Questions M1 (2067 questions)

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CAIE M1 2020 March Q6
9 marks Moderate -0.3
6 On a straight horizontal test track, driverless vehicles (with no passengers) are being tested. A car of mass 1600 kg is towing a trailer of mass 700 kg along the track. The brakes are applied, resulting in a deceleration of \(12 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The braking force acts on the car only. In addition to the braking force there are constant resistance forces of 600 N on the car and of 200 N on the trailer.
  1. Find the magnitude of the force in the tow-bar.
  2. Find the braking force.
  3. At the instant when the brakes are applied, the car has speed \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At this instant the car is 17.5 m away from a stationary van, which is directly in front of the car. Show that the car hits the van at a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  4. After the collision, the van starts to move with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the car and trailer continue moving in the same direction with speed \(2 \mathrm {~ms} ^ { - 1 }\). Find the mass of the van.
CAIE M1 2020 March Q7
10 marks Standard +0.3
7 A particle moves in a straight line through the point \(O\). The displacement of the particle from \(O\) at time \(t \mathrm {~s}\) is \(s \mathrm {~m}\), where $$\begin{array} { l l } s = t ^ { 2 } - 3 t + 2 & \text { for } 0 \leqslant t \leqslant 6 , \\ s = \frac { 24 } { t } - \frac { t ^ { 2 } } { 4 } + 25 & \text { for } t \geqslant 6 . \end{array}$$
  1. Find the value of \(t\) when the particle is instantaneously at rest during the first 6 seconds of its motion.
    At \(t = 6\), the particle hits a barrier at a point \(P\) and rebounds.
  2. Find the velocity with which the particle arrives at \(P\) and also the velocity with which the particle leaves \(P\).
  3. Find the total distance travelled by the particle in the first 10 seconds of its motion.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE M1 2021 March Q1
3 marks Moderate -0.8
1 Two particles \(P\) and \(Q\) of masses 0.2 kg and 0.3 kg respectively are free to move in a horizontal straight line on a smooth horizontal plane. \(P\) is projected towards \(Q\) with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same instant \(Q\) is projected towards \(P\) with speed \(1 \mathrm {~ms} ^ { - 1 } . Q\) comes to rest in the resulting collision. Find the speed of \(P\) after the collision.
CAIE M1 2021 March Q2
6 marks Moderate -0.3
2 A car of mass 1400 kg is travelling at constant speed up a straight hill inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = 0.1\). There is a constant resistance force of magnitude 600 N . The power of the car's engine is 22500 W .
  1. Show that the speed of the car is \(11.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    The car, moving with speed \(11.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), comes to a section of the hill which is inclined at \(2 ^ { \circ }\) to the horizontal.
  2. Given that the power and resistance force do not change, find the initial acceleration of the car up this section of the hill.
CAIE M1 2021 March Q3
5 marks Moderate -0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-05_518_616_255_767} A particle \(Q\) of mass 0.2 kg is held in equilibrium by two light inextensible strings \(P Q\) and \(Q R . P\) is a fixed point on a vertical wall and \(R\) is a fixed point on a horizontal floor. The angles which strings \(P Q\) and \(Q R\) make with the horizontal are \(60 ^ { \circ }\) and \(30 ^ { \circ }\) respectively (see diagram). Find the tensions in the two strings.
CAIE M1 2021 March Q4
6 marks Moderate -0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-06_661_1529_260_306} An elevator moves vertically, supported by a cable. The diagram shows a velocity-time graph which models the motion of the elevator. The graph consists of 7 straight line segments. The elevator accelerates upwards from rest to a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 1.5 s and then travels at this speed for 4.5 s , before decelerating to rest over a period of 1 s . The elevator then remains at rest for 6 s , before accelerating to a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards over a period of 2 s . The elevator travels at this speed for a period of 5 s , before decelerating to rest over a period of 1.5 s .
  1. Find the acceleration of the elevator during the first 1.5 s .
  2. Given that the elevator starts and finishes its journey on the ground floor, find \(V\).
  3. The combined weight of the elevator and passengers on its upward journey is 1500 kg . Assuming that there is no resistance to motion, find the tension in the elevator cable on its upward journey when the elevator is decelerating.
CAIE M1 2021 March Q5
9 marks Moderate -0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-08_286_661_260_742} A block of mass 5 kg is being pulled along a rough horizontal floor by a force of magnitude \(X \mathrm {~N}\) acting at \(30 ^ { \circ }\) above the horizontal (see diagram). The block starts from rest and travels 2 m in the first 5 s of its motion.
  1. Find the acceleration of the block.
  2. Given that the coefficient of friction between the block and the floor is 0.4 , find \(X\).
    The block is now placed on a part of the floor where the coefficient of friction between the block and the floor has a different value. The value of \(X\) is changed to 25, and the block is now in limiting equilibrium.
  3. Find the value of the coefficient of friction between the block and this part of the floor.
CAIE M1 2021 March Q6
11 marks Standard +0.3
6 A particle moves in a straight line. It starts from rest from a fixed point \(O\) on the line. Its velocity at time \(t \mathrm {~s}\) after leaving \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = t ^ { 2 } - 8 t ^ { \frac { 3 } { 2 } } + 10 t\).
  1. Find the displacement of the particle from \(O\) when \(t = 1\).
  2. Show that the minimum velocity of the particle is \(- 125 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2021 March Q7
10 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-12_439_1095_258_525} Two particles \(P\) and \(Q\) of masses 0.5 kg and \(m \mathrm {~kg}\) respectively are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the top of two inclined planes. The particles are initially at rest with \(P\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal and \(Q\) on a plane inclined at \(45 ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes. A force of magnitude 0.8 N is applied to \(P\) acting down the plane, causing \(P\) to move down the plane (see diagram).
  1. It is given that \(m = 0.3\), and that the plane on which \(Q\) rests is smooth. Find the tension in the string.
  2. It is given instead that the plane on which \(Q\) rests is rough, and that after each particle has moved a distance of 1 m , their speed is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done against friction in this part of the motion is 0.5 J . Use an energy method to find the value of \(m\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE M1 2022 March Q1
4 marks Moderate -0.8
1 A crane is used to raise a block of mass 600 kg vertically upwards at a constant speed through a height of 15 m . There is a resistance to the motion of the block, which the crane does 10000 J of work to overcome.
  1. Find the total work done by the crane.
  2. Given that the average power exerted by the crane is 12.5 kW , find the total time for which the block is in motion.
CAIE M1 2022 March Q2
5 marks Moderate -0.8
2 A particle \(P\) is projected vertically upwards from horizontal ground with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 } . P\) reaches a maximum height of 20 m above the ground.
  1. Find the value of \(u\).
  2. Find the total time for which \(P\) is at least 15 m above the ground.
CAIE M1 2022 March Q3
5 marks Standard +0.3
3 A car of mass \(m \mathrm {~kg}\) is towing a trailer of mass 300 kg down a straight hill inclined at \(3 ^ { \circ }\) to the horizontal at a constant speed. There are resistance forces on the car and on the trailer, and the total work done against the resistance forces in a distance of 50 m is 40000 J . The engine of the car is doing no work and the tow-bar is light and rigid.
  1. Find the value of \(m\).
    The resistance force on the trailer is 200 N .
  2. Find the tension in the tow-bar between the car and the trailer.
CAIE M1 2022 March Q4
6 marks Moderate -0.3
4 The total mass of a cyclist and her bicycle is 70 kg . The cyclist is riding with constant power of 180 W up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\). At an instant when the cyclist's speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), her acceleration is \(- 0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is a constant resistance to motion of magnitude \(F \mathrm {~N}\).
  1. Find the value of \(F\).
  2. Find the steady speed that the cyclist could maintain up the hill when working at this power. [2]
CAIE M1 2022 March Q5
7 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{19a41291-2692-48f4-86af-bb4930353959-08_645_611_258_767} Four coplanar forces act at a point. The magnitudes of the forces are \(10 \mathrm {~N} , F \mathrm {~N} , G \mathrm {~N}\) and \(2 F \mathrm {~N}\). The directions of the forces are as shown in the diagram.
  1. Given that the forces are in equilibrium, find the values of \(F\) and \(G\).
  2. Given instead that \(F = 3\), find the value of \(G\) for which the resultant of the forces is perpendicular to the 10 N force.
CAIE M1 2022 March Q6
11 marks Standard +0.8
6 A cyclist starts from rest at a fixed point \(O\) and moves in a straight line, before coming to rest \(k\) seconds later. The acceleration of the cyclist at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 2 t ^ { - \frac { 1 } { 2 } } - \frac { 3 } { 5 } t ^ { \frac { 1 } { 2 } }\) for \(0 < t \leqslant k\).
  1. Find the value of \(k\).
  2. Find the maximum speed of the cyclist.
  3. Find an expression for the displacement from \(O\) in terms of \(t\). Hence find the total distance travelled by the cyclist from the time at which she reaches her maximum speed until she comes to rest.
CAIE M1 2022 March Q7
12 marks Challenging +1.2
7 A bead, \(A\), of mass 0.1 kg is threaded on a long straight rigid wire which is inclined at \(\sin ^ { - 1 } \left( \frac { 7 } { 25 } \right)\) to the horizontal. \(A\) is released from rest and moves down the wire. The coefficient of friction between \(A\) and the wire is \(\mu\). When \(A\) has travelled 0.45 m down the wire, its speed is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(\mu = 0.25\).
    Another bead, \(B\), of mass 0.5 kg is also threaded on the wire. At the point where \(A\) has travelled 0.45 m down the wire, it hits \(B\) which is instantaneously at rest on the wire. \(A\) is brought to instantaneous rest in the collision. The coefficient of friction between \(B\) and the wire is 0.275 .
  2. Find the time from when the collision occurs until \(A\) collides with \(B\) again.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE M1 2020 November Q1
3 marks Easy -1.3
1 Two particles \(P\) and \(Q\), of masses 0.2 kg and 0.5 kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected towards \(Q\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Write down the momentum of \(P\).
  2. After the collision \(P\) continues to move in the same direction with speed \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(Q\) after the collision.
CAIE M1 2020 November Q2
5 marks Moderate -0.3
2 A car of mass 1800 kg is travelling along a straight horizontal road. The power of the car's engine is constant. There is a constant resistance to motion of 650 N .
  1. Find the power of the car's engine, given that the car's acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when its speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the steady speed which the car can maintain with the engine working at this power.
CAIE M1 2020 November Q3
5 marks Moderate -0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-05_446_851_260_646} A block of mass \(m \mathrm {~kg}\) is held in equilibrium below a horizontal ceiling by two strings, as shown in the diagram. One of the strings is inclined at \(45 ^ { \circ }\) to the horizontal and the tension in this string is \(T \mathrm {~N}\). The other string is inclined at \(60 ^ { \circ }\) to the horizontal and the tension in this string is 20 N . Find \(T\) and \(m\).
CAIE M1 2020 November Q4
5 marks Moderate -0.5
4
[diagram]
The diagram shows a velocity-time graph which models the motion of a car. The graph consists of four straight line segments. The car accelerates at a constant rate of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) from rest to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of \(T \mathrm {~s}\). It then decelerates at a constant rate for 5 seconds before travelling at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 27.5 s . The car then decelerates to rest at a constant rate over a period of 5 s .
  1. Find \(T\).
  2. Given that the distance travelled up to the point at which the car begins to move with constant speed is one third of the total distance travelled, find \(V\).
CAIE M1 2020 November Q5
8 marks Standard +0.3
5 A particle is projected vertically upwards with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) alongside a building of height \(h \mathrm {~m}\).
  1. Given that the particle is above the level of the top of the building for 4 s , find \(h\).
  2. One second after the first particle is projected, a second particle is projected vertically upwards from the top of the building with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Denoting the time after projection of the first particle by \(t \mathrm {~s}\), find the value of \(t\) for which the two particles are at the same height above the ground.
CAIE M1 2020 November Q6
8 marks Standard +0.3
6 A block of mass 5 kg is placed on a plane inclined at \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the plane is \(\mu\).
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-10_424_709_392_760} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
    \end{figure} When a force of magnitude 40 N is applied to the block, acting up the plane parallel to a line of greatest slope, the block begins to slide up the plane (see Fig. 6.1). Show that \(\mu < \frac { 1 } { 5 } \sqrt { 3 }\).
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-11_422_727_264_749} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
    \end{figure} When a force of magnitude 40 N is applied horizontally, in a vertical plane containing a line of greatest slope, the block does not move (see Fig. 6.2). Show that, correct to 3 decimal places, the least possible value of \(\mu\) is 0.152 .
CAIE M1 2020 November Q7
7 marks Moderate -0.3
7 A particle \(P\) moves in a straight line, starting from a point \(O\) with velocity \(1.72 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the particle, \(t \mathrm {~s}\) after leaving \(O\), is given by \(a = 0.1 t ^ { \frac { 3 } { 2 } }\).
  1. Find the value of \(t\) when the velocity of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the displacement of \(P\) from \(O\) when \(t = 2\), giving your answer correct to 2 decimal places. [3]
CAIE M1 2020 November Q8
9 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-14_388_1216_264_461} Two particles \(A\) and \(B\), of masses 0.3 kg and 0.5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to a horizontal plane and to the top of an inclined plane. The particles are initially at rest with \(A\) on the horizontal plane and \(B\) on the inclined plane, which makes an angle of \(30 ^ { \circ }\) with the horizontal. The string is taut and \(B\) can move on a line of greatest slope of the inclined plane. A force of magnitude 3.5 N is applied to \(B\) acting down the plane (see diagram).
  1. Given that both planes are smooth, find the tension in the string and the acceleration of \(B\).
  2. It is given instead that the two planes are rough. When each particle has moved a distance of 0.6 m from rest, the total amount of work done against friction is 1.1 J . Use an energy method to find the speed of \(B\) when it has moved this distance down the plane. [You should assume that the string is sufficiently long so that \(A\) does not hit the pulley when it moves 0.6 m .]
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE M1 2020 November Q1
3 marks Easy -1.2
1 A particle \(P\) is projected vertically upwards with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the ground. \(P\) reaches its greatest height after 3 s .
  1. Find \(v\).
  2. Find the greatest height of \(P\) above the ground.