the speed of \(A\) is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(1.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When they collide the line joining their centres makes an angle \(\alpha\) with the direction of motion of \(A\) and an angle \(\beta\) with the direction of motion of \(B\), where \(\tan \alpha = \frac { 4 } { 3 }\) and \(\tan \beta = \frac { 12 } { 5 }\) as shown in Fig. 1.
  1. Find the components of the velocities of \(A\) and \(B\) perpendicular and parallel to the line of centres immediately before the collision. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\).
  2. Find, to one decimal place, the speed of each sphere after the collision.
    (9)
Edexcel M4 2004 January Q3
14 marks Challenging +1.8
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{08aefb06-102f-4a1b-ae3c-2d0079b86045-3_933_1063_277_534}
\end{figure} Two uniform rods \(A B\) and \(A C\), each of mass \(2 m\) and length \(2 L\), are freely jointed at \(A\). The mid-points of the rods are \(D\) and \(E\) respectively. A light inextensible string of length \(s\) is fixed to \(E\) and passes round small, smooth light pulleys at \(D\) and \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs vertically. The points \(A , B\) and \(C\) lie in the same vertical plane with \(B\) and \(C\) on a smooth horizontal surface. The angles \(P A B\) and \(P A C\) are each equal to \(\theta ( \theta > 0 )\), as shown in Fig. 2.
  1. Find the length of \(A P\) in terms of \(s , L\) and \(\theta\).
  2. Show that the potential energy \(V\) of the system is given by $$V = 2 m g L ( 3 \cos \theta + \sin \theta ) + \text { constant } .$$
  3. Hence find the value of \(\theta\) for which the system is in equilibrium.
  4. Determine whether this position of equilibrium is stable or unstable.