Questions — Edexcel M1 (599 questions)

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Edexcel M1 2018 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4fd21e83-0bdf-4bb1-8a3f-76beada511ae-24_391_917_251_516} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A particle \(P\) of mass \(4 m\) is held at rest at the point \(X\) on the surface of a rough inclined plane which is fixed to horizontal ground. The point \(X\) is a distance \(h\) from the bottom of the inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\). The particle \(P\) is attached to one end of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of the plane. The other end of the string is attached to a particle \(Q\) of mass \(m\) which hangs freely at a distance \(d\), where \(d > h\), below the pulley, as shown in Figure 4. The string lies in a vertical plane through a line of greatest slope of the inclined plane. The system is released from rest with the string taut and \(P\) moves down the plane. For the motion of the particles before \(P\) hits the ground,
  1. state which of the information given above implies that the magnitudes of the accelerations of the two particles are the same,
  2. write down an equation of motion for each particle,
  3. find the acceleration of each particle. When \(P\) hits the ground, it immediately comes to rest. Given that \(Q\) comes to instantaneous rest before reaching the pulley,
  4. show that \(d > \frac { 28 h } { 25 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{4fd21e83-0bdf-4bb1-8a3f-76beada511ae-27_56_20_109_1950}
    END
Edexcel M1 Q1
  1. An aircraft moves along a straight horizontal runway with constant acceleration. It passes a point \(A\) on the runway with speed \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then passes the point \(B\) on the runway with speed \(34 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance from \(A\) to \(B\) is 150 m .
    1. Find the acceleration of the aircraft.
    2. Find the time taken by the aircraft in moving from \(A\) to \(B\).
    3. Find, to 3 significant figures, the speed of the aircraft when it passes the point mid-way between \(A\) and \(B\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ee3cbd24-55b1-4003-85bb-26d98f79a118-3_419_569_963_717} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} A particle has mass 2 kg . It is attached at \(B\) to the ends of two light inextensible strings \(A B\) and \(B C\). When the particle hangs in equilibrium, \(A B\) makes an angle of \(30 ^ { \circ }\) with the vertical, as shown in Fig. 1. The magnitude of the tension in \(B C\) is twice the magnitude of the tension in \(A B\).
  2. Find, in degrees to one decimal place, the size of the angle that \(B C\) makes with the vertical.
    (4 marks)
  3. Hence find, to 3 significant figures, the magnitude of the tension in \(A B\).
Edexcel M1 Q3
3. A racing car is travelling on a straight horizontal road. Its initial speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it accelerates for 4 s to reach a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then travels at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for a further 8 s . The total distance travelled by the car during this 12 s period is 600 m .
  1. Sketch a speed-time graph to illustrate the motion of the car during this 12 s period.
  2. Find the value of \(V\).
  3. Find the acceleration of the car during the initial 4 s period.
Edexcel M1 Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ee3cbd24-55b1-4003-85bb-26d98f79a118-4_179_729_449_671} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A plank \(A B\) has length 4 m . It lies on a horizontal platform, with the end \(A\) lying on the platform and the end \(B\) projecting over the edge, as shown in Fig. 2. The edge of the platform is at the point \(C\). Jack and Jill are experimenting with the plank. Jack has mass 40 kg and Jill has mass 25 kg . They discover that, if Jack stands at \(B\) and Jill stands at \(A\) and \(B C = 1.6 \mathrm {~m}\), the plank is in equilibrium and on the point of tilting about \(C\). By modelling the plank as a uniform rod, and Jack and Jill as particles,
  1. find the mass of the plank. They now alter the position of the plank in relation to the platform so that, when Jill stands at \(B\) and Jack stands at \(A\), the plank is again in equilibrium and on the point of tilting about \(C\).
  2. Find the distance \(B C\) in this position.
  3. State how you have used the modelling assumptions that
    1. the plank is uniform,
    2. the plank is a rod,
    3. Jack and Jill are particles.
Edexcel M1 Q5
5. A post is driven into the ground by means of a blow from a pile-driver. The pile-driver falls from rest from a height of 1.6 m above the top of the post.
  1. Show that the speed of the pile-driver just before it hits the post is \(5.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). (2 marks) The post has mass 6 kg and the pile-driver has mass 78 kg . When the pile-driver hits the top of the post, it is assumed that the there is no rebound and that both then move together with the same speed.
  2. Find the speed of the pile-driver and the post immediately after the pile-driver has hit the post. The post is brought to rest by the action of a resistive force from the ground acting for 0.06 s .
    By modelling this force as constant throughout this time,
  3. find the magnitude of the resistive force,
  4. find, to 2 significant figures, the distance travelled by the post and the pile-driver before they come to rest.
    (4 marks)
Edexcel M1 Q6
6. [In this question, the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due East and North respectively.] A coastguard station \(O\) monitors the movements of ships in a channel. At noon, the station's radar records two ships moving with constant speed. Ship \(A\) is at the point with position vector \(( - 5 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\) relative to \(O\) and has velocity \(( 2 \mathbf { i } + 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Ship \(B\) is at the point with position vector \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { km }\) and has velocity \(( - 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
  1. Given that the two ships maintain these velocities, show that they collide. The coast guard radios ship \(A\) and orders it to reduce its speed to move with velocity \(( \mathbf { i } + \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Given that \(A\) obeys this order and maintains this new constant velocity,
  2. find an expression for the vector \(\overrightarrow { A B }\) at time \(t\) hours after noon.
  3. find, to 3 significant figures, the distance between \(A\) and \(B\) at 1400 hours,
  4. find the time at which \(B\) will be due north of \(A\).
Edexcel M1 Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ee3cbd24-55b1-4003-85bb-26d98f79a118-6_271_683_367_717} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} A small parcel of mass 2 kg moves on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The parcel is pulled up a line of greatest slope of the plane by means of a light rope which it attached to it. The rope makes an angle of \(30 ^ { \circ }\) with the plane, as shown in Fig. 3. The coefficient of friction between the parcel and the plane is 0.4 Given that the tension in the rope is 24 N ,
  1. find, to 2 significant figures, the acceleration of the parcel. The rope now breaks. The parcel slows down and comes to rest.
  2. Show that, when the parcel comes to this position of rest, it immediately starts to move down the plane again.
  3. Find, to 2 significant figures, the acceleration of the parcel as it moves down the plane after it has come to this position of instantaneous rest.
Edexcel M1 2002 November Q1
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{14703bfa-abd8-4a8d-bc18-20d66eea409e-2_671_829_294_663}
\end{figure} A particle \(P\) of weight 6 N is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). A horizontal force of magnitude \(F\) newtons is applied to \(P\). The particle \(P\) is in equilibrium under gravity with the string making an angle of \(30 ^ { \circ }\) with the vertical, as shown in Fig. 1. Find, to 3 significant figures,
  1. the tension in the string,
  2. the value of \(F\).
Edexcel M1 2002 November Q2
2. A particle \(P\) of mass 1.5 kg is moving under the action of a constant force ( \(3 \mathbf { i } - 7.5 \mathbf { j }\) ) N. Initially \(P\) has velocity \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
  1. the magnitude of the acceleration of \(P\),
  2. the velocity of \(P\), in terms of \(\mathbf { i }\) and \(\mathbf { j }\), when \(P\) has been moving for 4 seconds.
Edexcel M1 2002 November Q3
3. A car accelerates uniformly from rest to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in \(T\) seconds. The car then travels at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(4 T\) seconds and finally decelerates uniformly to rest in a further 50 s .
  1. Sketch a speed-time graph to show the motion of the car. The total distance travelled by the car is 1220 m . Find
  2. the value of \(T\),
  3. the initial acceleration of the car.
Edexcel M1 2002 November Q4
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{14703bfa-abd8-4a8d-bc18-20d66eea409e-3_282_807_1194_648}
\end{figure} A uniform plank \(A B\) has weight 80 N and length \(x\) metres. The plank rests in equilibrium horizontally on two smooth supports at \(A\) and \(C\), where \(A C = 2 \mathrm {~m}\), as shown in Fig. 2. A rock of weight 20 N is placed at \(B\) and the plank remains in equilibrium. The reaction on the plank at \(C\) has magnitude 90 N . The plank is modelled as a rod and the rock as a particle.
  1. Find the value of \(x\).
  2. State how you have used the model of the rock as a particle. The support at \(A\) is now moved to a point \(D\) on the plank and the plank remains in equilibrium with the rock at \(B\). The reaction on the plank at \(C\) is now three times the reaction at \(D\).
  3. Find the distance \(A D\).
Edexcel M1 2002 November Q5
5. \section*{Figure 3}
\includegraphics[max width=\textwidth, alt={}]{14703bfa-abd8-4a8d-bc18-20d66eea409e-4_502_1154_339_552}
A suitcase of mass 10 kg slides down a ramp which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The suitcase is modelled as a particle and the ramp as a rough plane. The top of the plane is \(A\). The bottom of the plane is \(C\) and \(A C\) is a line of greatest slope, as shown in Fig. 3. The point \(B\) is on \(A C\) with \(A B = 5 \mathrm {~m}\). The suitcase leaves \(A\) with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and passes \(B\) with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the decleration of the suitcase,
  2. the coefficient of friction between the suitcase and the ramp. The suitcase reaches the bottom of the ramp.
  3. Find the greatest possible length of \(A C\).
Edexcel M1 2002 November Q6
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
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
  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
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
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
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
  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
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
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
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
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
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
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.