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
AQA M3 2014 June Q3
9 marks Moderate -0.3
3 A particle of mass 0.5 kg is moving in a straight line on a smooth horizontal surface.
The particle is then acted on by a horizontal force for 3 seconds. This force acts in the direction of motion of the particle and at time \(t\) seconds has magnitude \(( 3 t + 1 ) \mathrm { N }\). When \(t = 0\), the velocity of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the magnitude of the impulse of the force on the particle between the times \(t = 0\) and \(t = 3\).
  2. Hence find the velocity of the particle when \(t = 3\).
  3. Find the value of \(t\) when the velocity of the particle is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
AQA M3 2014 June Q4
14 marks Standard +0.3
4 Two boats, \(A\) and \(B\), are moving on straight courses with constant speeds. At noon, \(A\) and \(B\) have position vectors \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { km }\) and \(( - \mathbf { i } + \mathbf { j } ) \mathrm { km }\) respectively relative to a lighthouse. Thirty minutes later, the position vectors of \(A\) and \(B\) are ( \(- \mathbf { i } + 3 \mathbf { j }\) ) km and \(( 2 \mathbf { i } - \mathbf { j } ) \mathrm { km }\) respectively relative to the lighthouse.
  1. Find the velocity of \(A\) relative to \(B\) in the form \(( m \mathbf { i } + n \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\), where \(m\) and \(n\) are integers.
  2. The position vector of \(A\) relative to \(B\) at time \(t\) hours after noon is \(\mathbf { r } \mathrm { km }\). Show that $$\mathbf { r } = ( 2 - 10 t ) \mathbf { i } + ( 1 + 6 t ) \mathbf { j }$$
  3. Determine the value of \(t\) when \(A\) and \(B\) are closest together.
  4. Find the shortest distance between \(A\) and \(B\).
AQA M3 2014 June Q5
12 marks Standard +0.3
5 A small smooth ball is dropped from a height of \(h\) above a point \(A\) on a fixed smooth plane inclined at an angle \(\theta\) to the horizontal. The ball falls vertically and collides with the plane at the point \(A\). The ball rebounds and strikes the plane again at a point \(B\), as shown in the diagram. The points \(A\) and \(B\) lie on a line of greatest slope of the inclined plane. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-12_318_636_548_712}
  1. Explain whether or not the component of the velocity of the ball parallel to the plane is changed by the collision.
  2. The coefficient of restitution between the ball and the plane is \(e\). Find, in terms of \(h , \theta , e\) and \(g\), the components of the velocity of the ball parallel to and perpendicular to the plane immediately after the collision.
  3. Show that the distance \(A B\) is given by $$4 h e ( e + 1 ) \sin \theta$$
AQA M3 2014 June Q6
12 marks Challenging +1.2
6 Two smooth spheres, \(A\) and \(B\), have equal radii and masses 2 kg and 4 kg respectively. The spheres are moving on a smooth horizontal surface and collide. As they collide, \(A\) has velocity \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the line of centres of the spheres, and \(B\) has velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-16_291_844_607_468} Just after the collision, \(B\) moves in a direction perpendicular to the line of centres.
  1. Find the speed of \(A\) immediately after the collision.
  2. Find the acute angle, correct to the nearest degree, between the velocity of \(A\) and the line of centres immediately after the collision.
  3. Find the coefficient of restitution between the spheres.
  4. Find the magnitude of the impulse exerted on \(B\) during the collision.
AQA M3 2014 June Q7
15 marks Standard +0.3
7 Two small smooth spheres, \(A\) and \(B\), are the same size and have masses \(2 m\) and \(m\) respectively. Initially, the spheres are at rest on a smooth horizontal surface. The sphere \(A\) receives an impulse of magnitude \(J\) and moves with speed \(2 u\) directly towards \(B\).
  1. \(\quad\) Find \(J\) in terms of \(m\) and \(u\).
  2. The sphere \(A\) collides directly with \(B\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\). Find, in terms of \(u\), the speeds of \(A\) and \(B\) immediately after the collision.
  3. At the instant of collision, the centre of \(B\) is at a distance \(s\) from a fixed smooth vertical wall which is at right angles to the direction of motion of \(A\) and \(B\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-20_280_1114_1048_497} Subsequently, \(B\) collides with the wall. The radius of each sphere is \(r\).
    Show that the distance of the centre of \(A\) from the wall at the instant that \(B\) hits the wall is \(\frac { 3 s + 12 r } { 5 }\).
  4. The diagram below shows the positions of \(A\) and \(B\) when \(B\) hits the wall. \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-20_330_1109_1822_493} The sphere \(B\) collides with \(A\) again after rebounding from the wall. The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 5 }\). Find the distance of the centre of \(\boldsymbol { B }\) from the wall at the instant when \(A\) and \(B\) collide again.
    [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{79a08adc-ba78-4afb-96ef-ed595ad373d8-24_2488_1728_219_141}
AQA M3 2015 June Q1
6 marks Standard +0.3
1 A formula for calculating the lift force acting on the wings of an aircraft moving through the air is of the form $$F = k v ^ { \alpha } A ^ { \beta } \rho ^ { \gamma }$$ where \(F\) is the lift force in newtons, \(k\) is a dimensionless constant, \(v\) is the air velocity in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), \(A\) is the surface area of the aircraft's wings in \(\mathrm { m } ^ { 2 }\), and \(\rho\) is the density of the air in \(\mathrm { kg } \mathrm { m } ^ { - 3 }\).
By using dimensional analysis, find the values of the constants \(\alpha , \beta\) and \(\gamma\).
[0pt] [6 marks]
AQA M3 2015 June Q2
5 marks Standard +0.3
2 A projectile is launched from a point \(O\) on top of a cliff with initial velocity \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\alpha\) and moves in a vertical plane. During the motion, the position vector of the projectile relative to the point \(O\) is \(( x \mathbf { i } + y \mathbf { j } )\) metres where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical unit vectors respectively.
  1. Show that, during the motion, the equation of the trajectory of the projectile is given by $$y = x \tan \alpha - \frac { 4.9 x ^ { 2 } } { u ^ { 2 } \cos ^ { 2 } \alpha }$$
  2. When \(u = 21\) and \(\alpha = 55 ^ { \circ }\), the projectile hits a small buoy \(B\). The buoy is at a distance \(s\) metres vertically below \(O\) and at a distance \(s\) metres horizontally from \(O\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-04_601_935_964_548}
    1. Find the value of \(s\).
    2. Find the acute angle between the velocity of the projectile and the horizontal just before the projectile hits \(B\), giving your answer to the nearest degree.
      [0pt] [5 marks]
AQA M3 2015 June Q3
4 marks Moderate -0.3
3 A disc of mass 0.5 kg is moving with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a smooth horizontal surface when it receives a horizontal impulse in a direction perpendicular to its direction of motion. Immediately after the impulse, the disc has speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the magnitude of the impulse received by the disc.
  2. Before the impulse, the disc is moving parallel to a smooth vertical wall, as shown in the diagram. \section*{11/1/1/1/1/1/1/1/1/1/1/1/ Wall} $$\overbrace { 3 \mathrm {~ms} ^ { - 1 } } ^ { \underset { < } { \bigcirc } } \text { Disc }$$ After the impulse, the disc hits the wall and rebounds with speed \(3 \sqrt { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    Find the coefficient of restitution between the disc and the wall.
    [0pt] [4 marks]
AQA M3 2015 June Q4
2 marks Standard +0.3
4 Three uniform smooth spheres, \(A , B\) and \(C\), have equal radii and masses \(m , 2 m\) and \(6 m\) respectively. The spheres lie at rest in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). The sphere \(A\) is projected with speed \(u\) directly towards \(B\) and collides with it. \includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-10_218_1164_500_438} The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).
    1. Show that the speed of \(B\) immediately after the collision is \(\frac { 5 } { 9 } u\).
    2. Find, in terms of \(u\), the speed of \(A\) immediately after the collision.
  2. Subsequently, \(B\) collides with \(C\). The coefficient of restitution between \(B\) and \(C\) is \(e\). Show that \(B\) will collide with \(A\) again if \(e > k\), where \(k\) is a constant to be determined.
  3. Explain why it is not necessary to model the spheres as particles in this question.
    [0pt] [2 marks]
AQA M3 2015 June Q5
11 marks Challenging +1.2
5 Two smooth spheres, \(A\) and \(B\), have equal radii and masses 2 kg and 1 kg respectively. The spheres move on a smooth horizontal surface and collide. As they collide, \(A\) has velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at an angle \(\alpha\) to the line of centres of the spheres, and \(B\) has velocity \(2.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at an angle \(\beta\) to the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-14_458_1068_541_625} The coefficient of restitution between \(A\) and \(B\) is \(\frac { 4 } { 7 }\).
Given that \(\sin \alpha = \frac { 4 } { 5 }\) and \(\sin \beta = \frac { 12 } { 13 }\), find the speeds of \(A\) and \(B\) immediately after the collision.
[0pt] [11 marks]
AQA M3 2015 June Q6
18 marks Standard +0.8
6 A ship and a navy frigate are a distance of 8 km apart, with the frigate on a bearing of \(120 ^ { \circ }\) from the ship, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-16_451_549_411_760} The ship travels due east at a constant speed of \(50 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). The frigate travels at a constant speed of \(35 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
    1. Find the bearings, to the nearest degree, of the two possible directions in which the frigate can travel to intercept the ship.
      [0pt] [5 marks]
    2. Hence find the shorter of the two possible times for the frigate to intercept the ship.
      [0pt] [5 marks]
  2. The captain of the frigate would like the frigate to travel at less than \(35 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Find the minimum speed at which the frigate can travel to intercept the ship.
    [0pt] [3 marks] \(7 \quad\) A particle is projected from a point \(O\) on a plane which is inclined at an angle \(\theta\) to the horizontal. The particle is projected up the plane with velocity \(u\) at an angle \(\alpha\) above the horizontal. The particle strikes the plane for the first time at a point \(A\). The motion of the particle is in a vertical plane which contains the line \(O A\). \includegraphics[max width=\textwidth, alt={}, center]{bcd20c69-cace-408c-8961-169c19ff0231-20_469_624_502_685}
  3. Find, in terms of \(u , \theta , \alpha\) and \(g\), the time taken by the particle to travel from \(O\) to \(A\).
  4. The particle is moving horizontally when it strikes the plane at \(A\). By using the identity \(\sin ( P - Q ) = \sin P \cos Q - \cos P \sin Q\), or otherwise, show that $$\tan \alpha = k \tan \theta$$ where \(k\) is a constant to be determined.
    [0pt] [5 marks]
    \includegraphics[max width=\textwidth, alt={}]{bcd20c69-cace-408c-8961-169c19ff0231-24_2488_1728_219_141}
AQA M3 2016 June Q1
2 marks Easy -1.2
1 At a firing range, a man holds a gun and fires a bullet horizontally. The bullet is fired with a horizontal velocity of \(400 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The mass of the gun is 1.5 kg and the mass of the bullet is 30 grams.
  1. Find the speed of recoil of the gun.
  2. Find the magnitude of the impulse exerted by the man on the gun in bringing the gun to rest after the bullet is fired.
    [0pt] [2 marks]
AQA M3 2016 June Q2
3 marks Easy -1.2
2 A lunar mapping satellite of mass \(m _ { 1 }\) measured in kg is in an elliptic orbit around the moon, which has mass \(m _ { 2 }\) measured in kg . The effective potential, \(E\), of the satellite is given by $$E = \frac { K ^ { 2 } } { 2 m _ { 1 } r ^ { 2 } } - \frac { G m _ { 1 } m _ { 2 } } { r }$$ where \(r\) measured in metres is the distance of the satellite from the moon, \(G \mathrm { Nm } ^ { 2 } \mathrm {~kg} ^ { - 2 }\) is the universal gravitational constant, and \(K\) is the angular momentum of the satellite. By using dimensional analysis, find the dimensions of:
  1. \(E\),
  2. \(\quad K\).
    [0pt] [3 marks] \(3 \quad\) A ball is projected from a point \(O\) on horizontal ground with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(30 ^ { \circ }\) above the horizontal. The ball travels in a vertical plane through the point \(O\) and hits a point \(Q\) on a plane which is inclined at \(45 ^ { \circ }\) to the horizontal. The point \(O\) is 6 metres from \(P\), the foot of the inclined plane, as shown in the diagram. The points \(O , P\) and \(Q\) lie in the same vertical plane. The line \(P Q\) is a line of greatest slope of the inclined plane. \includegraphics[max width=\textwidth, alt={}, center]{d8c723df-d10a-4fdf-b5ca-ea12633f999a-06_406_1050_568_488}
  3. During its flight, the horizontal and upward vertical distances of the ball from \(O\) are \(x\) metres and \(y\) metres respectively. Show that \(x\) and \(y\) satisfy the equation $$y = x \frac { \sqrt { 3 } } { 3 } - \frac { x ^ { 2 } } { 30 }$$ Use \(\cos 30 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\) and \(\tan 30 ^ { \circ } = \frac { \sqrt { 3 } } { 3 }\).
  4. Find the distance \(P Q\).
AQA M3 2016 June Q4
3 marks Standard +0.3
4 A smooth uniform sphere \(A\), of mass \(m\), is moving with velocity \(8 u\) in a straight line on a smooth horizontal table. A smooth uniform sphere \(B\), of mass \(4 m\), has the same radius as \(A\) and is moving on the table with velocity \(u\). \includegraphics[max width=\textwidth, alt={}, center]{d8c723df-d10a-4fdf-b5ca-ea12633f999a-10_200_1148_456_447} The sphere \(A\) collides directly with the sphere \(B\).
The coefficient of restitution between \(A\) and \(B\) is \(e\).
    1. Find, in terms of \(u\) and \(e\), the velocities of \(A\) and \(B\) immediately after the collision.
    2. The direction of motion of \(A\) is reversed by the collision. Show that \(e > a\), where \(a\) is a constant to be determined.
  2. Subsequently, \(B\) collides with a fixed smooth vertical wall which is at right angles to the direction of motion of \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 3 }\). The sphere \(B\) collides with \(A\) again after rebounding from the wall.
    Show that \(e < b\), where \(b\) is a constant to be determined.
  3. Given that \(e = \frac { 4 } { 7 }\), find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) by the wall.
    [0pt] [3 marks]
AQA M3 2016 June Q5
11 marks Challenging +1.2
5 A ball is projected from a point \(O\) above a smooth plane which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The point \(O\) is at a perpendicular distance of 1 m from the inclined plane. The ball is projected with velocity \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(70 ^ { \circ }\) above the horizontal. The motion of the ball is in a vertical plane containing a line of greatest slope of the inclined plane. The ball strikes the inclined plane for the first time at a point \(A\). \includegraphics[max width=\textwidth, alt={}, center]{d8c723df-d10a-4fdf-b5ca-ea12633f999a-14_478_913_571_561}
    1. Find the time taken by the ball to travel from \(O\) to \(A\).
    2. Find the components of the velocity of the ball, parallel and perpendicular to the inclined plane, as it strikes the plane at \(A\).
  2. After striking \(A\), the ball rebounds and strikes the plane for a second time at a point further up than \(A\). The coefficient of restitution between the ball and the inclined plane is \(e\).
    Show that \(e < k\), where \(k\) is a constant to be determined.
    [0pt] [4 marks] \(6 \quad\) In this question use \(\cos 30 ^ { \circ } = \sin 60 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\).
    A smooth spherical ball, \(A\), is moving with speed \(u\) in a straight line on a smooth horizontal table when it hits an identical ball, \(B\), which is at rest on the table. Just before the collision, the direction of motion of \(A\) is parallel to a fixed smooth vertical wall. At the instant of collision, the line of centres of \(A\) and \(B\) makes an angle of \(60 ^ { \circ }\) with the wall, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{d8c723df-d10a-4fdf-b5ca-ea12633f999a-18_499_1036_721_593} The coefficient of restitution between \(A\) and \(B\) is \(e\).
  3. Show that the speed of \(B\) immediately after the collision is \(\frac { 1 } { 4 } u ( 1 + e )\) and find, in terms of \(u\) and \(e\), the components of the velocity of \(A\), parallel and perpendicular to the line of centres, immediately after the collision.
  4. Subsequently, \(B\) collides with the wall. After colliding with the wall, the direction of motion of \(B\) is parallel to the direction of motion of \(A\) after its collision with \(B\). Show that the coefficient of restitution between \(B\) and the wall is \(\frac { 1 + e } { 7 - e }\).
    [0pt] [7 marks]
AQA M3 2016 June Q7
5 marks Challenging +1.8
7 A quad-bike, a truck and a car are moving on a large, open, horizontal surface in a desert plain. Relative to the quad-bike, which is travelling due west at its maximum speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the truck is moving on a bearing of \(340 ^ { \circ }\). Relative to the car, which is travelling due east at a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the truck is moving on a bearing of \(300 ^ { \circ }\).
  1. Show that the speed of the truck is approximately \(24.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and that it is moving on a bearing of \(318 ^ { \circ }\), correct to the nearest degree.
  2. At the instant when the truck is at a distance of 400 metres from the quad-bike, the bearing of the truck from the quad-bike is \(060 ^ { \circ }\). The truck continues to move with the same velocity as in part (a). The quad-bike continues to move at a speed of \(10 \mathrm {~ms} ^ { - 1 }\). Find the bearing, to the nearest degree, on which the quad-bike should travel in order to approach the truck as closely as possible.
    [0pt] [5 marks]
    \includegraphics[max width=\textwidth, alt={}]{d8c723df-d10a-4fdf-b5ca-ea12633f999a-24_2032_1707_219_153}
Edexcel M3 Q1
7 marks Moderate -0.3
  1. A bird of mass 0.5 kg , flying around a vertical feeding post at a constant speed of \(6 \mathrm {~ms} ^ { - 1 }\), banks its wings to move in a horizontal circle of radius 2 m . The aerodynamic lift \(L\) newtons is perpendicular to the bird's wings, as shown.
    Modelling the bird as a particle, find, to the nearest degree, the \includegraphics[max width=\textwidth, alt={}, center]{430c3b75-57aa-42ff-867e-304b85e7d521-1_303_472_349_1505}
    angle that its wings make with the vertical.
  2. A thin elastic string, of modulus \(\lambda \mathrm { N }\) and natural length 20 cm , passes round two small, smooth pegs \(A\) and \(B\) on the same horizontal level to form a closed loop. \(A B = 10 \mathrm {~cm}\). The ends of the string are attached to a weight \(P\) of mass 0.7 kg .
    When \(P\) rests in equilibrium, \(A P B\) forms an equilateral triangle. \includegraphics[max width=\textwidth, alt={}, center]{430c3b75-57aa-42ff-867e-304b85e7d521-1_346_371_836_1560}
    1. Find the value of \(\lambda\).
    2. State one assumption that you have made about the weight \(P\), explaining how you have used this assumption in your solution.
    3. A particle \(P\) of mass 0.5 kg moves along a straight line. When \(P\) is at a distance \(x \mathrm {~m}\) from a fixed point \(O\) on the line, the force acting on it is directed towards \(O\) and has magnitude \(\frac { 8 } { x ^ { 2 } } \mathrm {~N}\). When \(x = 2\), the speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\).
      Find the speed of \(P\) when it is 0.5 m from \(O\).
    4. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\). The other end of the string is attached to a fixed point \(O . P\) is released from rest at \(O\) and falls vertically downwards under gravity. The greatest distance below \(O\) reached by \(P\) is \(2 l \mathrm {~m}\).
    5. Show that \(\lambda = 4 m g\).
    6. Find, in terms of \(l\) and \(g\), the speed with which \(P\) passes through the point \(A\), where
    $$O A = \frac { 5 l } { 4 } \mathrm {~m} .$$ (6 marks) \section*{MECHANICS 3 (A) TEST PAPER 1 Page 2}
Edexcel M3 Q5
12 marks Challenging +1.2
5.
\includegraphics[max width=\textwidth, alt={}]{430c3b75-57aa-42ff-867e-304b85e7d521-2_389_412_265_386}
A uniform solid right circular cone has height \(h\) and base radius \(r\). The top part of the cone is removed by cutting through the cone parallel to the base at a height \(\frac { h } { 2 }\).
  1. Show that the centre of mass of the remaining solid is at a height \(\frac { 11 h } { 56 }\) above the base, along its axis of symmetry. The remaining part of the solid is suspended from the point \(D\) on the circumference of its smaller circular face, and the axis of symmetry then makes an angle \(\alpha\) with the vertical, where \(\tan \alpha = \frac { 1 } { 2 }\).
  2. Find the value of the ratio \(h : r\).
Edexcel M3 Q6
15 marks Challenging +1.2
6. A light elastic string, of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\frac { m g } { 2 }\) newtons, has one end fastened to a fixed point \(O\). A particle \(P\), of mass \(m \mathrm {~kg}\), is attached to the other end of the string. \(P\) hangs in equilibrium at the point \(E\), vertically below \(O\), where \(O E = ( l + e ) \mathrm { m }\)
  1. Find the numerical value of the ratio \(e : l\). \(P\) is now pulled down a further distance \(\frac { 3 l } { 2 } \mathrm {~m}\) from \(E\) and is released from rest.
    In the subsequent motion, the string remains taut. At time \(t \mathrm {~s}\) after being released, \(P\) is at a distance \(x \mathrm {~m}\) below \(E\).
  2. Write down a differential equation for the motion of \(P\) and show that the motion is simple harmonic.
  3. Write down the period of the motion.
  4. Find the speed with which \(P\) first passes through \(E\) again.
  5. Show that the time taken by \(P\) after it is released to reach the point \(A\) above \(E\), where $$A E = \frac { 3 l } { 4 } \mathrm {~m} , \text { is } \frac { 2 \pi } { 3 } \sqrt { \frac { 2 l } { g } } \mathrm {~s} .$$
Edexcel M3 Q7
17 marks Challenging +1.2
  1. A particle \(P\) is attached to one end of a light inextensible string of length \(l \mathrm {~m}\). The other end of the string is attached to a fixed point \(O\). When \(P\) is hanging at rest vertically below \(O\), it is given a horizontal speed \(u \mathrm {~ms} ^ { - 1 }\) and starts to move in a vertical circle.
Given that the string becomes slack when it makes an angle of \(120 ^ { \circ }\) with the downward vertical through \(O\),
  1. show that \(u ^ { 2 } = \frac { 7 g l } { 2 }\).
  2. Find, in terms of \(l\), the greatest height above \(O\) reached by \(P\) in the subsequent motion.
    (7 marks)
Edexcel M3 Q1
7 marks Standard +0.3
  1. A particle of mass \(m \mathrm {~kg}\) moves in a horizontal straight line. Its initial speed is \(u \mathrm {~ms} ^ { - 1 }\) and the only force acting on it is a variable resistance of magnitude \(m k v \mathrm {~N}\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the particle after \(t\) seconds and \(k\) is a constant.
    Show that \(v = u e ^ { - k t }\).
  2. A particle \(P\) of mass \(m \mathrm {~kg}\) moves in a horizontal circle at one end of a light inextensible string of length 40 cm , as shown. The other end of the string is attached to a fixed point \(O\).
    The angular velocity of \(P\) is \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\).
    If the angle \(\theta\) which the string makes with the vertical must not \includegraphics[max width=\textwidth, alt={}, center]{e756c147-2711-4ee4-9ec6-7680fe84a0c5-1_314_401_722_1576}
    (7 marks)
    exceed \(60 ^ { \circ }\), calculate the greatest possible value of \(\omega\).
  3. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 0.5 m and modulus of elasticity \(\frac { m g } { 2 } \mathrm {~N}\). The other end of the string is attached to a fixed point \(O\) and \(P\) hangs vertically below \(O\).
    1. Find the stretched length of the string when \(P\) rests in equilibrium.
    2. Find the elastic potential energy stored in the string in the equilibrium position. \(P\), which is still attached to the string, is now held at rest at \(O\) and then lowered gently into its equilibrium position.
    3. Find the work done by the weight of the particle as it moves from \(O\) to the equilibrium position.
    4. Explain the discrepancy between your answers to parts (b) and (c).
    5. A particle \(P\), of mass \(m \mathrm {~kg}\), is attached to two light elastic strings, each of natural length \(l \mathrm {~m}\) and modulus of elasticity 3 mg N . The other ends of the strings are attached to the fixed points \(A\) and \(B\), where \(A B\) is horizontal and \(A B = 2 l \mathrm {~m}\). If \(P\) rests in equilibrium vertically below the mid-point of \(A B\), with each string making an angle \(\theta\) with the vertical, show that \includegraphics[max width=\textwidth, alt={}, center]{e756c147-2711-4ee4-9ec6-7680fe84a0c5-1_410_474_2052_1510}
    $$\cot \theta - \cos \theta = \frac { 1 } { 6 } .$$ \section*{MECHANICS 3 (A) TEST PAPER 2 Page 2}
Edexcel M3 Q5
18 marks Standard +0.3
  1. A small bead \(P\), of mass \(m \mathrm {~kg}\), can slide on a smooth circular ring, with centre \(O\) and radius \(r \mathrm {~m}\), which is fixed in a vertical plane. \(P\) is projected from the lowest point \(L\) of the ring with speed \(\sqrt { } ( 3 g r ) \mathrm { ms } ^ { - 1 }\). When \(P\) has reached a position such that \(O P\) makes an angle \(\theta\) with the downward vertical, as shown, its speed is \(v \mathrm {~ms} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{e756c147-2711-4ee4-9ec6-7680fe84a0c5-2_355_337_262_1590}
    1. Show that \(v ^ { 2 } = g r ( 1 + 2 \cos \theta )\).
    2. Show that the magnitude of the reaction \(R N\) of the ring on the bead is given by
    $$R = m g ( 1 + 3 \cos \theta ) .$$
  2. Find the values of \(\cos \theta\) when
    1. \(P\) is instantaneously at rest, (ii) the reaction \(R\) is instantaneously zero.
  3. Hence show that the ratio of the heights of \(P\) above \(L\) in cases (i) and (ii) is \(9 : 8\).
Edexcel M3 Q6
18 marks Standard +0.8
6. A light elastic string, of natural length 0.8 m , has one end fastened to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass 0.5 kg . When \(P\) hangs in equilibrium, the length of the string is 1.5 m .
  1. Calculate the modulus of elasticity of the string. \(P\) is displaced to a point 0.5 m vertically below its equilibrium position and released from rest.
  2. Show that the subsequent motion of \(P\) is simple harmonic, with period 1.68 s .
  3. Calculate the maximum speed of \(P\) during its motion.
  4. Show that the time taken for \(P\) to first reach a distance 0.25 m from the point of release is 0.28 s , to 2 significant figures.
Edexcel M3 Q7
16 marks Standard +0.8
7. (a) Show that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance \(\frac { 3 r } { 8 }\) from the centre \(O\) of the plane face. The figure shows the vertical cross-section of a rough solid hemisphere at rest on a rough inclined plane inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 10 }\).
(b) Indicate on a copy of the figure the three forces acting on the hemisphere, clearly stating what they are, and paying \includegraphics[max width=\textwidth, alt={}, center]{e756c147-2711-4ee4-9ec6-7680fe84a0c5-2_356_520_2042_1457}
(c) Given that the plane face containing the diameter \(A B\) makes an angle \(\alpha\) with the vertical, show that \(\cos \alpha = \frac { 4 } { 5 }\).
Edexcel M3 Q1
7 marks Standard +0.8
  1. One end of a light inextensible string of length \(2 r \mathrm {~m}\) is attached to a fixed point \(O\). A particle of mass \(m \mathrm {~kg}\) is attached to the other end \(Q\) of the string, so that it can move in a vertical plane. The string is held taut and horizontal and the particle is projected vertically downwards with a speed \(\sqrt { } ( g r ) \mathrm { ms } ^ { - 1 }\). When the string is vertical it begins to wrap round a small, smooth peg \(X\) at a distance \(r \mathrm {~m}\) vertically below \(O\). The particle continues to move.
    1. Find the speed of the particle when it reaches \(O\), in terms of \(g\) and \(r\).
    2. Show that, when \(Q X\) is horizontal, the tension in the string is 3 mgN .
    3. A particle moving along the \(x\)-axis describes simple harmonic motion about the origin \(O\). The period of its motion is \(\frac { \pi } { 2 }\) seconds. When it is at a distance 1 m from \(O\), its speed is \(3 \mathrm {~ms} ^ { - 1 }\). Calculate
    4. the amplitude of its motion,
    5. the maximum acceleration of the particle,
    6. the least time that it takes to move from \(O\) to a point 0.25 m from \(O\).
    7. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of a light elastic string of natural length \(8 l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\). The two ends of the string are attached to fixed points \(A\) and \(B\) on the same horizontal level, where \(A B = 81 \mathrm {~m} . P\) is released from rest at the mid-point of \(A B\).
    8. If \(P\) comes to instantaneous rest at a depth \(3 / \mathrm { m }\) below \(A B\), find an expression for \(\lambda\) in terms of \(m\) and \(g\).
    9. Using this value of \(\lambda\), show that the speed \(v \mathrm {~ms} ^ { - 1 }\) of \(P\) when it passes through the point \(2 l \mathrm {~m}\) below \(A B\) is given by \(v ^ { 2 } = 4 ( 24 \sqrt { 5 } - 53 ) g l\).
    10. A particle \(P\) of mass 0.8 kg moves along a straight line \(O L\) and is acted on by a resistive force of magnitude \(R \mathrm {~N}\) directed towards the fixed point \(O\). When the displacement of \(P\) from \(O\) is \(x \mathrm {~m} , R = \frac { 0 \cdot 8 x v ^ { 2 } } { 1 + x ^ { 2 } }\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of \(P\) at that instant.
    11. Write down a differential equation for the motion of \(P\).
    Given that \(v = 2\) when \(x = 0\),
  2. find the speed with which \(P\) passes through the point \(A\), where \(O A = 1 \mathrm {~m}\). \section*{MECHANICS 3 (A) TEST PAPER 3 Page 2}