Questions — Edexcel M1 (663 questions)

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Edexcel M1 Q6
15 marks Standard +0.3
6. A small ring, of mass \(m \mathrm {~kg}\), can slide along a straight wire which is fixed at an angle of \(45 ^ { \circ }\) to the horizontal as shown. The coefficient of friction between the ring and the wire is \(\frac { 2 } { 7 }\).
The ring rests in equilibrium on the wire and is just prevented from \includegraphics[max width=\textwidth, alt={}, center]{cc75a4a5-1c3a-4e36-acfd-21f6246f2a38-2_273_296_1192_1617}
sliding down the wire when a horizontal string is attached to it, as shown
  1. Show that the tension in the string has magnitude \(\frac { 5 m g } { 9 } \mathrm {~N}\). The string is now removed and the ring starts to slide down the wire.
  2. Find the time that elapses before the ring has moved 10 cm along the wire.
Edexcel M1 Q7
15 marks Standard +0.8
7. Two cyclists, Alice and Bobbie, travel from \(P\) to \(Q\) along a straight path. Alice starts from rest at \(P\) just as Bobbie passes her at \(3.5 \mathrm {~ms} ^ { - 1 }\). Bobbie continues at this speed while Alice accelerates at \(0.2 \mathrm {~ms} ^ { - 2 }\) for \(T\) seconds until she attains her maximum speed. At this moment both cyclists immediately start to slow down, with constant but different decelerations, and they come to rest at \(Q 80\) seconds after Alice started moving.
  1. Sketch, on the same diagram, the velocity-time graphs for the two cyclists. By using the fact that both cyclists cover the same distance, find
  2. the value of \(T\),
  3. the distance between \(P\) and \(Q\),
  4. the magnitude of Bobbie's deceleration.
Edexcel M1 Q1
3 marks Moderate -0.3
  1. A car accelerates from 0 to \(108 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in 7.5 seconds. Find its acceleration in \(\mathrm { ms } ^ { - 2 }\). ( 3 marks)
  2. A book rests on a rough desk-lid which is hinged at one end. When the lid is raised so that it makes an angle of \(15 ^ { \circ }\) with the horizontal, the book is just on the point of sliding down a line of greatest slope. Modelling the book as a particle, find
    1. the coefficient of friction between the book and the desk-lid,
    2. the acceleration with which the book starts to move if it is released from rest when the lid is inclined at \(20 ^ { \circ }\) to the horizontal.
    3. A particle \(P\) is projected vertically upwards from ground level at time \(t = 0\) with speed 20 \(\mathrm { ms } ^ { - 1 }\). Two seconds later another particle \(Q\) is projected vertically upwards with speed 30 \(\mathrm { ms } ^ { - 1 }\) from a point on the same horizontal ground.
    4. Taking the upward direction as positive, write down expressions in terms of \(g\) and \(t\) for the velocities of \(P\) and of \(Q\) at time \(t\) seconds after \(P\) is projected.
    5. Find the value of \(t\) when both particles are moving with the same speed.
    6. A jet of water issues from a cylindrical pipe with a circular cross-section of radius \(2 \cdot 75 \mathrm {~cm}\). The water strikes a vertical wall at a speed of \(9 \mathrm {~ms} ^ { - 1 }\). Taking the density of water to be \(1000 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\), calculate
    7. the momentum destroyed each second by the impact with the wall,
    8. the magnitude of the force exerted by the water on the wall.
    9. State one modelling assumption that you have made.
    10. Two particles \(A\) and \(B\), of mass 1 kg and \(m \mathrm {~kg}\) respectively, where \(m > 1\), are attached to the ends of a light inextensible string which passes over a small fixed smooth pulley. The particles are released from rest and move with the string taut and vertical.
    11. Show that the acceleration of the system is equal to \(\frac { ( m - 1 ) g } { m + 1 }\).
    12. Find the tension in the string, in terms of \(m\) and \(g\), expressing your answer as a single algebraic fraction in its simplest form.
    When the system is released from rest, both particles are \(52 \cdot 5 \mathrm {~cm}\) above ground level and 60 cm below the level of the pulley. \(B\) hits the ground after half a second.
  3. Find the value of \(m\).
  4. Find the speed with which \(B\) hits the ground. \section*{MECHANICS 1 (A) TEST PAPER 9 Page 2}
Edexcel M1 Q6
17 marks Standard +0.3
  1. At noon, two boats \(P\) and \(Q\) have position vectors \(( \mathbf { i } + 7 \mathbf { j } ) \mathrm { km }\) and \(( 3 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km }\) respectively relative to an origin \(O\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions due East and due North respectively. \(P\) is moving with constant velocity \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and \(Q\) is moving with constant velocity \(( 6 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
    1. Find the position vector of each boat at time \(t\) hours after noon, giving your answers in the form \(\mathrm { f } ( t ) \mathrm { i } + \mathrm { g } ( t ) \mathrm { j }\), where \(\mathrm { f } ( t )\) and \(\mathrm { g } ( t )\) are linear functions of \(t\) to be found.
    2. Find, in terms of \(t\), the distance between the boats \(t\) hours after noon.
    3. Calculate the time when the boats are closest together and find the distance between them at this time.
    4. A particle starts from rest and accelerates at a uniform rate over a distance of 12 m . It then travels at a constant speed of \(u \mathrm {~ms} ^ { - 1 }\) for a further 30 seconds. Finally it decelerates uniformly to rest at \(1.6 \mathrm {~ms} ^ { - 2 }\).
    5. Sketch the velocity-time graph for this motion.
    6. Show that the total time for which the particle is in motion is
    $$\frac { 5 u } { 8 } + 30 + \frac { 24 } { u } \text { seconds. }$$
  2. Find, in terms of \(u\), the total distance travelled by the particle during the motion.
  3. Given that the total time for the motion is \(39 \cdot 5\) seconds, show that \(5 u ^ { 2 } - 76 u + 192 = 0\).
  4. Find the two possible values of \(u\) and the total distance travelled in each case.
Edexcel M1 Q1
8 marks Moderate -0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8a0ff401-83da-4539-a9e9-68736c57df2a-2_520_1278_207_333} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a distance-time graph for a car journey from Birmingham to Newquay which included a stop for lunch at a service station near Exeter. During the first part of the journey three-quarters of the total distance, \(d\), was covered in 3 hours. After a 1 hour stop, the remaining distance was completed in 2 hours.
  1. Calculate, in the form \(k : 1\), the ratio of the average speed during the first 3 hours of the journey to the average speed during the last 2 hours of the journey.
    (4 marks)
    Given that the average speed of the car over the whole journey (excluding the stop) was \(80 \mathrm { kmh } ^ { - 1 }\),
  2. find the average speed of the car on the first part of the journey.
    (4 marks)
Edexcel M1 Q2
8 marks Moderate -0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8a0ff401-83da-4539-a9e9-68736c57df2a-2_291_613_1599_516} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows a washing line suspended at either end by vertical rigid poles. A jacket of mass 0.7 kg is suspended in equilibrium part of the way along the line. The sections of the washing line on either side of the jacket make angles of \(35 ^ { \circ }\) and \(40 ^ { \circ }\) with the horizontal.
  1. Find the tension in the washing line on each side of the jacket.
  2. Explain why, in practice, the angles are likely to be very similar in value.
Edexcel M1 Q3
9 marks Moderate -0.3
3. In a simple model for the motion of a car, its velocity, \(\mathbf { v }\), at time \(t\) seconds, is given by $$\mathbf { v } = \left( 3 t ^ { 2 } - 2 t + 8 \right) \mathbf { i } + ( 5 t + 6 ) \mathbf { j } \mathrm { ms } ^ { - 1 }$$
  1. Calculate the speed of the car when \(t = 0\).
  2. Find the values of \(t\) for which the velocity of the car is parallel to the vector \(( \mathbf { i } + \mathbf { j } )\).
  3. Why would this model not be appropriate for large values of \(t\) ?
Edexcel M1 Q4
10 marks Standard +0.3
4. The force \(\mathbf { F } _ { \mathbf { 1 } } = ( 5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) acts at the point \(A\) on a lamina where the position vector of \(A\), relative to a fixed origin \(O\), is \(( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\).
  1. Calculate the magnitude and the sense of the moment of the force about \(O\). Another force \(\mathbf { F } _ { 2 } = ( p \mathbf { i } + q \mathbf { j } )\), acts at the point \(B\) with position vector ( \({ } ^ { - } \mathbf { i } + 4 \mathbf { j }\) ) m so that the resultant moment of the two forces, \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\), about \(O\) is zero. Given also that the moment of \(\mathbf { F } _ { 2 }\) about \(A\) is 34 Ns in a clockwise sense,
  2. find the values of \(p\) and \(q\).
Edexcel M1 Q5
13 marks Moderate -0.3
5. A car and a motorbike are at rest adjacent to one another at a set of traffic lights on a long, straight stretch of road. They set off simultaneously at time \(t = 0\). The motorcyclist accelerates uniformly at \(6 \mathrm {~ms} ^ { - 2 }\) until he reaches a speed of \(30 \mathrm {~ms} ^ { - 1 }\) which he then maintains. The car driver accelerates uniformly for 9 seconds until she reaches \(36 \mathrm {~ms} ^ { - 1 }\) and then remains at this speed.
  1. Find the acceleration of the car.
  2. Draw on the same diagram speed-time graphs to illustrate the movements of both vehicles.
  3. Find the value of \(t\) when the car again draws level with the motorcyclist.
Edexcel M1 Q6
13 marks Standard +0.3
6. Corinne and her brother Dermot are lifted by their parents onto the two ends of a rope which is slung over a large, horizontal branch. When their parents let go of them Dermot, whose mass is 54 kg , begins to descend with an acceleration of \(1 \mathrm {~ms} ^ { - 2 }\). By modelling the children as a pair of particles connected by a light inextensible string, and the branch as a smooth pulley,
  1. show that Corinne's mass is 44 kg ,
  2. calculate the tension in the rope,
  3. find the force on the branch. In a more sophisticated model, the branch is assumed to be rough.
  4. Explain what effect this would have on the initial acceleration of the children.
    (1 mark)
Edexcel M1 Q7
14 marks Standard +0.3
7. Two particles \(A\) and \(B\), of mass \(3 M \mathrm {~kg}\) and \(2 M \mathrm {~kg}\) respectively, are moving towards each other on a rough horizontal track. Just before they collide, \(A\) has speed \(3 \mathrm {~ms} ^ { - 1 }\) and \(B\) has speed \(5 \mathrm {~ms} ^ { - 1 }\). Immediately after the impact, the direction of motion of both particles has been reversed and they are both travelling at the same speed, \(v\).
  1. Show that \(v = 1 \mathrm {~ms} ^ { - 1 }\). The magnitude of the impulse exerted on \(A\) during the collision is 24 Ns.
  2. Find the value of \(M\). Given that the coefficient of friction between \(A\) and the track is 0.1 ,
  3. find the time taken from the moment of impact until \(A\) comes to rest. END
Edexcel M1 Q1
4 marks Moderate -0.8
  1. Three forces \(( - 5 \mathbf { i } + 4 p \mathbf { j } ) \mathrm { N } , ( 2 q \mathbf { i } + 3 \mathbf { j } ) \mathrm { N }\) and \(( \mathbf { i } + \mathbf { j } ) \mathrm { N }\) act on a particle \(A\) of mass 2 kg .
Given that \(A\) is in equilibrium, find the values of \(p\) and \(q\).
Edexcel M1 Q2
7 marks Moderate -0.8
2. An underground train accelerates uniformly from rest at station \(A\) to a velocity of \(24 \mathrm {~ms} ^ { - 1 }\). It maintains this speed for 84 seconds, until it decelerates uniformly to rest at station \(B\). The total journey time is 116 seconds and the magnitudes of the acceleration and deceleration are equal.
  1. Find the time it takes the train to accelerate from rest to \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Illustrate this information on a velocity-time graph.
  3. Using your graph, or otherwise, find the distance between the two stations.
Edexcel M1 Q3
8 marks Moderate -0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{60b9db45-b48e-40a1-bd22-909e11877bc3-2_442_805_1023_719} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows the forces acting on a particle, \(P\). These consist of a 20 N force to the South, a 6 N force to the East, an 18 N force \(30 ^ { \circ }\) West of North and two unknown forces \(X\) and \(Y\) which act to the North-East and North respectively. Given that \(P\) is in equilibrium,
  1. show that \(X\) has magnitude \(3 \sqrt { } 2 \mathrm {~N}\),
  2. find the exact value of \(Y\).
Edexcel M1 Q4
8 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{60b9db45-b48e-40a1-bd22-909e11877bc3-3_275_842_194_408} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows a uniform plank \(A B\) of mass 50 kg and length 5 m which overhangs a river by 2 m . When a boy of mass 20 kg stands at \(A\), his sister can walk to within 0.3 m of \(B\), at which point the plank is in limiting equilibrium.
  1. What is the mass of the girl?
  2. Find the smallest extra weight which must be placed at \(A\) to enable the girl to walk right to the end \(B\).
  3. How have you used the fact that the plank is uniform?
Edexcel M1 Q5
8 marks Moderate -0.3
5. A cricket ball of mass 0.3 kg is approaching a batsman at \({ } ^ { - } 30 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The batsman hits the ball with a 1.5 kg bat moving with velocity \(15 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\). Contact between bat and ball lasts for 0.2 seconds. Immediately after this, bat and ball move with velocities \(5 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) and \(v \mathbf { i } \mathrm {~ms} ^ { - 1 }\) respectively.
  1. Suggest a suitable model for the cricket ball.
  2. Calculate the value of \(v\).
  3. Find the magnitude of the force with which the batsman hits the ball.
Edexcel M1 Q6
10 marks Moderate -0.3
6. A boy kicks a football vertically upwards from a height of 0.6 m above the ground with a speed of \(10.5 \mathrm {~ms} ^ { - 1 }\). The ball is modelled as a particle and air resistance is ignored.
  1. Find the greatest height above the ground reached by the ball.
  2. Calculate the length of time for which the ball is more than 2 m above the ground.
Edexcel M1 Q7
11 marks Standard +0.3
7. A particle has an initial velocity of \(( \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and is accelerating uniformly in the direction \(( 2 \mathbf { i } + \mathbf { j } )\) where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors. Given that the magnitude of the acceleration is \(3 \sqrt { } 5 \mathrm {~ms} ^ { - 2 }\),
  1. show that, after \(t\) seconds, the velocity vector of the particle is $$[ ( 6 t + 1 ) \mathbf { i } + ( 3 t - 5 ) \mathbf { j } ] \mathrm { ms } ^ { - 1 }$$
  2. Using your answer to part (a), or otherwise, find the value of \(t\) for which the speed of the particle is at its minimum.
    (5 marks)
Edexcel M1 Q8
19 marks Standard +0.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{60b9db45-b48e-40a1-bd22-909e11877bc3-4_442_924_877_443} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows two particles \(A\) and \(B\), of mass \(5 M\) and \(3 M\) respectively, attached to the ends of a light inextensible string of length 4 m . The string passes over a smooth pulley which is fixed to the edge of a rough horizontal table 2 m high. Particle \(A\) lies on the table at a distance of 3 m from the pulley, whilst particle \(B\) hangs freely over the edge of the table 1 m above the ground. The coefficient of friction between \(A\) and the table is \(\frac { 3 } { 20 }\). The system is released from rest with the string taut.
  1. Show that the initial acceleration of the system is \(\frac { 9 } { 32 } \mathrm {~g} \mathrm {~ms} ^ { - 2 }\).
  2. Find, in terms of \(g\), the speed of \(A\) immediately before \(B\) hits the ground. When \(B\) hits the ground, it comes to rest and the string becomes slack.
  3. Calculate how far particle \(A\) is from the pulley when it comes to rest. END
Edexcel M1 Q1
7 marks Moderate -0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0de1908-cf67-460f-9473-b2dfded95b33-2_257_693_239_447} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a particle \(P\) of mass 4 kg on a smooth plane inclined at \(15 ^ { \circ }\) to the horizontal. \(P\) is held in equilibrium by a horizontal force, \(F\).
  1. Show that the normal reaction exerted by the plane on \(P\) is 40.6 N correct to 3 significant figures.
  2. Calculate the value of \(F\).
Edexcel M1 Q2
7 marks Moderate -0.3
2. During trials of a bullet-proof vest, a shotgun of mass 2 kg is used to fire a bullet of mass 30 g horizontally at the vest. The initial speed of the bullet is \(100 \mathrm {~ms} ^ { - 1 }\).
  1. Calculate the initial speed of recoil of the gun. The bullet hits the vest horizontally at a speed of \(80 \mathrm {~ms} ^ { - 1 }\) and is brought uniformly to rest in a distance of 2 cm .
  2. Find the magnitude of the force exerted by the vest on the bullet in bringing it to rest.
    (4 marks)
Edexcel M1 Q3
7 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0de1908-cf67-460f-9473-b2dfded95b33-2_387_460_1626_726} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows 4 points \(A , B , C\) and \(D\) arranged such that they form the corners of a square of side 2 m . Forces of \(5 \mathrm {~N} , 3 \mathrm {~N} , 2 \mathrm {~N}\) and 4 N act in the directions \(\overrightarrow { A B } , \overrightarrow { B C } , \overrightarrow { D C }\) and \(\overrightarrow { D A }\) respectively.
  1. Calculate the magnitude and sense of the resultant moment about \(A\). An additional force of magnitude \(X\) Newtons is added in the direction \(\overrightarrow { C A }\). The resultant moment of all the forces about \(D\) is now zero.
  2. Find, in the form \(k \sqrt { } 2\), the value of \(X\).
Edexcel M1 Q4
11 marks Standard +0.3
4. A lift of mass 70 kg is supported by a cable which remains taut at all times. A man of mass 90 kg gets into the lift and it begins to descend vertically from rest with constant acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate, giving your answers correct to 3 significant figures,
  1. the magnitude of the force which the lift exerts on the man,
  2. the tension in the cable. Prior to slowing down, the lift is moving at \(2 \mathrm {~ms} ^ { - 1 }\). It then uniformly decelerates until it is brought to rest.
  3. Find the impulse exerted by the cable on the lift in bringing the lift to rest.
  4. Given that it takes 2 seconds to come to rest, use your answer to part (c) to calculate the magnitude of the force exerted by the cable on the lift in bringing the lift to rest.
    (2 marks)
Edexcel M1 Q5
11 marks Moderate -0.3
5. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively. At midday a motor boat \(A\) is 6 km east of a fixed origin \(O\) and is moving with constant velocity ( \({ } ^ { - } 4 \mathbf { i } + \mathbf { j }\) ) \(\mathrm { km } \mathrm { h } ^ { - 1 }\). At the same time, another boat \(B\) is 3 km north of \(O\) and is moving with uniform velocity \(( 4 \mathbf { i } - 3 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
  1. Show that, at time \(T\) hours after midday, the position vector of \(A\) is \([ ( 6 - 4 T ) \mathbf { i } + T \mathbf { j } ] \mathrm { km }\) and find a similar expression for the position vector of \(B\) at this time.
  2. Hence show that, at time \(T\), the position vector of \(B\) relative to \(A\) is $$[ ( 8 T - 6 ) \mathbf { i } + ( 3 - 4 T ) \mathbf { j } ] \mathrm { km }$$
  3. By using your answer to part (b), or otherwise, show that the boats would collide if they continued at the same velocities and find the time at which the collision would occur.
Edexcel M1 Q6
13 marks Standard +0.3
6. A student attempts to sketch the acceleration-time graph of a parachutist who jumps from a plane at a height of 2200 m above the ground. The student assumes that the parachutist falls freely from rest under gravity until she is 240 m from the ground at which point she opens her parachute. The student makes the assumption that, at this point, the velocity of the parachutist is immediately reduced to a value which remains constant until she reaches the ground 140 seconds after she left the plane. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0de1908-cf67-460f-9473-b2dfded95b33-4_314_1013_598_383} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} The student decides to ignore air resistance and his sketch is shown in Figure 3. The value \(t _ { 1 }\) is used by the student to denote the time at which the parachute is opened. Using the model proposed by the student, calculate
  1. the speed of the parachutist immediately before she opens her parachute,
  2. the value of \(t _ { 1 }\),
  3. the speed of the parachutist after the parachute is opened.
  4. Comment on two features of the student's model which are unrealistic and say what effect taking account of these would have had on the values which you calculated in parts (a) and (b).
    (4 marks)