Questions M4 (327 questions)

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OCR M4 2010 June Q6
12 marks Challenging +1.2
  1. Show that \(\theta = 0\) is a position of stable equilibrium.
  2. Show that the kinetic energy of the system is \(4 m a ^ { 2 } \dot { \theta } ^ { 2 }\).
  3. By differentiating the energy equation, then making suitable approximations for \(\sin \theta\) and \(\cos \theta\), find the approximate period of small oscillations about the equilibrium position \(\theta = 0\). \section*{[Question 7 is printed overleaf.]}
OCR M4 2013 June Q5
14 marks Standard +0.8
  1. Find the magnitude and bearing of the velocity of \(U\) relative to \(P\).
  2. Find the shortest distance between \(P\) and \(U\) in the subsequent motion.
    (ii) Plane \(Q\) is flying with constant velocity \(160 \mathrm {~ms} ^ { - 1 }\) in the direction which brings it as close as possible to \(U\).
    1. Find the bearing of the direction in which \(Q\) is flying.
    2. Find the shortest distance between \(Q\) and \(U\) in the subsequent motion. \includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-3_771_769_262_646} A square frame \(A B C D\) consists of four uniform rods \(A B , B C , C D , D A\), rigidly joined at \(A , B , C , D\). Each rod has mass 0.6 kg and length 1.5 m . The frame rotates freely in a vertical plane about a fixed horizontal axis passing through the mid-point \(O\) of \(A D\). At time \(t\) seconds the angle between \(A D\) and the horizontal, measured anticlockwise, is \(\theta\) radians (see diagram).
      1. Show that the moment of inertia of the frame about the axis through \(O\) is \(3.15 \mathrm {~kg} \mathrm {~m} ^ { 2 }\).
      2. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - 5.6 \sin \theta\).
      3. Deduce that the frame can make small oscillations which are approximately simple harmonic, and find the period of these oscillations. The frame is at rest with \(A D\) horizontal. A couple of constant moment 25 Nm about the axis is then applied to the frame.
      4. Find the angular speed of the frame when it has rotated through 1.2 radians.
OCR M4 2015 June Q5
15 marks Challenging +1.8
  1. Taking \(H\) as the reference level for gravitational potential energy, show that the total potential energy \(V\) of the system is given by $$V = m g \left( 2 \lambda r \cos \theta - 2 r \cos ^ { 2 } \theta - \lambda a \right)$$
  2. Find the set of possible values of \(\lambda\) so that there is more than one position of equilibrium.
  3. For the case \(\lambda = \frac { 3 } { 2 }\), determine whether each equilibrium position is stable or unstable.
Edexcel M4 Q1
13 marks Challenging +1.2
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-03_457_638_233_598} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A fixed smooth plane is inclined to the horizontal at an angle of \(45 ^ { \circ }\). A particle \(P\) is moving horizontally and strikes the plane. Immediately before the impact, \(P\) is moving in a vertical plane containing a line of greatest slope of the inclined plane. Immediately after the impact, \(P\) is moving in a direction which makes an angle of \(30 ^ { \circ }\) with the inclined plane, as shown in Figure 1. Find the fraction of the kinetic energy of \(P\) which is lost in the impact.
Edexcel M4 Q2
8 marks Challenging +1.8
2. At time \(t = 0\), a particle \(P\) of mass \(m\) is projected vertically upwards with speed \(\sqrt { \frac { g } { k } }\), where \(k\) is a constant. At time \(t\) the speed of \(P\) is \(v\). The particle \(P\) moves against air resistance whose magnitude is modelled as being \(m k v ^ { 2 }\) when the speed of \(P\) is \(v\). Find, in terms of \(k\), the distance travelled by \(P\) until its speed first becomes half of its initial speed.
Edexcel M4 Q3
10 marks Challenging +1.2
  1. At noon a motorboat \(P\) is 2 km north-west of another motorboat \(Q\). The motorboat \(P\) is moving due south at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The motorboat \(Q\) is pursuing motorboat \(P\) at a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and sets a course in order to get as close to motorboat \(P\) as possible.
    1. Find the course set by \(Q\), giving your answer as a bearing to the nearest degree.
    2. Find the shortest distance between \(P\) and \(Q\).
    3. Find the distance travelled by \(Q\) from its position at noon to the point of closest approach.
Edexcel M4 Q4
10 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-08_479_807_246_571} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A light inextensible string of length \(2 a\) has one end attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(m\). A second light inextensible string of length \(L\), where \(L > \frac { 12 a } { 5 }\), has one of its ends attached to \(P\) and passes over a small smooth peg fixed at a point \(B\). The line \(A B\) is horizontal and \(A B = 2 a\). The other end of the second string is attached to a particle of mass \(\frac { 7 } { 20 } m\), which hangs vertically below \(B\), as shown in Figure 2.
  1. Show that the potential energy of the system, when the angle \(P A B = 2 \theta\), is $$\frac { 1 } { 5 } m g a ( 7 \sin \theta - 10 \sin 2 \theta ) + \text { constant. }$$
  2. Show that there is only one value of \(\cos \theta\) for which the system is in equilibrium and find this value.
  3. Determine the stability of the position of equilibrium.
Edexcel M4 Q5
8 marks Standard +0.8
  1. Two small smooth spheres \(A\) and \(B\), of mass 2 kg and 1 kg respectively, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of \(A\) is \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(- 2 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(\mathbf { j } \mathrm { m } \mathrm { s } ^ { - 1 }\).
    1. Show that the velocity of \(B\) immediately after the collision is \(2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    2. Find the impulse of \(B\) on \(A\) in the collision, giving your answer as a vector, and hence show that the line of centres is parallel to \(\mathbf { i } + \mathbf { j }\).
    3. Find the coefficient of restitution between \(A\) and \(B\).
Edexcel M4 Q6
14 marks Challenging +1.2
6. A light elastic spring \(A B\) has natural length \(2 a\) and modulus of elasticity \(2 m n ^ { 2 } a\), where \(n\) is a constant. A particle \(P\) of mass \(m\) is attached to the end \(A\) of the spring. At time \(t = 0\), the spring, with \(P\) attached, lies at rest and unstretched on a smooth horizontal plane. The other end \(B\) of the spring is then pulled along the plane in the direction \(A B\) with constant acceleration \(f\). At time \(t\) the extension of the spring is \(x\).
  1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + n ^ { 2 } x = f .$$
  2. Find \(x\) in terms of \(n , f\) and \(t\). Hence find
  3. the maximum extension of the spring,
  4. the speed of \(P\) when the spring first reaches its maximum extension.
    1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors due east and due north respectively]
    A man cycles at a constant speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on level ground and finds that when his velocity is \(u \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the velocity of the wind appears to be \(v ( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(v\) is a positive constant. When the man cycles with velocity \(\frac { 1 } { 5 } u ( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), the velocity of the wind appears to be \(w \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(w\) is a positive constant. Find, in terms of \(u\), the true velocity of the wind.
Edexcel M4 Q7
12 marks Challenging +1.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cf941854-3a33-4d9d-9fa0-ce9a63227599-38_451_1077_315_370} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a framework \(A B C\), consisting of two uniform rods rigidly joined together at \(B\) so that \(\angle A B C = 90 ^ { \circ }\). The \(\operatorname { rod } A B\) has length \(2 a\) and mass \(4 m\), and the \(\operatorname { rod } B C\) has length \(a\) and mass \(2 m\). The framework is smoothly hinged at \(A\) to a fixed point, so that the framework can rotate in a fixed vertical plane. One end of a light elastic string, of natural length \(2 a\) and modulus of elasticity \(3 m g\), is attached to \(A\). The string passes through a small smooth ring \(R\) fixed at a distance \(2 a\) from \(A\), on the same horizontal level as \(A\) and in the same vertical plane as the framework. The other end of the string is attached to \(B\). The angle \(A R B\) is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\).
  1. Show that the potential energy \(V\) of the system is given by $$V = 8 a m g \sin 2 \theta + 5 a m g \cos 2 \theta + \text { constant }$$
  2. Find the value of \(\theta\) for which the system is in equilibrium.
  3. Determine the stability of this position of equilibrium.
    1. A smooth uniform sphere \(S\), of mass \(m\), is moving on a smooth horizontal plane when it collides obliquely with another smooth uniform sphere \(T\), of the same radius as \(S\) but of mass \(2 m\), which is at rest on the plane. Immediately before the collision the velocity of \(S\) makes an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), with the line joining the centres of the spheres. Immediately after the collision the speed of \(T\) is \(V\). The coefficient of restitution between the spheres is \(\frac { 3 } { 4 }\).
    2. Find, in terms of \(V\), the speed of \(S\)
      1. immediately before the collision,
      2. immediately after the collision.
    3. Find the angle through which the direction of motion of \(S\) is deflected as a result of the collision.
Edexcel M4 2002 January Q1
4 marks Moderate -0.8
A river of width 40 m flows with uniform and constant speed between straight banks. A swimmer crosses as quickly as possible and takes 30 s to reach the other side. She is carried 25 m downstream. Find
  1. the speed of the river, [2]
  2. the speed of the swimmer relative to the water. [2]
Edexcel M4 2002 January Q2
8 marks Challenging +1.2
A ball of mass \(m\) is thrown vertically upwards from the ground with an initial speed \(u\). When the speed of the ball is \(v\), the magnitude of the air resistance is \(mkv\), where \(k\) is a positive constant. By modelling the ball as a particle, find, in terms of \(u\), \(k\) and \(g\), the time taken for the ball to reach its greatest height. [8]
Edexcel M4 2002 January Q3
10 marks Challenging +1.2
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 \(\theta\), \(\theta < 45°\), 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 \(e = \tan^2 \theta\). [6]
  2. Show that the magnitude of the impulse exerted by \(P\) on the plane is \(mu \sec \theta\). [4]
Edexcel M4 2002 January Q4
11 marks Challenging +1.2
A pilot flying an aircraft at a constant speed of 2000 kmh\(^{-1}\) detects an enemy aircraft 100 km away on a bearing of 045°. The enemy aircraft is flying at a constant velocity of 1500 kmh\(^{-1}\) due west. Find
  1. the course, as a bearing to the nearest degree, that the pilot should set up in order to intercept the enemy aircraft,
  2. the time, to the nearest s, that the pilot will take to reach the enemy aircraft. [11]
Edexcel M4 2002 January Q5
12 marks Challenging +1.2
\includegraphics{figure_1} A smooth uniform sphere \(S\) of mass \(m\) is moving on a smooth horizontal table. The sphere \(S\) collides with another smooth uniform sphere \(T\), of the same radius as \(S\) but of mass \(km\), \(k > 1\), which is at rest on the table. The coefficient of restitution between the spheres is \(e\). Immediately before the spheres collide the direction of motion of \(S\) makes an angle \(\theta\) with the line joining their centres, as shown in Fig. 1. Immediately after the collision the directions of motion of \(S\) and \(T\) are perpendicular.
  1. Show that \(e = \frac{1}{k}\). [6]
Given that \(k = 2\) and that the kinetic energy lost in the collision is one quarter of the initial kinetic energy,
  1. find the value of \(\theta\). [6]
Edexcel M4 2002 January Q6
15 marks Standard +0.8
\includegraphics{figure_2} In a simple model of a shock absorber, a particle \(P\) of mass \(m\) kg is attached to one end of a light elastic horizontal spring. The other end of the spring is fixed at \(A\) and the motion of \(P\) takes place along a fixed horizontal line through \(A\). The spring has natural length \(L\) metres and modulus of elasticity \(2mL\) newtons. The whole system is immersed in a fluid which exerts a resistance on \(P\) of magnitude \(3mv\) newtons, where \(v\) m s\(^{-1}\) is the speed of \(P\) at time \(t\) seconds. The compression of the spring at time \(t\) seconds is \(x\) metres, as shown in Fig. 2.
  1. Show that $$\frac{\text{d}^2 x}{\text{d}t^2} + 3\frac{\text{d}x}{\text{d}t} + 2x = 0.$$ [4]
Given that when \(t = 0\), \(x = 2\) and \(\frac{\text{d}x}{\text{d}t} = -4\),
  1. find \(x\) in terms of \(t\). [8]
  2. Sketch the graph of \(x\) against \(t\). [2]
  3. State, with a reason, whether the model is realistic. [1]
Edexcel M4 2002 January Q7
15 marks Challenging +1.8
\includegraphics{figure_3} A uniform rod \(AB\), of mass \(m\) and length \(2a\), can rotate freely in a vertical plane about a fixed smooth horizontal axis through \(A\). The fixed point \(C\) is vertically above \(A\) and \(AC = 4a\). A light elastic string, of natural length \(2a\) and modulus of elasticity \(\frac{1}{4}mg\), joins \(B\) to \(C\). The rod \(AB\) makes an angle \(\theta\) with the upward vertical at \(A\), as shown in Fig. 3.
  1. Show that the potential energy of the system is $$-mga[\cos \theta + \sqrt{(5 - 4 \cos \theta)}] + \text{constant}.$$ [9]
  2. Hence determine the values of \(\theta\) for which the system is in equilibrium. [6]
Edexcel M4 2003 January Q1
6 marks Standard +0.8
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 m s\(^{-1}\). The boy can walk at a maximum speed of 1 m 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 Standard +0.8
Boat \(A\) is sailing due east at a constant speed of 10 km 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 km 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. [7]
Edexcel M4 2003 January Q3
11 marks Challenging +1.2
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 \(mkv^2\), where \(k\) is a positive constant. Find the speed of the pebble after it has fallen a distance \(D\) through the liquid. [11]
Edexcel M4 2003 January Q4
16 marks Challenging +1.8
\includegraphics{figure_1} Figure 1 shows a uniform rod \(AB\), of mass \(m\) and length \(4a\), resting on a smooth fixed sphere of radius \(a\). A light elastic string, of natural length \(a\) and modulus of elasticity \(\frac{1}{4}mg\), 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 \(\angle BAC = 2\theta\), where \(\theta < \frac{\pi}{4}\). Given that \(AC\) is always horizontal,
  1. show that the potential energy of the system is $$\frac{mga}{8}(16\sin 2\theta + 3\cot^2 \theta - 6\cot \theta) + \text{constant}.$$ [7]
  2. show that there is a value of \(\theta\) for which the system is in equilibrium such that \(0.535 < \theta < 0.545\). [6]
  3. Determine whether this position of equilibrium is stable or unstable. [3]
Edexcel M4 2003 January Q5
17 marks Challenging +1.3
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 $$\frac{\text{d}^2 x}{\text{d}t^2} + 2\frac{\text{d}x}{\text{d}t} + 2x = 12\cos 2t - 6\sin 2t.$$ When \(t = 0\), \(P\) is at rest at \(O\).
  1. Find, in terms of \(t\), the displacement of \(P\) from \(O\). [11]
  2. Show that \(P\) comes to instantaneous rest when \(t = \frac{\pi}{4}\). [2]
  3. Find, in metres to 3 significant figures, the displacement of \(P\) from \(O\) when \(t = \frac{\pi}{4}\). [2]
  4. Find the approximate period of the motion for large values of \(t\). [2]
Edexcel M4 2003 January Q6
18 marks Challenging +1.8
\includegraphics{figure_2} A small ball \(Q\) of mass \(2m\) 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 \(BC\) 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 \(BC\), as shown in Fig. 2, where \(\tan \alpha = \frac{5}{12}\). The line of centres of \(P\) and \(Q\) is parallel to \(BC\). After the collision \(Q\) moves towards \(C\) with speed \(\frac{5}{4}u\).
  1. Show that, after the collision, the velocity components of \(P\) parallel and perpendicular to \(CB\) are \(\frac{1}{4}u\) and \(\frac{5}{12}u\) respectively. [4]
  2. Find the coefficient of restitution between \(P\) and \(Q\). [2]
  3. Show that when \(Q\) reaches \(C\), \(P\) is at a distance \(\frac{4}{5}d_1\) from \(W\). [3]
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,
  1. show that the time between the two collisions of the balls is \(\frac{15d_1}{u}\). [4]
  2. find the ratio \(d_1 : d_2\). [5]
Edexcel M4 2004 January Q1
5 marks Standard +0.3
A particle \(P\) of mass 3 kg moves in a straight line on a smooth horizontal plane. When the speed of \(P\) is \(v\) m s\(^{-1}\), the resultant force acting on \(P\) is a resistance to motion of magnitude \(2v\) N. Find the distance moved by \(P\) while slowing down from 5 m s\(^{-1}\) to 2 m s\(^{-1}\). [5]
Edexcel M4 2004 January Q2
13 marks Standard +0.8
\includegraphics{figure_1} 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 m s\(^{-1}\) and the speed of \(B\) is 1.3 m 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. [4]
The coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{2}\).
  1. Find, to one decimal place, the speed of each sphere after the collision. [9]