Questions M4 (327 questions)

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Edexcel M4 2017 June Q5
9 marks Challenging +1.2
5. A cyclist riding due north at a steady speed of \(12 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) notices that the wind appears to come from the north-west. At the same time, another cyclist, moving on a bearing of \(120 ^ { \circ }\) and also riding at a steady speed of \(12 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), notices that the wind appears to come from due south. The velocity of the wind is assumed to be constant. Find
  1. the wind speed,
  2. the direction from which the wind is blowing, giving your answer as a bearing.
Edexcel M4 2017 June Q6
13 marks Standard +0.8
6. A particle \(P\) of mass 0.2 kg is suspended from a fixed point by a light elastic spring. The spring has natural length 0.8 m and modulus of elasticity 7 N . At time \(t = 0\) the particle is released from rest from a point 0.2 metres vertically below its equilibrium position. The motion of \(P\) is resisted by a force of magnitude \(2 v\) newtons, where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of \(P\). At time \(t\) seconds, \(P\) is \(x\) metres below its equilibrium position.
  1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 10 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 43.75 x = 0\)
  2. Find \(x\) in terms of \(t\).
  3. Find the value of \(t\) when \(P\) first comes to instantaneous rest.
Edexcel M4 2017 June Q7
13 marks Challenging +1.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{23e6a9ae-bf00-45a3-b462-e18760d9af45-24_655_890_239_529} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows four uniform rods, each of mass \(m\) and length \(2 a\). The rods are freely hinged at their ends to form a rhombus \(A B C D\). Point \(A\) is attached to a fixed point on a ceiling and the rhombus hangs freely with \(C\) vertically below \(A\). A light elastic spring of natural length \(2 a\) and modulus of elasticity \(7 m g\) connects the points \(A\) and \(C\). A particle of mass \(3 m\) is attached to point \(C\).
  1. Show that, when \(A D\) is at an angle \(\theta\) to the downward vertical, the potential energy \(V\) of the system is given by $$V = 28 m g a \cos ^ { 2 } \theta - 48 m g a \cos \theta + \text { constant }$$ Given that \(\theta > 0\)
  2. find the value of \(\theta\) for which the system is in equilibrium,
  3. determine the stability of this position of equilibrium.
Edexcel M4 2018 June Q1
11 marks Challenging +1.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0f141c7-ecd0-4f62-bfad-76c81c2d6396-02_538_881_278_534} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform rod \(A B\) has mass \(m\) and length 4a. The end \(A\) of the rod is freely hinged to a fixed point. One end of a light elastic string, of natural length \(a\) and modulus \(\frac { 1 } { 4 } m g\), is attached to the end \(B\) of the rod. The other end of the string is attached to a small light smooth ring \(R\). The ring can move freely on a smooth horizontal wire which is fixed at a height \(a\) above \(A\), and in a vertical plane through \(A\). The angle between the rod and the horizontal is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\), as shown in Figure 1. Given that the elastic string is vertical,
  1. show that the potential energy of the system is $$2 m g a \left( \sin ^ { 2 } \theta - \sin \theta \right) + \text { constant }$$
  2. Show that when \(\theta = \frac { \pi } { 6 }\) the rod is in stable equilibrium.
Edexcel M4 2018 June Q2
8 marks Standard +0.8
2. A small ball \(B\), moving on a smooth horizontal plane, collides with a fixed smooth vertical wall. Immediately before the collision the angle between the direction of motion of \(B\) and the wall is \(\alpha\). The coefficient of restitution between \(B\) and the wall is \(\frac { 3 } { 4 }\). The kinetic energy of \(B\) immediately after the collision is \(60 \%\) of its kinetic energy immediately before the collision. Find, in degrees, the size of angle \(\alpha\).
Edexcel M4 2018 June Q3
7 marks Challenging +1.2
3. When a man walks due West at a constant speed of \(4 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), the wind appears to be blowing from due South. When he runs due North at a constant speed of \(8 \mathrm {~km} \mathrm {~h} ^ { - 1 }\), the speed of the wind appears to be \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
The velocity of the wind relative to the Earth is constant with magnitude \(w \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
Find the two possible values of \(w\).
Edexcel M4 2018 June Q4
11 marks Challenging +1.2
4. A particle \(P\) of mass 0.5 kg moves in a horizontal straight line. At time \(t\) seconds \(( t \geqslant 0 )\), the displacement of \(P\) from a fixed point \(O\) of the line is \(x\) metres, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(P\) is moving in the direction of \(x\) increasing. A force of magnitude \(k x\) newtons acts on \(P\) in the direction \(P O\). The motion of \(P\) is also subject to a resistance of magnitude \(\lambda v\) newtons. Given that $$x = ( 1.5 + 10 t ) \mathrm { e } ^ { - 4 t }$$ find
  1. the value of \(k\) and the value of \(\lambda\),
  2. the distance from \(P\) to \(O\) when \(P\) is instantaneously at rest.
Edexcel M4 2018 June Q5
11 marks Standard +0.8
5. A horizontal square field, \(P Q R S\), has sides of length 75 m . Ali is at corner \(P\) of the field and Beth is at corner \(Q\) of the field. Ali starts to walk in a straight line along the diagonal of the field from \(P\) to \(R\) at a constant speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Beth sees Ali start to walk, waits 10 seconds, and then walks from \(Q\) to intercept Ali. Beth walks in a straight line at a constant speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the time from the instant Beth leaves \(Q\) until the instant that she intercepts Ali,
  2. the direction Beth should take.
Edexcel M4 2018 June Q6
14 marks Challenging +1.2
6. A particle of mass \(m\) is projected vertically upwards in a resisting medium. As the particle moves upwards, the speed \(v\) of the particle is given by $$v ^ { 2 } = k g \left( 5 \mathrm { e } ^ { - \frac { x } { 2 k } } - 4 \right)$$ where \(x\) is the distance of the particle above the point of projection and \(k\) is a positive constant.
  1. Show that the magnitude of the resistance to the motion of the particle is \(\frac { m v ^ { 2 } } { 4 k }\).
    (4)
  2. Find, in terms of \(k\), the greatest height reached by the particle above the point of projection.
  3. Show that the time taken by the particle to reach its greatest height above the point of projection is \(\sqrt { \frac { 4 k } { g } } \arctan \left( \frac { 1 } { 2 } \right)\)
Edexcel M4 2018 June Q7
13 marks Standard +0.8
7. Two smooth uniform spheres \(A\) and \(B\), of mass 2 kg and 3 kg respectively, and of equal radius, are moving on a smooth horizontal plane when they collide. Immediately before the collision the velocity of \(A\) is \(( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Immediately after the collision the velocity of \(A\) is \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Show that, at the instant when \(A\) and \(B\) collide, their line of centres is parallel to \(- \mathbf { i } + \mathbf { j }\).
  2. Find the velocity of \(B\) immediately after the collision.
  3. Find the coefficient of restitution between \(A\) and \(B\).
Edexcel M4 Q1
6 marks Standard +0.3
  1. A smooth sphere \(S\) is moving on a smooth horizontal plane with speed \(u\) when it collides with a smooth fixed vertical wall. At the instant of collision the direction of motion of \(S\) makes an angle of \(30 ^ { \circ }\) with the wall. The coefficient of restitution between \(S\) and the wall is \(\frac { 1 } { 3 }\).
Find the speed of \(S\) immediately after the collision.
Edexcel M4 Q2
8 marks Challenging +1.2
2. A car of mass 1000 kg , moving along a straight horizontal road, is driven by an engine which produces a constant power of 12 kW . The only resistance to the motion of the car is air resistance of magnitude \(10 v ^ { 2 } \mathrm {~N}\) where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car. Find the distance travelled by the car as its speed increases from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(8 marks)
Edexcel M4 Q3
10 marks Challenging +1.2
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d57ea92a-4d6a-46bf-a6aa-bbd5083e8726-3_469_1163_1217_443}
\end{figure} A smooth uniform sphere \(A\), moving on a smooth horizontal table, collides with a second identical sphere \(B\) which is at rest on the table. When the spheres collide the line joining their centres makes an angle of \(30 ^ { \circ }\) with the direction of motion of \(A\), as shown in Fig. 1. The coefficient of restitution between the spheres is \(e\). The direction of motion of \(A\) is deflected through an angle \(\theta\) by the collision. Show that \(\tan \theta = \frac { ( 1 + e ) \sqrt { 3 } } { 5 - 3 e }\).
(10 marks)
Edexcel M4 Q4
10 marks Standard +0.8
4. A body falls vertically from rest and is subject to air resistance of a magnitude which is proportional to its speed. Given that its terminal speed is \(100 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the time it takes for the body to attain a speed of \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(10 marks)
Edexcel M4 Q5
12 marks Standard +0.8
5. A particle \(P\) of mass \(m\) is fixed to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(2 m a n ^ { 2 }\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at a point which is a distance \(2 a\) vertically below \(O\). The air resistance is modelled as having magnitude \(2 m n v\), where \(v\) is the speed of \(P\).
  1. Show that, when the extension of the string is \(x\), $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 n \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 n ^ { 2 } x = g$$
  2. Find \(x\) in terms of \(t\).
Edexcel M4 Q6
12 marks Challenging +1.8
6. Two particles \(P\) and \(Q\) have constant velocity vectors \(\mathbf { v } _ { P }\) and \(\mathbf { v } _ { Q }\) respectively. The magnitude of the velocity of \(P\) relative to \(Q\) is equal to the speed of \(P\). If the direction of motion of one of the particles is reversed, the magnitude of the velocity of \(P\) relative to \(Q\) is doubled. Find
  1. the ratio of the speeds of \(P\) and \(Q\),
  2. the cosine of the angle between the directions of motion of \(P\) and \(Q\).
Edexcel M4 Q7
17 marks Challenging +1.8
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d57ea92a-4d6a-46bf-a6aa-bbd5083e8726-5_955_855_349_573}
\end{figure} A smooth wire \(A B\), in the shape of a circle of radius \(r\), is fixed in a vertical plane with \(A B\) vertical. A small smooth ring \(R\) of mass \(m\) is threaded on the wire and is connected by a light inextensible string to a particle \(P\) of mass \(m\). The length of the string is greater than the diameter of the circle. The string passes over a small smooth pulley which is fixed at the highest point \(A\) of the wire and angle \(R \hat { A } P = \theta\), as shown in Fig. 2.
  1. Show that the potential energy of the system is given by $$2 m g r \left( \cos \theta - \cos ^ { 2 } \theta \right) + \text { constant. }$$
  2. Hence determine the values of \(\theta , \theta \geq 0\), for which the system is in equilibrium. (6 marks)
  3. Determine the stability of each position of equilibrium. END
OCR M4 2002 January Q1
4 marks Moderate -0.8
1 A wheel rotating about a fixed axis is slowing down with constant angular deceleration. Initially the angular speed is \(24 \mathrm { rad } \mathrm { s } ^ { - 1 }\). In the first 5 seconds the wheel turns through 96 radians.
  1. Find the angular deceleration.
  2. Find the total angle the wheel turns through before coming to rest.
OCR M4 2002 January Q2
5 marks Challenging +1.2
2 A uniform solid of revolution is formed by rotating the region bounded by the \(x\)-axis, the line \(x = 1\) and the curve \(y = x ^ { 2 }\) for \(0 \leqslant x \leqslant 1\), about the \(x\)-axis. The units are metres, and the density of the solid is \(5400 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\). Find the moment of inertia of this solid about the \(x\)-axis.
OCR M4 2002 January Q3
6 marks Challenging +1.2
3 A uniform rectangular lamina \(A B C D\) of mass 0.6 kg has sides \(A B = 0.4 \mathrm {~m}\) and \(A D = 0.3 \mathrm {~m}\). The lamina is free to rotate about a fixed horizontal axis which passes through \(A\) and is perpendicular to the lamina.
  1. Find the moment of inertia of the lamina about the axis.
  2. Find the approximate period of small oscillations in a vertical plane.
OCR M4 2002 January Q4
8 marks Challenging +1.2
4 A uniform circular disc has mass \(m\), radius \(a\) and centre \(C\). The disc is free to rotate in a vertical plane about a fixed horizontal axis passing through a point \(A\) on the disc, where \(C A = \frac { 1 } { 3 } a\).
  1. Find the moment of inertia of the disc about this axis. The disc is released from rest with \(C A\) horizontal.
  2. Find the initial angular acceleration of the disc.
  3. State the direction of the force acting on the disc at \(A\) immediately after release, and find its magnitude.
OCR M4 2002 January Q5
8 marks Challenging +1.2
5 The region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = \ln 5\) and the curve \(y = \mathrm { e } ^ { x }\) for \(0 \leqslant x \leqslant \ln 5\), is occupied by a uniform lamina.
  1. Show that the centre of mass of this lamina has \(x\)-coordinate $$\frac { 5 } { 4 } \ln 5 - 1$$
  2. Find the \(y\)-coordinate of the centre of mass.
OCR M4 2002 January Q6
8 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{98647526-b52a-4316-9a09-48d756b8f599-3_117_913_251_630} An arm on a fairground ride is modelled as a uniform rod \(A B\), of mass 75 kg and length 7.2 m , with a particle of mass 124 kg attached at \(B\). The arm can rotate about a fixed horizontal axis perpendicular to the rod and passing through the point \(P\) on the rod, where \(A P = 1.2 \mathrm {~m}\).
  1. Show that the moment of inertia of the arm about the axis is \(5220 \mathrm {~kg} \mathrm {~m} ^ { 2 }\).
  2. The arm is released from rest with \(A B\) horizontal, and a frictional couple of constant moment 850 N m opposes the motion. Find the angular speed of the arm when \(B\) is first vertically below \(P\).
OCR M4 2002 January Q7
9 marks Standard +0.3
7 At midnight, ship \(A\) is 70 km due north of ship \(B\). Ship \(A\) travels with constant velocity \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in the direction with bearing \(140 ^ { \circ }\). Ship \(B\) travels with constant velocity \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in the direction with bearing \(025 ^ { \circ }\).
  1. Find the magnitude and direction of the velocity of \(A\) relative to \(B\).
  2. Find the distance between the ships when they are at their closest, and find the time when this occurs.
OCR M4 2002 January Q8
12 marks Challenging +1.2
8 \includegraphics[max width=\textwidth, alt={}, center]{98647526-b52a-4316-9a09-48d756b8f599-3_493_748_1393_708} The diagram shows a uniform rod \(A B\), of mass \(m\) and length \(2 a\), free to rotate in a vertical plane about a fixed horizontal axis through \(A\). A light elastic string has natural length \(a\) and modulus of elasticity \(\frac { 1 } { 2 } m g\). The string joins \(B\) to a light ring \(R\) which slides along a smooth horizontal wire fixed at a height \(a\) above \(A\) and in the same vertical plane as \(A B\). The string \(B R\) remains vertical. The angle between \(A B\) and the horizontal is denoted by \(\theta\), where \(0 < \theta < \pi\).
  1. Taking the reference level for gravitational potential energy to be the horizontal through \(A\), show that the total potential energy of the system is $$m g a \left( \sin ^ { 2 } \theta - \sin \theta \right) .$$
  2. Find the three values of \(\theta\) for which the system is in equilibrium.
  3. For each position of equilibrium, determine whether it is stable or unstable.