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Edexcel M1 2002 November Q6
11 marks Moderate -0.8
6. A railway truck \(P\) of mass 1500 kg is moving on a straight horizontal track. The truck \(P\) collides with a truck \(Q\) of 2500 kg at a point \(A\). Immediately before the collision, \(P\) and \(Q\) are moving in the same direction with speeds \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Immediately after the collision, the direction of motion of \(P\) is unchanged and its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By modelling the trucks as particles,
  1. show that the speed of \(Q\) immediately after the collision is \(8.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After the collision at \(A\), the truck \(P\) is acted upon by a constant braking force of magnitude 500 N . The truck \(P\) comes to rest at the point \(B\).
  2. Find the distance \(A B\). After the collision \(Q\) continues to move with constant speed \(8.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find the distance between \(P\) and \(Q\) at the instant when \(P\) comes to rest.
Edexcel M1 2002 November Q7
11 marks Moderate -0.8
7. Two helicopters \(P\) and \(Q\) are moving in the same horizontal plane. They are modelled as particles moving in straight lines with constant speeds. At noon \(P\) is at the point with position vector \(( 20 \mathbf { i } + 35 \mathbf { j } ) \mathrm { km }\) with respect to a fixed origin \(O\). At time \(t\) hours after noon the position vector of \(P\) is \(\mathbf { p } \mathrm { km }\). When \(t = \frac { 1 } { 2 }\) the position vector of \(P\) is \(( 50 \mathbf { i } - 25 \mathbf { j } ) \mathrm { km }\). Find
  1. the velocity of \(P\) in the form \(( a \mathbf { i } + b \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\),
  2. an expression for \(\mathbf { p }\) in terms of \(t\). At noon \(Q\) is at \(O\) and at time \(t\) hours after noon the position vector of \(Q\) is \(\mathbf { q } \mathrm { km }\). The velocity of \(Q\) has magnitude \(120 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in the direction of \(4 \mathbf { i } - 3 \mathbf { j }\). Find
  3. an expression for \(\mathbf { q }\) in terms of \(t\),
  4. the distance, to the nearest km , between \(P\) and \(Q\) when \(t = 2\). \section*{8.} \section*{Figure 4}
    \includegraphics[max width=\textwidth, alt={}]{14703bfa-abd8-4a8d-bc18-20d66eea409e-6_695_1153_322_562}
    Two particles \(A\) and \(B\), of mass \(m \mathrm {~kg}\) and 3 kg respectively, are connected by a light inextensible string. The particle \(A\) is held resting on a smooth fixed plane inclined at \(30 ^ { \circ }\) to the horizontal. The string passes over a smooth pulley \(P\) fixed at the top of the plane. The portion \(A P\) of the string lies along a line of greatest slope of the plane and \(B\) hangs freely from the pulley, as shown in Fig. 4. The system is released from rest with \(B\) at a height of 0.25 m above horizontal ground. Immediately after release, \(B\) descends with an acceleration of \(\frac { 2 } { 5 } g\). Given that \(A\) does not reach \(P\), calculate
  5. the tension in the string while \(B\) is descending,
  6. the value of \(m\). The particle \(B\) strikes the ground and does not rebound. Find
  7. the magnitude of the impulse exerted by \(B\) on the ground,
  8. the time between the instant when \(B\) strikes the ground and the instant when \(A\) reaches its highest point.
Edexcel M1 2003 November Q1
6 marks Moderate -0.8
  1. A small ball is projected vertically upwards from a point \(A\). The greatest height reached by the ball is 40 m above \(A\). Calculate
    1. the speed of projection,
    2. the time between the instant that the ball is projected and the instant it returns to \(A\).
    3. A railway truck \(S\) of mass 2000 kg is travelling due east along a straight horizontal track with constant speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The truck \(S\) collides with a truck \(T\) which is travelling due west along the same track as \(S\) with constant speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The magnitude of the impulse of \(T\) on \(S\) is 28800 Ns.
    4. Calculate the speed of \(S\) immediately after the collision.
    5. State the direction of motion of \(S\) immediately after the collision.
    Given that, immediately after the collision, the speed of \(T\) is \(3.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and that \(T\) and \(S\) are moving in opposite directions,
  2. calculate the mass of \(T\).
    (4)
Edexcel M1 2003 November Q3
9 marks Moderate -0.8
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{ebbf20a5-63f8-4c80-a8c4-477e1ea1eee7-2_421_1011_1738_614}
\end{figure} A heavy suitcase \(S\) of mass 50 kg is moving along a horizontal floor under the action of a force of magnitude \(P\) newtons. The force acts at \(30 ^ { \circ }\) to the floor, as shown in Fig. 1, and \(S\) moves in a straight line at constant speed. The suitcase is modelled as a particle and the floor as a rough horizontal plane. The coefficient of friction between \(S\) and the floor is \(\frac { 3 } { 5 }\). Calculate the value of \(P\).
Edexcel M1 2003 November Q4
12 marks Standard +0.3
4. A car starts from rest at a point \(S\) on a straight racetrack. The car moves with constant acceleration for 20 s , reaching a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then travels at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 120 s . Finally it moves with constant deceleration, coming to rest at a point \(F\).
  1. In the space below, sketch a speed-time graph to illustrate the motion of the car. The distance between \(S\) and \(F\) is 4 km .
  2. Calculate the total time the car takes to travel from \(S\) to \(F\).
    (3) A motorcycle starts at \(S , 10 \mathrm {~s}\) after the car has left \(S\). The motorcycle moves with constant acceleration from rest and passes the car at a point \(P\) which is 1.5 km from \(S\). When the motorcycle passes the car, the motorcycle is still accelerating and the car is moving at a constant speed. Calculate
  3. the time the motorcycle takes to travel from \(S\) to \(P\),
  4. the speed of the motorcycle at \(P\).
    (2)
Edexcel M1 2003 November Q5
12 marks Moderate -0.3
5. A particle \(P\) of mass 3 kg is moving under the action of a constant force \(\mathbf { F }\) newtons. At \(t = 0\), \(P\) has velocity \(( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At \(t = 4 \mathrm {~s}\), the velocity of \(P\) is \(( - 5 \mathbf { i } + 11 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
  1. the acceleration of \(P\), in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
  2. the magnitude of \(\mathbf { F }\). At \(t = 6 \mathrm {~s} , P\) is at the point \(A\) with position vector ( \(6 \mathbf { i } - 29 \mathbf { j }\) ) m relative to a fixed origin \(O\). At this instant the force \(\mathbf { F }\) newtons is removed and \(P\) then moves with constant velocity. Three seconds after the force has been removed, \(P\) is at the point \(B\).
  3. Calculate the distance of \(B\) from \(O\).
    (6) \section*{6.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{ebbf20a5-63f8-4c80-a8c4-477e1ea1eee7-4_298_1221_358_411}
    \end{figure} A non-uniform rod \(A B\) has length 5 m and weight 200 N . The rod rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(A C = 1.5 \mathrm {~m}\) and \(D B = 1 \mathrm {~m}\), as shown in Fig. 2. The centre of mass of \(A B\) is \(x\) metres from \(A\). A particle of weight \(W\) newtons is placed on the rod at \(A\). The rod remains in equilibrium and the magnitude of the reaction of \(C\) on the rod is 160 N .
  4. Show that \(50 x - W = 100\). The particle is now removed from \(A\) and placed on the rod at \(B\). The rod remains in equilibrium and the reaction of \(C\) on the rod now has magnitude 50 N .
  5. Obtain another equation connecting \(W\) and \(x\).
  6. Calculate the value of \(x\) and the value of \(W\). \section*{7.} \section*{Figure 3}
    \includegraphics[max width=\textwidth, alt={}]{ebbf20a5-63f8-4c80-a8c4-477e1ea1eee7-5_688_1477_379_328}
    Figure 3 shows two particles \(A\) and \(B\), of mass \(m \mathrm {~kg}\) and 0.4 kg respectively, connected by a light inextensible string. Initially \(A\) is held at rest on a fixed smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The string passes over a small light smooth pulley \(P\) fixed at the top of the plane. The section of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. The particle \(B\) hangs freely below \(P\). The system is released from rest with the string taut and \(B\) descends with acceleration \(\frac { 1 } { 5 } g\).
  7. Write down an equation of motion for \(B\).
  8. Find the tension in the string.
  9. Prove that \(m = \frac { 16 } { 35 }\).
  10. State where in the calculations you have used the information that \(P\) is a light smooth pulley. On release, \(B\) is at a height of one metre above the ground and \(A P = 1.4 \mathrm {~m}\). The particle \(B\) strikes the ground and does not rebound.
  11. Calculate the speed of \(B\) as it reaches the ground.
  12. Show that \(A\) comes to rest as it reaches \(P\). \section*{END}
Edexcel M1 2004 November Q1
5 marks Moderate -0.8
  1. A man is driving a car on a straight horizontal road. He sees a junction \(S\) ahead, at which he must stop. When the car is at the point \(P , 300 \mathrm {~m}\) from \(S\), its speed is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car continues at this constant speed for 2 s after passing \(P\). The man then applies the brakes so that the car has constant deceleration and comes to rest at \(S\).
    1. Sketch, in the space below, a speed-time graph to illustrate the motion of the car in moving from \(P\) to \(S\).
    2. Find the time taken by the car to travel from \(P\) to \(S\).
      (3)
    \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{31c17a67-4fcf-4402-b00e-239ce9f20964-2_421_460_884_758}
    \end{figure} The particles have mass 3 kg and \(m \mathrm {~kg}\), where \(m < 3\). They are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The particles are held in position with the string taut and the hanging parts of the string vertical, as shown in Figure 1. The particles are then released from rest. The initial acceleration of each particle has magnitude \(\frac { 3 } { 7 } g\). Find
  2. the tension in the string immediately after the particles are released,
  3. the value of \(m\). \section*{3.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{31c17a67-4fcf-4402-b00e-239ce9f20964-3_241_1202_388_420}
    \end{figure} A plank of wood \(A B\) has mass 10 kg and length 4 m . It rests in a horizontal position on two smooth supports. One support is at the end \(A\). The other is at the point \(C , 0.4 \mathrm {~m}\) from \(B\), as shown in Figure 2. A girl of mass 30 kg stands at \(B\) with the plank in equilibrium. By modelling the plank as a uniform rod and the girl as a particle,
  4. find the reaction on the plank at \(A\). The girl gets off the plank. A boulder of mass \(m \mathrm {~kg}\) is placed on the plank at \(A\) and a man of mass 80 kg stands on the plank at \(B\). The plank remains in equilibrium and is on the point of tilting about \(C\). By modelling the plank again as a uniform rod, and the man and the boulder as particles,
  5. find the value of \(m\).
    (4)
Edexcel M1 2004 November Q4
8 marks Moderate -0.5
4. A tent peg is driven into soft ground by a blow from a hammer. The tent peg has mass 0.2 kg and the hammer has mass 3 kg . The hammer strikes the peg vertically. Immediately before the impact, the speed of the hammer is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It is assumed that, immediately after the impact, the hammer and the peg move together vertically downwards.
  1. Find the common speed of the peg and the hammer immediately after the impact. Until the peg and hammer come to rest, the resistance exerted by the ground is assumed to be constant and of magnitude \(R\) newtons. The hammer and peg are brought to rest 0.05 s after the impact.
  2. Find, to 3 significant figures, the value of \(R\).
Edexcel M1 2004 November Q5
10 marks Moderate -0.8
5. A particle \(P\) moves in a horizontal plane. The acceleration of \(P\) is \(( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). At time \(t = 0\), the velocity of \(P\) is \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Find, to the nearest degree, the angle between the vector \(\mathbf { j }\) and the direction of motion of \(P\) when \(t = 0\). At time \(t\) seconds, the velocity of \(P\) is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
  2. an expression for \(\mathbf { v }\) in terms of \(t\), in the form \(a \mathbf { i } + b \mathbf { j }\),
  3. the speed of \(P\) when \(t = 3\),
  4. the time when \(P\) is moving parallel to \(\mathbf { i }\).
Edexcel M1 2004 November Q6
11 marks Moderate -0.8
6. Two cars \(A\) and \(B\) are moving in the same direction along a straight horizontal road. At time \(t = 0\), they are side by side, passing a point \(O\) on the road. Car \(A\) travels at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Car \(B\) passes \(O\) with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and has constant acceleration of \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find
  1. the speed of \(B\) when it has travelled 78 m from \(O\),
  2. the distance from \(O\) of \(A\) when \(B\) is 78 m from \(O\),
  3. the time when \(B\) overtakes \(A\).
    (5) \section*{7.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{31c17a67-4fcf-4402-b00e-239ce9f20964-5_271_926_392_639}
    \end{figure} A sledge has mass 30 kg . The sledge is pulled in a straight line along horizontal ground by means of a rope. The rope makes an angle \(20 ^ { \circ }\) with the horizontal, as shown in Figure 3. The coefficient of friction between the sledge and the ground is 0.2 . The sledge is modelled as a particle and the rope as a light inextensible string. The tension in the rope is 150 N . Find, to 3 significant figures,
  4. the normal reaction of the ground on the sledge,
  5. the acceleration of the sledge. When the sledge is moving at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rope is released from the sledge.
  6. Find, to 3 significant figures, the distance travelled by the sledge from the moment when the rope is released to the moment when the sledge comes to rest.
    (6) \section*{8.} \section*{Figure 4}
    \includegraphics[max width=\textwidth, alt={}]{31c17a67-4fcf-4402-b00e-239ce9f20964-6_513_570_340_753}
    A heavy package is held in equilibrium on a slope by a rope. The package is attached to one end of the rope, the other end being held by a man standing at the top of the slope. The package is modelled as a particle of mass 20 kg . The slope is modelled as a rough plane inclined at \(60 ^ { \circ }\) to the horizontal and the rope as a light inextensible string. The string is assumed to be parallel to a line of greatest slope of the plane, as shown in Figure 4. At the contact between the package and the slope, the coefficient of friction is 0.4 .
  7. Find the minimum tension in the rope for the package to stay in equilibrium on the slope.
    (8) The man now pulls the package up the slope. Given that the package moves at constant speed,
  8. find the tension in the rope.
  9. State how you have used, in your answer to part (b), the fact that the package moves
    1. up the slope,
    2. at constant speed.
Edexcel M1 Specimen Q1
7 marks Moderate -0.8
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{e590030f-0c46-42ab-80b8-3627d3c36908-2_367_605_315_751}
\end{figure} A tennis ball \(P\) is attached to one end of a light inextensible string, the other end of the string being attached to a the top of a fixed vertical pole. A girl applies a horizontal force of magnitude 50 N to \(P\), and \(P\) is in equilibrium under gravity with the string making an angle of \(40 ^ { \circ }\) with the pole, as shown in Fig. 1. By modelling the ball as a particle find, to 3 significant figures,
  1. the tension in the string,
  2. the weight of \(P\).
Edexcel M1 Specimen Q2
7 marks Easy -1.2
2. A car starts from rest at a point \(O\) and moves in a straight line. The car moves with constant acceleration \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it passes the point \(A\) when it is moving with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then moves with constant acceleration \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 6 s until it reaches the point \(B\). Find
  1. the speed of the car at \(B\),
  2. the distance \(O B\). \section*{3.} \section*{Figure 2} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e590030f-0c46-42ab-80b8-3627d3c36908-3_386_970_412_575} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} A non-uniform plank of wood \(A B\) has length 6 m and mass 90 kg . The plank is smoothly supported at its two ends \(A\) and \(B\), with \(A\) and \(B\) at the same horizontal level. A woman of mass 60 kg stands on the plank at the point \(C\), where \(A C = 2 \mathrm {~m}\), as shown in Fig. 2. The plank is in equilibrium and the magnitudes of the reactions on the plank at \(A\) and \(B\) are equal. The plank is modelled as a non-uniform rod and the woman as a particle. Find
  3. the magnitude of the reaction on the plank at \(B\),
  4. the distance of the centre of mass of the plank from \(A\).
  5. State briefly how you have used the modelling assumption that
    1. the plank is a rod,
    2. the woman is a particle.
Edexcel M1 Specimen Q4
12 marks Moderate -0.8
4. A train \(T\), moves from rest at Station \(A\) with constant acceleration \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it reaches a speed of \(36 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In maintains this constant speed for 90 s before the brakes are applied, which produce constant retardation \(3 \mathrm {~ms} ^ { - 2 }\). The train \(T _ { 1 }\) comes to rest at station \(B\).
  1. Sketch a speed-time graph to illustrate the journey of \(T _ { 1 }\) from \(A\) to \(B\).
  2. Show that the distance between \(A\) and \(B\) is 3780 m . \includegraphics[max width=\textwidth, alt={}, center]{e590030f-0c46-42ab-80b8-3627d3c36908-4_538_734_689_635} A second train \(T _ { 2 }\) takes 150 s to move form rest at \(A\) to rest at \(B\). Figure 3 shows the speed-time graph illustrating this journey.
  3. Explain briefly one way in which \(T _ { 1 }\) 's journey differs from \(T _ { 2 }\) 's journey.
  4. Find the greatest speed, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), attained by \(T _ { 2 }\) during its journey.
Edexcel M1 Specimen Q5
12 marks Moderate -0.8
5. A truck of mass 3 tonnes moves on straight horizontal rails. It collides with truck \(B\) of mass 1 tonne, which is moving on the same rails. Immediately before the collision, the speed of \(A\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the speed of \(B\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the trucks are moving towards each other. In the collision, the trucks couple to form a single body \(C\), which continues to move on the rails.
  1. Find the speed and direction of \(C\) after the collision.
  2. Find, in Ns, the magnitude of the impulse exerted by \(B\) on \(A\) in the collision.
  3. State a modelling assumption which you have made about the trucks in your solution Immediately after the collision, a constant braking force of magnitude 250 N is applied to \(C\). It comes to rest in a distance \(d\) metres.
  4. Find the value of \(d\).
    (4)
Edexcel M1 Specimen Q6
13 marks Standard +0.3
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{e590030f-0c46-42ab-80b8-3627d3c36908-5_345_1255_1329_265}
\end{figure} A particle of mass \(m\) rests on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The particle is attached to one end of a light inextensible string which lies in a line of greatest slope of the plane and passes over a small light smooth pulley \(P\) fixed at the top of the plane. The other end of the string is attached to a particle \(B\) of mass \(3 m\), and \(B\) hangs freely below \(P\), as shown in Fig. 4. The particles are released from rest with the string taut. The particle \(B\) moves down with acceleration of magnitude \(\frac { 1 } { 2 } g\). Find
  1. the tension in the string,
  2. the coefficient of friction between \(A\) and the plane.
Edexcel M1 Specimen Q7
15 marks Moderate -0.3
7. Two cars \(A\) and \(B\) are moving on straight horizontal roads with constant velocities. The velocity of \(A\) is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due east, and the velocity of \(B\) is \(( 10 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors directed due east and due north respectively. Initially \(A\) is at the fixed origin \(O\), and the position vector of \(B\) is \(300 \mathbf { i }\) m relative to \(O\). At time \(t\) seconds, the position vectors of \(A\) and \(B\) are \(\mathbf { r }\) metres and \(\mathbf { s }\) metres respectively.
  1. Find expressions for \(\mathbf { r }\) and \(\mathbf { s }\) in terms of \(t\).
  2. Hence write down an expression for \(\overrightarrow { A B }\) in terms of \(t\).
  3. Find the time when the bearing of \(B\) from \(A\) is \(045 ^ { \circ }\).
  4. Find the time when the cars are again 300 m apart. END
Edexcel M2 2014 January Q1
8 marks Moderate -0.3
  1. A particle \(P\) of mass 2 kg is moving with velocity \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse. Immediately after the impulse is applied, \(P\) has velocity \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
    1. Find the magnitude of the impulse.
    2. Find the angle between the direction of the impulse and the direction of motion of \(P\) immediately before the impulse is applied.
    3. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where
    $$v = ( t - 2 ) ( 3 t - 10 ) , \quad t \geqslant 0$$ When \(t = 0 , P\) is at the origin \(O\).
  2. Find the acceleration of \(P\) when \(t = 3\)
  3. Find the total distance travelled by \(P\) in the first 3 seconds of its motion.
  4. Show that \(P\) never returns to \(O\).
Edexcel M2 2014 January Q3
12 marks Standard +0.3
  1. A car has mass 550 kg . When the car travels along a straight horizontal road there is a constant resistance to the motion of magnitude \(R\) newtons, the engine of the car is working at a rate of \(P\) watts and the car maintains a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the car travels up a line of greatest slope of a hill which is inclined at \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\), with the engine working at a rate of \(P\) watts, it maintains a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The non-gravitational resistance to motion when the car travels up the hill is a constant force of magnitude \(R\) newtons.
    1. (i) Find the value of \(R\).
      (ii) Find the value of \(P\).
    2. Find the acceleration of the car when it travels along the straight horizontal road at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with the engine working at 50 kW .
Edexcel M2 2014 January Q4
11 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ad09e19e-c4f3-4b93-9e9a-4987def62f26-07_542_700_219_628} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform lamina \(A B C D\) is formed by removing the isosceles triangle \(A D C\) of height \(h\) metres, where \(h < 2 \sqrt { 3 }\), from a uniform lamina \(A B C\) in the shape of an equilateral triangle of side 4 m , as shown in Figure 1. The centre of mass of \(A B C D\) is at \(D\).
  1. Show that \(h = \sqrt { } 3\) The weight of the lamina \(A B C D\) is \(W\) newtons. The lamina is freely suspended from \(A\). A horizontal force of magnitude \(F\) newtons is applied at \(B\) so that the lamina is in equilibrium with \(A B\) vertical. The horizontal force acts in the vertical plane containing the lamina.
  2. Find \(F\) in terms of \(W\).
Edexcel M2 2014 January Q5
11 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ad09e19e-c4f3-4b93-9e9a-4987def62f26-09_620_776_219_584} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a uniform rod \(A B\), of mass \(m\) and length \(2 a\), with the end \(B\) resting on rough horizontal ground. The rod is held in equilibrium at an angle \(\theta\) to the vertical by a light inextensible string. One end of the string is attached to the rod at the point \(C\), where \(A C = \frac { 2 } { 3 } a\). The other end of the string is attached to the point \(D\), which is vertically above \(B\), where \(B D = 2 a\).
  1. By taking moments about \(D\), show that the magnitude of the frictional force acting on the rod at \(B\) is \(\frac { 1 } { 2 } m g \sin \theta\)
  2. Find the magnitude of the normal reaction on the rod at \(B\). The rod is in limiting equilibrium when \(\tan \theta = \frac { 4 } { 3 }\)
  3. Find the coefficient of friction between the rod and the ground.
Edexcel M2 2014 January Q6
11 marks Standard +0.8
  1. \hspace{0pt} [In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ad09e19e-c4f3-4b93-9e9a-4987def62f26-11_375_1008_354_475} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The point \(O\) is a fixed point on a horizontal plane. A ball is projected from \(O\) with velocity \(( 3 \mathbf { i } + v \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 } , v > 3\). The ball moves freely under gravity and passes through the point A before reaching its maximum height above the horizontal plane, as shown in Figure 3. The ball passes through \(A\) at time \(\frac { 15 } { 49 } \mathrm {~s}\) after projection. The initial kinetic energy of the ball is \(E\) joules. When the ball is at \(A\) it has kinetic energy \(\frac { 1 } { 2 } E\) joules.
  1. Find the value of \(v\). At another point \(B\) on the path of the ball the kinetic energy is also \(\frac { 1 } { 2 } E\) joules. The ball passes through \(B\) at time \(T\) seconds after projection.
  2. Find the value of \(T\).
Edexcel M2 2014 January Q7
11 marks Standard +0.3
7. Three particles \(A , B\) and \(C\), each of mass \(m\), lie at rest in a straight line \(L\) on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). Particles \(A\) and \(B\) are projected directly towards each other with speeds \(5 u\) and \(4 u\) respectively. Particle \(C\) is projected directly away from \(B\) with speed \(3 u\). In the subsequent motion, \(A , B\) and \(C\) move along \(L\). Particles \(A\) and \(B\) collide directly. The coefficient of restitution between \(A\) and \(B\) is \(e\).
  1. Find (i) the speed of \(A\) immediately after the collision,
    (ii) the speed of \(B\) immediately after the collision. Given that the direction of motion of \(A\) is reversed in the collision between \(A\) and \(B\), and that there is no collision between \(B\) and \(C\),
  2. find the set of possible values of \(e\).
Edexcel M2 2015 January Q1
7 marks Moderate -0.3
  1. A particle \(P\) of mass 0.6 kg is moving with velocity ( \(4 \mathbf { i } - 2 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) when it receives an impulse \(\mathbf { I } \mathrm { N }\) s. Immediately after receiving the impulse, \(P\) has velocity ( \(2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Find
  1. the magnitude of \(\mathbf { I }\),
  2. the kinetic energy lost by \(P\) as a result of receiving the impulse.
Edexcel M2 2015 January Q2
9 marks Moderate -0.3
2. A car of mass 500 kg is moving at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The resistance to motion from non-gravitational forces is modelled as a constant force of magnitude 150 N .
  1. Find the rate of working of the engine of the car. When the car is travelling up the road at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the engine is switched off. The car then comes to instantaneous rest, without braking, having moved a distance \(d\) metres up the road from the point where the engine was switched off. The resistance to motion from non-gravitational forces is again modelled as a constant force of magnitude 150 N .
  2. Use the work-energy principle to find the value of \(d\).
Edexcel M2 2015 January Q3
12 marks Standard +0.3
  1. At time \(t\) seconds \(( t \geqslant 0 )\) a particle \(P\) has position vector \(\mathbf { r }\) metres, with respect to a fixed origin \(O\), where
$$\mathbf { r } = \left( \frac { 1 } { 8 } t ^ { 4 } - 2 \lambda t ^ { 2 } + 5 \right) \mathbf { i } + \left( 5 t ^ { 2 } - \lambda t \right) \mathbf { j }$$ and \(\lambda\) is a constant. When \(t = 4 , P\) is moving parallel to the vector \(\mathbf { j }\).
  1. Show that \(\lambda = 2\)
  2. Find the speed of \(P\) when \(t = 4\)
  3. Find the acceleration of \(P\) when \(t = 4\) When \(t = 0 , P\) is at the point \(A\). When \(t = 4 , P\) is at the point \(B\).
  4. Find the distance \(A B\).