Questions — Edexcel (9685 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
Edexcel M1 Q5
11 marks Standard +0.3
5. Two flies \(P\) and \(Q\), are crawling vertically up a wall. At time \(t = 0\), the flies are at the same height above the ground, with \(P\) crawling at a steady speed of \(4 \mathrm { cms } ^ { - 1 }\). \(Q\) starts from rest at time \(t = 0\) and accelerates uniformly to a speed of \(6 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\) in 6 seconds. Fly \(Q\) then maintains this speed.
  1. Find the value of \(t\) when the two flies are moving at the same speed.
  2. Sketch on the same diagram, speed-time graphs to illustrate the motion of the two flies. Given that the distance of the two flies from the top of the wall at time \(t = 0\) is \(x \mathrm {~cm}\) and that \(Q\) reaches the top of the wall first,
  3. show that \(x > 36\).
Edexcel M1 Q6
14 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4fe54579-ac77-46f9-85e1-2e95963d6b3e-4_288_1275_201_410} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows a uniform plank \(A B\) of length 8 m and mass 50 kg suspended horizontally by two light vertical inextensible strings attached at either end of the plank. The maximum tension that either string can support is 40 gN . A rock of mass \(M \mathrm {~kg}\) is placed on the plank at \(A\) and rolled along the plank to \(B\) without either string breaking.
  1. Explain, with the aid of a sketch-graph, how the tension in the string at \(A\) varies with \(x\), the distance of the rock from \(A\).
  2. Show that \(M \leq 15\). The first rock is removed and a second rock of mass 20 kg is placed on the plank.
  3. Find the fraction of the plank on which the rock can be placed without one of the strings breaking.
Edexcel M1 Q7
14 marks Standard +0.3
7. At 6 a.m. a cargo ship has position vector \(( 7 \mathbf { i } + 56 \mathbf { j } ) \mathrm { km }\) relative to a fixed origin \(O\) on the coast and moves with constant velocity \(( 9 \mathbf { i } - 6 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }\). A ferry sails from \(O\) at 6 a.m. and moves with constant velocity \(( 12 \mathbf { i } + 18 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and due north respectively.
  1. Show that the position vector of the cargo ship \(t\) hours after 6 a.m. is given by $$[ ( 7 + 9 t ) \mathbf { i } + ( 56 - 6 t ) \mathbf { j } ] \mathrm { km }$$ and find the position vector of the ferry in terms of \(t\).
  2. Show that if both vessels maintain their course and speed, they will collide and find the time and position vector at which this occurs.
    (6 marks)
    At 8 a.m. the captain of the ferry realises that a collision is imminent and changes course so that the ferry now has velocity \(( 21 \mathbf { i } + 6 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }\).
  3. Find the distance between the two ships at the time when they would have collided.
Edexcel M1 Q1
7 marks Moderate -0.8
  1. In a safety test, a car of mass 800 kg is driven directly at a wall at a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The constant force exerted by the wall on the car in bringing it to rest is 60 kN .
    1. Calculate the magnitude of the impulse exerted by the wall on the car.
    2. Find the time it takes for the car to come to rest.
    3. Show that the deceleration of the car is \(75 \mathrm {~ms} ^ { - 2 }\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2def617f-3c25-4458-8500-8e20ba7c1e53-2_531_661_678_539} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Figure 1 shows an aerial view of a revolving door consisting of 4 panels, each of width 1.2 m and set at \(90 ^ { \circ }\) intervals, which are free to rotate about a fixed central column, \(O\). The revolving door is situated outside a lecture theatre and four students are trying to push the door. Two of the students are pushing panels \(O A\) and \(O D\) clockwise (as viewed from above) with horizontal forces of 70 N and 90 N respectively, whilst the other two are pushing panels \(O B\) and \(O C\) anti-clockwise with horizontal forces of 80 N and 60 N respectively.
  2. Calculate the total moment about \(O\) when the four students are pushing the panels at their outer edge, 1.2 m from \(O\).
    (3 marks)
    The student at \(C\) moves her hand 0.2 m closer to \(O\) and the student at \(D\) moves his hand \(x \mathrm {~m}\) closer to \(O\). Given that the students all push in the same directions and with the same forces as in part (a), and that the door is in equilibrium,
  3. find the value of \(x\).
Edexcel M1 Q3
10 marks Moderate -0.3
3. During a cricket match, the batsman hits the ball and begins running with constant velocity \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to try and score a run. When the batsman is at the fixed origin \(O\), the ball is thrown by a member of the opposing team with velocity \(\left( { } ^ { - } 8 \mathbf { i } + 24 \mathbf { j } \right) \mathrm { ms } ^ { - 1 }\) from the point with position vector \(( 30 \mathbf { i } - 60 \mathbf { j } ) \mathrm { m }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors. At time \(t\) seconds after the ball is thrown, the position vectors of the batsman and the ball are \(\mathbf { r }\) metres and s metres respectively. In a model of the situation, the ball is assumed to travel horizontally and air resistance is considered to be negligible.
  1. Find expressions for \(\mathbf { r }\) and \(\mathbf { s }\) in terms of \(t\).
  2. Show that the ball hits the batsman and find the position vector of the batsman when this occurs.
  3. Write down two reasons why the assumptions used in these calculations are unlikely to provide a realistic model.
    (2 marks)
Edexcel M1 Q4
10 marks Standard +0.3
4. In a physics experiment, two balls \(A\) and \(B\), of mass \(4 m\) and \(3 m\) respectively, are travelling towards one another on a straight horizontal track. Both balls are travelling with speed \(2 \mathrm {~ms} ^ { - 1 }\) immediately before they collide. As a result of the impact, \(A\) is brought to rest and the direction of motion of \(B\) is reversed.
Modelling the track as smooth and the balls as particles,
  1. find the speed of \(B\) immediately after the collision. A student notices that after the collision, \(B\) comes to rest 0.2 m from \(A\).
  2. Show that the coefficient of friction between \(B\) and the track is 0.113 , correct to 3 decimal places.
Edexcel M1 Q5
12 marks Standard +0.3
5. A cyclist is riding up a hill inclined at an angle of \(5 ^ { \circ }\) to the horizontal. She produces a driving force of 50 N and experiences resistive forces which total 20 N . Given that the combined mass of the cyclist and her bicycle is 70 kg ,
  1. find, correct to 2 decimal places, the magnitude of the deceleration of the cyclist. When the cyclist reaches the top of the hill, her speed is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She subsequently accelerates uniformly so that in the fifth second after she has reached the top of the hill, she travels 12 m .
  2. Find her speed at the end of the fifth second.
    (8 marks)
Edexcel M1 Q6
14 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2def617f-3c25-4458-8500-8e20ba7c1e53-4_451_734_964_607} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows a particle \(A\) of mass 5 kg , lying on a smooth horizontal table which is 0.9 m above the floor. A light inextensible string of length 0.7 m connects \(A\) to a particle \(B\) of mass 2 kg . The string passes over a smooth pulley which is fixed to the edge of the table and \(B\) hangs vertically 0.4 m below the pulley. When the system is released from rest,
  1. show that the magnitude of the force exerted on the pulley is \(\frac { 10 \sqrt { 2 } } { 7 } \mathrm {~g} \mathrm {~N}\),
  2. find the speed with which \(A\) hits the pulley. When \(A\) hits the pulley, the string breaks and \(B\) subsequently falls freely under gravity.
  3. Find the speed with which \(B\) hits the ground.
Edexcel M1 Q7
14 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2def617f-3c25-4458-8500-8e20ba7c1e53-5_355_682_237_550} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows a block of mass 25 kg held in equilibrium on a plane inclined at an angle of \(35 ^ { \circ }\) to the horizontal by means of a string which is at an angle of \(15 ^ { \circ }\) to the line of greatest slope of the plane. In an initial model of the situation, the plane is assumed to be smooth. Giving your answers correct to 3 significant figures,
  1. show that the tension in the string is 145 N ,
  2. find the magnitude of the reaction between the plane and the block. In a more refined model, the plane is assumed to be rough.
    Given that the tension in the string can be increased to 200 N before the block begins to move up the slope,
  3. find, correct to 3 significant figures, the magnitude of the frictional force and state the direction in which it acts.
    (4 marks)
  4. Without performing any further calculations, state whether the reaction calculated in part (b) will increase, decrease or remain the same in the refined model. Give a reason for your answer.
Edexcel M1 Q1
7 marks Moderate -0.8
  1. Two particles \(P\) and \(Q\), of mass \(m\) and \(k m\) respectively, are travelling in opposite directions on a straight horizontal path with speeds \(3 u\) and \(2 u\) respectively. \(P\) and \(Q\) collide and, as a result, the direction of motion of both particles is reversed and their speeds are halved.
    1. Find the value of \(k\).
    2. Write down an expression in terms of \(m\) and \(u\) for the magnitude of the impulse which \(P\) exerts on \(Q\) during the collision.
      (3 marks)
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{54642aff-2042-494e-ba4a-8332bd47a751-2_222_1170_790_372} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Figure 1 shows a plank \(A B\) of mass 40 kg and length 6 m , which rests on supports at each of its ends. The plank is wedge-shaped, being thicker at end \(A\) than at end \(B\). A woman of mass 60 kg stands on the plank at a distance of 2 m from \(B\).
  2. Suggest suitable modelling assumptions which can be made about
    1. the plank,
    2. the woman. Given that the reactions at each support are of equal magnitude,
  3. find the magnitude of the reaction on the support at \(A\),
  4. calculate the distance of the centre of mass of the plank from \(A\).
Edexcel M1 Q3
9 marks Moderate -0.5
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{54642aff-2042-494e-ba4a-8332bd47a751-3_515_903_194_475} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows a cable car \(C\) of mass 1 tonne which has broken down. The cable car is suspended in equilibrium by two perpendicular cables \(A C\) and \(B C\) which are attached to fixed points \(A\) and \(B\), at the same horizontal level on either side of a valley. The cable \(A C\) is inclined at an angle \(\alpha\) to the horizontal where \(\tan \alpha = \frac { 3 } { 4 }\).
  1. Show that the tension in the cable \(A C\) is 5880 N and find the tension in the cable BC. A gust of wind then blows along the valley.
  2. Explain the effect that this will have on the tension in the two cables.
Edexcel M1 Q4
10 marks Moderate -0.8
4. Andrew hits a tennis ball vertically upwards towards his sister Barbara who is leaning out of a window 7.5 m above the ground to try to catch it. When the ball leaves Andrew's racket, it is 1.9 m above the ground and travelling at \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Barbara fails to catch the ball on its way up but succeeds as the ball comes back down. Modelling the ball as a particle and assuming that air resistance can be neglected,
  1. find the maximum height above the ground which the ball reaches.
  2. find how long Barbara has to wait from the moment that the ball first passes her until she catches it.
Edexcel M1 Q5
11 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{54642aff-2042-494e-ba4a-8332bd47a751-4_380_326_237_705} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows two particles \(A\) and \(B\) of masses \(m\) and \(k m\) respectively, connected by a light inextensible string which passes over a smooth fixed pulley. When the system is released from rest with both particles 0.5 m above the ground, particle \(A\) moves vertically upwards with acceleration \(\frac { 1 } { 4 } \mathrm {~g} \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Write down, with a brief justification, the magnitude and direction of the acceleration of \(B\).
  2. Find the value of \(k\). Given that \(A\) does not hit the pulley,
  3. calculate, correct to 3 significant figures, the speed with which \(B\) hits the ground.
    (3 marks)
Edexcel M1 Q6
12 marks Standard +0.3
6. Two trains \(A\) and \(B\) leave the same station, \(O\), at 10 a.m. and travel along straight horizontal tracks. \(A\) travels with constant speed \(80 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) due east and \(B\) travels with constant speed \(52 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in the direction ( \(5 \mathbf { i } + 12 \mathbf { j }\) ) where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors due east and due north respectively.
  1. Show that the velocity of \(B\) is \(( 20 \mathbf { i } + 48 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
  2. Find the displacement vector of \(B\) from \(A\) at 10:15 a.m. Given that the trains are 23 km apart \(t\) minutes after 10 a.m.
  3. find the value of \(t\) correct to the nearest whole number.
Edexcel M1 Q7
17 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{54642aff-2042-494e-ba4a-8332bd47a751-5_485_1191_194_333} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Figure 4 shows two golf balls \(P\) and \(Q\) being held at the top of planes inclined at \(30 ^ { \circ }\) and \(60 ^ { \circ }\) to the vertical respectively. Both planes slope down to a common hole at \(H\), which is 3 m vertically below \(P\) and \(Q\). \(P\) is released from rest and travels down the line of greatest slope of the plane it is on which is assumed to be smooth.
  1. Find the acceleration of \(P\) down the slope.
  2. Show that the time taken for \(P\) to reach the hole is 0.904 seconds, correct to 3 significant figures. \(Q\) travels down the line of greatest slope of the plane it is on which is rough. The coefficient of friction between \(Q\) and the plane is \(\mu\). Given that the acceleration of \(Q\) down the slope is \(3 \mathrm {~ms} ^ { - 2 }\),
  3. find, correct to 3 significant figures, the value of \(\mu\). In order for the two balls to arrive at the hole at the same time, \(Q\) must be released \(t\) seconds before \(P\).
  4. Find the value of \(t\) correct to 2 decimal places.
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
4 marks Standard +0.3
\begin{enumerate} \item A constant force acts on a particle of mass 200 grams, moving it 50 cm in a straight line on a rough horizontal surface at a constant speed. The coefficient of friction between the particle and the surface is \(\frac { 1 } { 4 }\).
Calculate, in J , the work done by the force. \item A stone, of mass 0.9 kg , is projected vertically upwards with speed \(10 \mathrm {~ms} ^ { - 1 }\) in a medium which exerts a constant resistance to motion. It comes to rest after rising a distance of 3.75 m . Find the magnitude of the non-gravitational resisting force acting on the stone. \item A particle \(P\), of mass 0.4 kg , moves in a straight line such that, at time \(t\) seconds after passing through a fixed point \(O\), its distance from \(O\) is \(x\) metres, where \(x = 3 t ^ { 2 } + 8 t\).
  1. Show that \(P\) never returns to \(O\).
  2. Find the value of \(t\) when \(P\) has velocity \(20 \mathrm {~ms} ^ { - 1 }\).
  3. Show that the force acting on \(P\) is constant, and find its magnitude. \item Two smooth spheres \(A\) and \(B\), of masses \(2 m\) and \(3 m\) respectively, are moving on a smooth horizontal table with velocities \(( 3 \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and \(( 4 \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors. They collide, after which \(A\) has velocity \(( 5 \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
Edexcel M2 Q6
12 marks Standard +0.3
  1. A uniform wire \(A B C D\) is bent into the shape shown, where the sections \(A B , B C\) and \(C D\) are straight and of length \(3 a , 10 a\) and \(5 a\) respectively and \(A D\) is parallel to \(B C\).
    1. Show that the cosine of angle \(B C D\) is \(\frac { 4 } { 5 }\). \includegraphics[max width=\textwidth, alt={}, center]{70e0f6f0-9016-45a1-9ae2-b908f6b3911e-2_261_479_283_1462}
    2. Find the distances of the centre of mass of the bent wire from (i) \(A B\), (ii) \(B C\).
    The wire is hung over a smooth peg at \(B\) and rests in equilibrium.
  2. Find, to the nearest \(0.1 ^ { \circ }\), the angle between \(B C\) and the vertical in this position.
Edexcel M2 Q7
12 marks Standard +0.3
7. Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are moving towards each other along a straight line. \(P\) has speed \(4 \mathrm {~ms} ^ { - 1 }\). They collide directly. After the collision the direction of motion of both particles has been reversed, and \(Q\) has speed \(2 \mathrm {~ms} ^ { - 1 }\). The coefficient of restitution between \(P\) and \(Q\) is \(\frac { 1 } { 3 }\). Find
  1. the speed of \(Q\) before the collision,
  2. the speed of \(P\) after the collision,
  3. the kinetic energy, in J , lost in the impact.
Edexcel M2 Q8
15 marks Moderate -0.3
8. In a fairground game, a contestant bowls a ball at a coconut 6 metres away on the same horizontal level. The ball is thrown with an initial speed of \(8 \mathrm {~ms} ^ { - 1 }\) in a direction making an angle of \(30 ^ { \circ }\) with the horizontal. \includegraphics[max width=\textwidth, alt={}, center]{70e0f6f0-9016-45a1-9ae2-b908f6b3911e-2_293_641_1604_1254}
  1. Find the time taken by the ball to travel 6 m horizontally.
  2. Showing your method clearly, decide whether or not the ball will hit the coconut.
  3. Find the greatest height reached by the ball above the level from which it was thrown.
  4. Find the maximum horizontal distance from which it is possible to hit the coconut if the ball is thrown with the same initial speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  5. State two assumptions that you have made about the ball and the forces which act on it as it travels towards the coconut.
Edexcel M2 Q1
6 marks Moderate -0.3
  1. A ball, of mass \(m \mathrm {~kg}\), is moving with velocity \(( 5 \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) when it receives an impulse of \(( - 2 \mathbf { i } - 4 \mathbf { j } )\) Ns. Immediately after the impulse is applied, the ball has velocity \(( 3 \mathbf { i } + k \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Find the values of the constants \(k\) and \(m\).
  2. A particle \(P\), initially at rest at the point \(O\), moves in a straight line such that at time \(t\) seconds after leaving \(O\) its acceleration is \(( 12 t - 15 ) \mathrm { ms } ^ { - 2 }\). Find
    1. the velocity of \(P\) at time \(t\) seconds after it leaves \(O\),
    2. the value of \(t\) when the speed of \(P\) is \(36 \mathrm {~ms} ^ { - 1 }\).
    3. A non-uniform ladder \(A B\), of length \(3 a\), has its centre of mass at \(G\), where \(A G = 2 a\). The ladder rests in limiting equilibrium with the end \(B\) against a smooth vertical wall and the end \(A\) resting on rough horizontal ground. The angle between \(A B\) and the horizontal in this position is \(\alpha\), where \(\tan \alpha = \frac { 14 } { 9 }\). Calculate the coefficient of friction between the ladder and the ground.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6198c550-927b-4554-9ddf-ef166fc9f2dd-1_355_330_1019_1615} \captionsetup{labelformat=empty} \caption{(7 marks)}
    \end{figure}
Edexcel M2 Q4
9 marks Standard +0.3
  1. A particle \(P\) starts from the point \(O\) and moves such that its position vector \(\mathbf { r } \mathbf { m }\) relative to \(O\) after \(t\) seconds is given by \(\mathbf { r } = a t ^ { 2 } \mathbf { i } + b t \mathbf { j }\).
    60 seconds after \(P\) leaves \(O\) it is at the point \(Q\) with position vector \(( 90 \mathbf { i } + 30 \mathbf { j } ) \mathrm { m }\).
    1. Find the values of the constants \(a\) and \(b\).
    2. Find the speed of \(P\) when it is at \(Q\).
    3. Sketch the path followed by \(P\) for \(0 \leq t \leq 60\).
    4. A lorry of mass 4200 kg can develop a maximum power of 84 kW . On any road the lorry experiences a non-gravitational resisting force which is directly proportional to its speed. When the lorry is travelling at \(20 \mathrm {~ms} ^ { - 1 }\) the resisting force has magnitude 2400 N . Find the maximum speed of the lorry when it is
    5. travelling on a horizontal road,
    6. climbing a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 7 }\).
    \section*{MECHANICS 2 (A)TEST PAPER 3 Page 2}
Edexcel M2 Q6
11 marks Standard +0.3
  1. Two railway trucks, \(P\) and \(Q\), of equal mass, are moving towards each other with speeds \(4 u\) and \(5 u\) respectively along a straight stretch of rail which may be modelled as being smooth. They collide and move apart. The coefficient of restitution between \(P\) and \(Q\) is \(e\).
    1. Find, in terms of \(u\) and \(e\), the speed of \(Q\) after the collision.
    2. Show that \(e > \frac { 1 } { 9 }\). \(Q\) now hits a fixed buffer and rebounds along the track. \(P\) continues to move with the speed that it had immediately after it collided with \(Q\).
    3. Prove that it is impossible for a further collision between \(P\) and \(Q\) to occur.
    4. A uniform lamina is in the form of a trapezium \(A B C D\), as shown. \(A B\) and \(D C\) are perpendicular to \(B C . A B = 17 \mathrm {~cm} , B C = 21 \mathrm {~cm}\) and \(C D = 8 \mathrm {~cm}\).
    5. Find the distances of the centre of mass of the lamina from \includegraphics[max width=\textwidth, alt={}, center]{6198c550-927b-4554-9ddf-ef166fc9f2dd-2_273_426_948_1537}
      1. \(A B\),
      2. \(B C\).
    The lamina is freely suspended from \(C\) and rests in equilibrium.
  2. Find the angle between \(C D\) and the vertical.
Edexcel M2 Q8
15 marks Moderate -0.3
8. A stone, of mass 1.5 kg , is projected horizontally with speed \(4 \mathrm {~ms} ^ { - 1 }\) from a height of 7 m above horizontal ground.
  1. Show that the stone travels about 4.78 m horizontally before it hits the ground.
  2. Find the height of the stone above the ground when it has travelled half of this horizontal distance.
  3. Calculate the potential energy lost by the stone as it moves from its point of projection to the ground.
  4. Showing your method clearly, use your answer to part (c) to find the speed with which the stone hits the ground.
  5. State two modelling assumptions that you have made in answering this question.