Questions — Edexcel M2 (623 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 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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 M2 Q1
4 marks Standard +0.2
  1. A car of mass 1200 kg decelerates from \(30 \mathrm {~ms} ^ { - 1 }\) to \(20 \mathrm {~ms} ^ { - 1 }\) in 6 seconds at a constant rate.
    1. Find the magnitude, in N , of the decelerating force.
    2. Find the loss, in J , in the car's kinetic energy.
    3. A particle moves in a straight line from \(A\) to \(B\) in 5 seconds. At time \(t\) seconds after leaving \(A\), the velocity of the particle is \(\left( 32 t - 3 t ^ { 2 } \right) \mathrm { ms } ^ { - 1 }\).
    4. Calculate the straight-line distance \(A B\).
    5. Find the acceleration of the particle when \(t = 3\).
    6. Eddie, whose mass is 71 kg , rides a bicycle of mass 25 kg up a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\). When Eddie is working at a rate of 600 W , he is moving at a constant speed of \(6 \mathrm {~ms} ^ { - 1 }\).
      Find the magnitude of the non-gravitational resistance to his motion.
    7. A boat leaves the point \(O\) and moves such that, \(t\) seconds later, its position vector relative to \(O\) is \(\left( t ^ { 2 } - 2 \right) \mathbf { i } + 2 t \mathbf { j }\), where the vectors \(\mathbf { i }\) and \(\mathbf { j }\) both have magnitude 1 metre and are directed parallel and perpendicular to the shoreline through \(O\).
    8. Find the speed with which the boat leaves \(O\).
    9. Show that the boat has constant acceleration and state the magnitude of this acceleration.
    10. Find the value of \(t\) when the boat is 40 m from \(O\).
    11. Comment on the limitations of the given model of the boat's motion.
    \includegraphics[max width=\textwidth, alt={}]{996976f3-2a97-4c68-8c97-f15a3bfde9a2-1_446_595_1965_349}
    The diagram shows a body which may be modelled as a uniform lamina. The body is suspended from the point marked \(A\) and rests in equilibrium.
  2. Calculate, to the nearest degree, the angle which the edge \(A B\) then makes with the vertical.
    (8 marks) Frank suggests that the angle between \(A B\) and the vertical would be smaller if the lamina were made from lighter material.
  3. State, with a brief explanation, whether Frank is correct.
    (2 marks) \section*{MECHANICS 2 (A) TEST PAPER 1 Page 2}
Edexcel M2 Q6
10 marks Standard +0.3
  1. A uniform rod \(A B\), of mass 0.8 kg and length \(10 a\), is supported at the end \(A\) by a light inextensible vertical string and rests in limiting equilibrium on a rough fixed peg at \(C\), where \(A C = 7 a\). \includegraphics[max width=\textwidth, alt={}, center]{996976f3-2a97-4c68-8c97-f15a3bfde9a2-2_319_638_228_1293}
  2. Two particles \(A\) and \(B\), of mass \(m\) and \(k m\) respectively, are moving in the same direction on a smooth horizontal surface. \(A\) has speed \(4 u\) and \(B\) has speed \(u\). The coefficient of restitution between \(A\) and \(B\) is \(e \quad A\) collides directly with \(B\), and in the collision the direction of \(A\) 's motion is reversed. Immediately after the impact, \(B\) has speed \(2 u\).
    1. Show that the speed of \(A\) immediately after the impact is \(u ( 3 e - 2 )\).
    2. Deduce the range of possible values of \(e\).
    3. Show that \(4 < k \leq 5\).
    4. A ball is projected from ground level with speed \(34 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 8 } { 15 }\).
    5. Find the greatest height reached by the ball above ground level.
    While it is descending, the ball hits a horizontal ledge 6 metres above ground level.
  3. Find the horizontal distance travelled by the ball before it hits the ledge.
  4. Find the speed of the ball at the instant when it hits the ledge.
Edexcel M2 Q1
6 marks Moderate -0.8
  1. A ship, of mass 5000 tonnes, is moving through the sea at a constant speed of \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
    1. Calculate the momentum of the ship, in the form \(a \times 10 ^ { n }\), where \(0 \leq a < 10\) and \(n\) is an integer. State the units of your answer.
    Given that there is a constant force of magnitude 4000 N acting against the ship due to air and water resistances,
  2. find the rate, in kW , at which the ship's engines are working.
Edexcel M2 Q2
7 marks Standard +0.8
2. Two small smooth spheres \(P\) and \(Q\) are moving along a straight line in opposite directions, with equal speeds, and collide directly. Immediately after the impact, the direction of \(P\) 's motion has been reversed and its speed has been halved. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
  1. Express the speed of \(Q\) after the impact in the form \(a u ( b e + c )\), where \(a , b\) and \(c\) are constants to be found.
  2. Deduce the range of values of \(e\) for which the direction of motion of \(Q\) remains unaltered.
Edexcel M2 Q3
8 marks Standard +0.3
3. \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane. At a certain instant, a particle \(P\) of mass 1.8 kg is moving with velocity \(( 24 \mathrm { i } - 7 \mathrm { j } ) \mathrm { ms } ^ { - 1 }\).
  1. Calculate the kinetic energy of \(P\) at this instant. \(P\) is now subjected to a constant retardation. After 10 seconds, the velocity of \(P\) is \(( - 12 \mathbf { i } + 3 \cdot 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
  2. Calculate the work done by the retarding force over the 10 seconds.
Edexcel M2 Q4
9 marks Standard +0.3
4. A small block of wood, of mass 0.5 kg , slides down a line of \includegraphics[max width=\textwidth, alt={}, center]{3c084e42-d304-4b77-afee-7e4bd801a03c-1_219_501_2042_338}
greatest slope of a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 2 } { 5 }\). The block is given an initial impulse of magnitude 2 Ns , and reaches the bottom of the plane with kinetic energy 19 J.
  1. Find, in J , the change in the potential energy of the block as it moves down the plane.
  2. Hence find the distance travelled by the block down the plane.
  3. State two modelling assumptions that you have made. \section*{MECHANICS 2 (A) TEST PAPER 6 Page 2}
Edexcel M2 Q5
9 marks Standard +0.3
5.
\includegraphics[max width=\textwidth, alt={}]{3c084e42-d304-4b77-afee-7e4bd801a03c-2_278_483_246_386}
A uniform rod \(X Y\), of length \(2 a\) and mass \(m\), is connected to a vertical wall by a smooth hinge at the end \(X\). A horizontal light inelastic string connects the mid-point of \(X Y\) to the wall and the rod is in equilibrium in this position.
  1. Draw a diagram to show all the forces acting on the rod. Given that the tension in the horizontal string is of magnitude \(2 m g\),
  2. find the angle which \(X Y\) makes with the vertical.
Edexcel M2 Q6
10 marks Standard +0.3
6. \includegraphics[max width=\textwidth, alt={}, center]{3c084e42-d304-4b77-afee-7e4bd801a03c-2_424_492_813_379} The diagram shows a uniform lamina \(A B C D E F\).
  1. Calculate the distance of the centre of mass of the lamina from (i) \(A F\), (ii) \(A B\). The lamina is hung over a smooth peg at \(D\) and rests in equilibrium in a vertical plane.
  2. Find the angle between \(C D\) and the vertical.
Edexcel M2 Q7
11 marks Moderate -0.3
7. A particle \(P\) moves in a straight line so that its displacement \(s\) metres from a fixed point \(O\) at time \(t\) seconds is given by the formula \(s = t ^ { 3 } - 7 t ^ { 2 } + 8 t\).
  1. Find the values of \(t\) when the velocity of \(P\) equals zero, and briefly describe what is happening to \(P\) at these times.
  2. Find the distance travelled by \(P\) between the times \(t = 3\) and \(t = 5\).
  3. Find the value of \(t\) when the acceleration of \(P\) is \(- 2 \mathrm {~ms} ^ { - 2 }\). Briefly explain the significance of a negative acceleration at this time.
Edexcel M2 Q8
15 marks Standard +0.3
8. A particle \(P\) is projected from a point \(O\) with initial velocity \(( 3 \cdot 5 \mathbf { i } + 12 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and moves under gravity. \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the horizontal and vertical directions respectively.
  1. Find the initial speed of \(P\).
  2. Show that the position vector \(\mathbf { r } \mathbf { m }\) of \(P\) at time \(t\) seconds after projection is given by $$\mathbf { r } = 3 \cdot 5 t \mathbf { i } + \left( 12 t - 4 \cdot 9 t ^ { 2 } \right) \mathbf { j } .$$
  3. Find the horizontal distance of \(P\) from \(O\) at each of the times when it is 4.4 m vertically above the level of \(O\). In a refined model of the motion of \(P\), the position vector of \(P\) at time \(t\) seconds is taken to be $$\mathbf { r } = 3 \cdot 5 t \mathbf { i } + \left( 12 t - t ^ { 3 } \right) \mathbf { j } \mathbf { ~ m } .$$
  4. Using this model, find the position vector of the highest point reached by \(P\).
Edexcel M2 Q1
5 marks Moderate -0.3
  1. A snooker ball \(A\) is moving on a horizontal table with velocity \(( 5 \mathbf { i } + 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
It collides with another ball \(B\), whose mass is twice the mass of \(A\).
After the collision, \(A\) has velocity \(( - 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and \(B\) has velocity \(( \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
Find the velocity of \(B\) before the collision.
Edexcel M2 Q2
6 marks Standard +0.3
2. Charlotte, whose mass is 55 kg , is running up a straight hill inclined at \(6 ^ { \circ }\) to the horizontal. She passes two points \(P\) and \(Q , 80\) metres apart, with speeds \(2 \cdot 5 \mathrm {~ms} ^ { - 1 }\) and \(1 \cdot 5 \mathrm {~ms} ^ { - 1 }\) respectively.
Calculate, in J to the nearest whole number, the total work done by Charlotte as she runs from \(P\) to \(Q\).
Edexcel M2 Q3
7 marks Moderate -0.8
3. A particle \(P\) moves in a horizontal plane such that, at time \(t\) seconds, its velocity is \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where \(\mathbf { v } = 2 t \mathbf { i } - t ^ { \frac { 1 } { 2 } } \mathbf { j }\). When \(t = 0 , P\) is at the point with position vector \(- 10 \mathbf { i } + \mathbf { j }\) relative to a fixed origin \(O\).
  1. Find the position vector \(\mathbf { r }\) of \(P\) at time \(t\) seconds.
  2. Find the distance \(O P\) when \(t = 4\).
Edexcel M2 Q4
7 marks Standard +0.8
4. A small stone, of mass 600 grams, is released from rest a height of 2 metres above ground level and falls under gravity. The time it takes to reach the ground is \(T\) seconds. The stone is then again released from rest at the surface of a tank containing a 2 metre depth of liquid and reaches the bottom after \(2 T\) seconds. It may be assumed that the resisting force acting on the stone is constant.
  1. Find the magnitude of the resisting force exerted on the stone by the liquid.
  2. Find the speed with which the stone hits the bottom of the tank.
Edexcel M2 Q5
7 marks Standard +0.3
5. \includegraphics[max width=\textwidth, alt={}, center]{9e1d8a2f-0c35-4398-98ff-083ec76653ec-1_367_529_2122_383} A sign-board consists of a rectangular sheet of metal, of mass \(M\), which is 3 metres wide and 1 metre high, attached to two thin metal supports, each of mass \(m\) and length 2 metres. The board stands on horizontal ground.
  1. Calculate the height above the ground of the centre of mass of the sign-board, in terms of \(M\) and \(m\). Given now that the centre of mass of the sign-board is \(2 \cdot 2\) metres above the ground, (b) find the ratio \(M : m\), in its simplest form. \section*{MECHANICS 2 (A) TEST PAPER 9 Page 2}
Edexcel M2 Q6
12 marks Standard +0.3
  1. A ball is hit with initial speed \(u \mathrm {~ms} ^ { - 1 }\), at an angle \(\theta\) above the horizontal, from a point at a height of \(h \mathrm {~m}\) above horizontal ground. The ball, which is modelled as a particle moving freely under gravity, hits the ground at a horizontal distance \(d \mathrm {~m}\) from the point of projection.
    1. Prove that \(\frac { g d ^ { 2 } } { 2 u ^ { 2 } } \sec ^ { 2 } \theta - d \tan \theta - h = 0\).
    Given further that \(u = 14 , h = 7\) and \(d = 14\), and assuming the result \(\sec ^ { 2 } \theta = 1 + \tan ^ { 2 } \theta\),
  2. find the value of \(\theta\).
Edexcel M2 Q7
14 marks Standard +0.3
7. A cyclist is pedalling along a horizontal cycle track at a constant speed of \(5 \mathrm {~ms} ^ { - 1 }\). The air resistance opposing her motion has magnitude 42 N . The combined mass of the cyclist and her machine is 84 kg .
  1. Find the rate, in W , at which the cyclist is working. The cyclist now starts to ascend a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 21 }\), at a constant speed.
    She continues to work at the same rate as before, against the same air resistance.
  2. Find the constant speed at which she ascends the hill. In fact the air resistance is not constant, and a revised model takes account of this by assuming that the air resistance is proportional to the cyclist's speed.
  3. Use this model to find an improved estimate of the speed at which she ascends the hill, if her rate of working still remains constant.
Edexcel M2 Q8
17 marks Standard +0.3
8. Two ships \(A\) and \(B\), of masses \(m\) and km respectively, are moving towards each other in heavy fog along the same straight line, both with speed \(u\). The ships collide and immediately after the collision they drift away from each other, both their directions of motion having been reversed. The speed of \(A\) after the impact is \(\frac { 1 } { 5 } u\) and the speed of \(B\) after the impact is \(v\).
  1. Show that \(v = u \left( \frac { 6 } { 5 k } - 1 \right)\). The coefficient of restitution between \(A\) and \(B\) is \(e\).
  2. Show that \(v = u \left( 2 e - \frac { 1 } { 5 } \right)\).
  3. Use your answers to parts (a) and (b) to find the rational numbers \(p\) and \(q\) such that \(p \leq k < q\).
    (9 marks)
Edexcel M2 Q1
4 marks Moderate -0.8
  1. Particles of mass \(2 m , 3 m\) and \(5 m\) are placed at the points in the \(x - y\) plane with coordinates \(( - 1,5 ) , ( 0,6 )\) and \(( 3 , - 2 )\) respectively.
    Find the coordinates of the centre of mass of this system of particles.
  2. A lorry of mass 3800 kg is pulling a trailer of mass 1200 kg along a straight horizontal road. At a particular moment, the lorry and trailer are moving at a speed of \(10 \mathrm {~ms} ^ { - 1 }\) and accelerating at \(0.8 \mathrm {~ms} ^ { - 2 }\). The resistances to the motion of the lorry and the trailer are constant and of magnitude 1600 N and 600 N respectively.
    Find the rate, in kW , at which the engine of the lorry is working.
  3. A bullet of mass 0.05 kg is fired with speed \(u \mathrm {~ms} ^ { - 1 }\) from a gun, which recoils at a speed of \(0.008 u \mathrm {~ms} ^ { - 1 }\) in the opposite direction to that in which the bullet is fired.
    1. Find the mass of the gun.
    2. Find, in terms of \(u\), the kinetic energy given to the bullet and to the gun at the instant of firing.
    3. If the total kinetic energy created in firing the gun is 5100 J , find the value of \(u\).
    4. The acceleration of a particle \(P\) at time \(t \mathrm {~s}\) is \(\mathbf { a } \mathrm { ms } ^ { - 2 }\), where \(\mathbf { a } = 4 \mathrm { e } ^ { t } \mathbf { i } - \mathrm { e } ^ { t } \mathbf { j }\). When \(t = 0 , P\) has velocity \(4 \mathrm { i } \mathrm { ms } ^ { - 1 }\).
    5. Find the speed of \(P\) when \(t = 2\).
    6. Find the time at which the direction of motion of \(P\) is parallel to the vector \(5 \mathbf { i } - \mathbf { j }\).
    \includegraphics[max width=\textwidth, alt={}]{63133ab4-9381-4777-a575-1207219948b7-1_323_383_1992_429}
    A uniform plank \(A B\), of mass 3 kg and length 2 m , rests in equilibrium with the point \(P\) in contact with a smooth cylinder. The end \(B\) rests on a rough horizontal surface and the coefficient of friction between the plank and the surface is \(\frac { 1 } { 3 } . A B\) makes an angle of \(60 ^ { \circ }\) with the horizontal.
    If the plank is in limiting equilibrium in this position, find
  4. the magnitude of the force exerted by the cylinder on the plank at \(P\),
  5. the distance \(A P\). \section*{MECHANICS 2 (A) TEST PAPER 10 Page 2}
Edexcel M2 Q6
11 marks Moderate -0.3
  1. Two smooth spheres \(A\) and \(B\) have equal radii and masses 0.4 kg and 0.8 kg respectively. They are moving in opposite directions along the same straight line, with speeds \(3 \mathrm {~ms} ^ { - 1 }\) and 2 \(\mathrm { ms } ^ { - 1 }\) respectively, and collide directly. The coefficient of restitution between \(A\) and \(B\) is 0.8 .
    1. Calculate the speeds of \(A\) and \(B\) after the impact, stating in each case whether the direction of motion has been reversed.
    2. Find the kinetic energy, in J, lost in the impact.
    3. A point of light, \(P\), is moving along a straight line in such a way that, \(t\) seconds after passing through a fixed point \(O\) on the line, its velocity is \(v \mathrm {~ms} ^ { - 1 }\), where \(v = \frac { 1 } { 2 } t ^ { 2 } - 4 t + 10\). Calculate
    4. the velocity of \(P 6\) seconds after it passes \(O\),
    5. the magnitude of the acceleration of \(P\) when \(t = 1\),
    6. the minimum speed of \(P\),
    7. the distance travelled by \(P\) in the fourth second after it passes \(O\).
    8. A bullet is fired out of a window at a height of 5.2 m above horizontal ground. The initial velocity of the bullet is \(392 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the vertical, where \(\sin \alpha = \frac { 1 } { 20 }\), as shown.
      Find \includegraphics[max width=\textwidth, alt={}, center]{63133ab4-9381-4777-a575-1207219948b7-2_335_490_1343_1419}
    9. the range of times after firing during which the bullet is 15 m or more above ground level,
    10. the greatest height above the ground reached by the bullet,
    11. the horizontal distance travelled by the bullet before it reaches its highest point.
    Certain modelling assumptions have been made about the bullet.
  2. State these assumptions and suggest a way in which the model could be refined.
  3. State, with a reason, whether you think this refinement would make a significant difference to the answers.
    (2 marks)
Edexcel M2 Q1
6 marks Moderate -0.3
  1. Two identical particles are approaching each other along a straight horizontal track. Just before they collide, they are moving with speeds \(5 \mathrm {~ms} ^ { - 1 }\) and \(3 \mathrm {~ms} ^ { - 1 }\) respectively. The coefficient of restitution between the particles is \(\frac { 1 } { 2 }\).
Find the speeds of the particles immediately after the impact.
Edexcel M2 Q2
8 marks Standard +0.3
2. A particle \(P\) of mass 3 kg moves such that at time \(t\) seconds its position vector, \(\mathbf { r }\) metres, relative to a fixed origin, \(O\), is given by $$\mathbf { r } = \left( t ^ { 2 } - 3 t \right) \mathbf { i } + \frac { 1 } { 6 } t ^ { 3 } \mathbf { j }$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.
  1. Find the velocity of \(P\) when \(t = 0\).
  2. Find the kinetic energy lost by \(P\) in the interval \(0 \leq t \leq 2\).
Edexcel M2 Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0e751be-f095-4a56-8ee9-8433cc4873e9-2_424_360_1155_648} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a uniform ladder of mass 15 kg and length 8 m which rests against a smooth vertical wall at \(A\) with its lower end on rough horizontal ground at \(B\). The coefficient of friction between the ladder and the ground is \(\frac { 1 } { 3 }\) and the ladder is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = 2\). A man of mass 75 kg ascends the ladder until he reaches a point \(P\). The ladder is then on the point of slipping.
  1. Write down suitable models for
    1. the ladder,
    2. the man.
  2. Find the distance \(A P\).
Edexcel M2 Q4
11 marks Standard +0.3
4. A particle \(P\) moves in a straight horizontal line such that its acceleration at time \(t\) seconds is proportional to \(\left( 3 t ^ { 2 } - 5 \right)\). Given that at time \(t = 0 , P\) is at rest at the origin \(O\) and that at time \(t = 3\), its velocity is \(3 \mathrm {~ms} ^ { - 1 }\),
  1. find, in \(\mathrm { m } \mathrm { s } ^ { - 2 }\), the acceleration of \(P\) in terms of \(t\),
  2. show that the displacement of the particle, \(s\) metres, from \(O\) at time \(t\) is given by $$s = \frac { 1 } { 16 } t ^ { 2 } \left( t ^ { 2 } - 10 \right)$$ (4 marks)
Edexcel M2 Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0e751be-f095-4a56-8ee9-8433cc4873e9-3_591_609_785_623} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows a uniform plane lamina \(A B C D E G\) in the shape of a letter ' \(L\) ' consisting of a rectangle \(A B F G\) joined to another rectangle \(C D E F\). The sides \(A B\) and \(D E\) are both 8 cm long and the sides \(E G\) and \(G A\) are of length 24 cm and 32 cm respectively.
  1. Show that the centre of mass of the lamina lies on the line \(B F\).
  2. Find the distance of the centre of mass from the line \(A B\). The uniform lamina in Figure 2 is a model of the letter ' \(L\) ' in a sign above a shop. The letter is normally suspended from a wall at \(A\) and \(B\) so that \(A B\) is horizontal but the fixing at \(B\) has broken and the letter hangs in equilibrium from the point \(A\).
  3. Find, in degrees to one decimal place, the acute angle \(A G\) makes with the vertical.