Questions M1 (2067 questions)

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CAIE M1 2018 June Q5
7 marks Standard +0.8
5 A particle of mass 20 kg is on a rough plane inclined at an angle of \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force of magnitude \(P \mathrm {~N}\) acting on the particle, in a direction parallel to a line of greatest slope of the plane. The greatest possible value of \(P\) is twice the least possible value of \(P\). Find the value of the coefficient of friction between the particle and the plane.
CAIE M1 2018 June Q6
8 marks Standard +0.3
6 A particle \(P\) moves in a straight line passing through a point \(O\). At time \(t \mathrm {~s}\), the acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), of \(P\) is given by \(a = 6 - 0.24 t\). The particle comes to instantaneous rest at time \(t = 20\).
  1. Find the value of \(t\) at which the particle is again at instantaneous rest.
  2. Find the distance the particle travels between the times of instantaneous rest.
CAIE M1 2018 June Q7
14 marks Standard +0.3
7
[diagram]
As shown in the diagram, a particle \(A\) of mass 1.6 kg lies on a horizontal plane and a particle \(B\) of mass 2.4 kg lies on a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The particles are connected by a light inextensible string which passes over a small smooth pulley \(P\) fixed at the top of the inclined plane. The distance \(A P\) is 2.5 m and the distance of \(B\) from the bottom of the inclined plane is 1 m . There is a barrier at the bottom of the inclined plane preventing any further motion of \(B\). The part \(B P\) of the string is parallel to a line of greatest slope of the inclined plane. The particles are released from rest with both parts of the string taut.
  1. Given that both planes are smooth, find the acceleration of \(A\) and the tension in the string.
  2. It is given instead that the horizontal plane is rough and that the coefficient of friction between \(A\) and the horizontal plane is 0.2 . The inclined plane is smooth. Find the total distance travelled by \(A\).
    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 2019 June Q1
3 marks Moderate -0.5
1 \includegraphics[max width=\textwidth, alt={}, center]{555678d3-f37d-4822-a005-de8c6094dc50-03_563_503_262_820} Given that \(\tan \alpha = \frac { 12 } { 5 }\) and \(\tan \theta = \frac { 4 } { 3 }\), show that the coplanar forces shown in the diagram are in equilibrium.
CAIE M1 2019 June Q2
7 marks Moderate -0.5
2 A particle \(P\) is projected vertically upwards with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 3 m above horizontal ground.
  1. Find the time taken for \(P\) to reach its greatest height.
  2. Find the length of time for which \(P\) is higher than 23 m above the ground.
  3. \(P\) is higher than \(h \mathrm {~m}\) above the ground for 1 second. Find \(h\).
CAIE M1 2019 June Q3
7 marks Standard +0.3
3 A lorry has mass 12000 kg .
  1. The lorry moves at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.08\). At this speed, the magnitude of the resistance to motion on the lorry is 1500 N . Show that the power of the lorry's engine is 55.5 kW .
    When the speed of the lorry is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the magnitude of the resistance to motion is \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant.
  2. Show that \(k = 60\).
  3. The lorry now moves at a constant speed on a straight level road. Given that its engine is still working at 55.5 kW , find the lorry's speed.
CAIE M1 2019 June Q4
9 marks Standard +0.3
4 A particle of mass 1.3 kg rests on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 12 } { 5 }\). The coefficient of friction between the particle and the plane is \(\mu\).
  1. A force of magnitude 20 N parallel to a line of greatest slope of the plane is applied to the particle and the particle is on the point of moving up the plane. Show that \(\mu = 1.6\).
    The force of magnitude 20 N is now removed.
  2. Find the acceleration of the particle.
  3. Find the work done against friction during the first 2 s of motion.
CAIE M1 2019 June Q5
10 marks Standard +0.3
5 A particle \(P\) moves in a straight line from a fixed point \(O\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) at time \(t \mathrm {~s}\) is given by $$v = t ^ { 2 } - 8 t + 12 \quad \text { for } 0 \leqslant t \leqslant 8$$
  1. Find the minimum velocity of \(P\).
  2. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 8\). \includegraphics[max width=\textwidth, alt={}, center]{555678d3-f37d-4822-a005-de8c6094dc50-12_401_1102_260_520} Two particles \(A\) and \(B\), of masses 0.4 kg and 0.2 kg respectively, are connected by a light inextensible string. Particle \(A\) is held on a smooth plane inclined at an angle of \(\theta ^ { \circ }\) to the horizontal. The string passes over a small smooth pulley \(P\) fixed at the top of the plane, and \(B\) hangs freely 0.5 m above horizontal ground (see diagram). The particles are released from rest with both sections of the string taut.
  3. Given that the system is in equilibrium, find \(\theta\).
  4. It is given instead that \(\theta = 20\). In the subsequent motion particle \(A\) does not reach \(P\) and \(B\) remains at rest after reaching the ground.
    1. Find the tension in the string and the acceleration of the system.
    2. Find the speed of \(A\) at the instant \(B\) reaches the ground.
    3. Use an energy method to find the total distance \(A\) moves up the plane before coming to instantaneous rest.
      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 2019 June Q1
5 marks Easy -1.2
1 A bus moves in a straight line between two bus stops. The bus starts from rest and accelerates at \(2.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 5 s . The bus then travels for 24 s at constant speed and finally slows down, with a constant deceleration, stopping in a further 6 s . Sketch a velocity-time graph for the motion and hence find the distance between the two bus stops.
CAIE M1 2019 June Q2
6 marks Moderate -0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{539be201-7bfc-4ba0-8378-c7aec4473ac7-03_577_691_262_724} Coplanar forces of magnitudes \(12 \mathrm {~N} , 24 \mathrm {~N}\) and 30 N act at a point in the directions shown in the diagram.
  1. Find the components of the resultant of the three forces in the \(x\)-direction and in the \(y\)-direction. Component in \(x\)-direction \(\_\_\_\_\) Component in \(y\)-direction. \(\_\_\_\_\)
  2. Hence find the direction of the resultant.
CAIE M1 2019 June Q3
7 marks Standard +0.3
3 A car of mass 1400 kg is travelling up a hill inclined at an angle of \(4 ^ { \circ }\) to the horizontal. There is a constant resistance to motion of magnitude 1550 N acting on the car.
  1. Given that the engine of the car is working at 30 kW , find the speed of the car at an instant when its acceleration is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. The greatest possible constant speed at which the car can travel up the hill is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the maximum possible power of the engine. \includegraphics[max width=\textwidth, alt={}, center]{539be201-7bfc-4ba0-8378-c7aec4473ac7-06_643_419_255_863} Two particles \(A\) and \(B\), of masses 1.3 kg and 0.7 kg respectively, are connected by a light inextensible string which passes over a smooth fixed pulley. Particle \(A\) is 1.75 m above the floor and particle \(B\) is 1 m above the floor (see diagram). The system is released from rest with the string taut, and the particles move vertically. When the particles are at the same height the string breaks.
CAIE M1 2019 June Q5
12 marks Standard +0.3
5 A particle of mass 18 kg is on a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The particle is projected up a line of greatest slope of the plane with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Given that the plane is smooth, use an energy method to find the distance the particle moves up the plane before coming to instantaneous rest.
  2. Given instead that the plane is rough and the coefficient of friction between the particle and the plane is 0.25 , find the speed of the particle as it returns to its starting point.
CAIE M1 2019 June Q6
10 marks Standard +0.3
6 A particle \(P\) moves in a straight line. The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of \(P\) at time \(t \mathrm {~s}\) is given by \(a = 6 t - 12\). The displacement of \(P\) from a fixed point \(O\) on the line is \(s \mathrm {~m}\). It is given that \(s = 5\) when \(t = 1\) and \(s = 1\) when \(t = 3\).
  1. Show that \(s = t ^ { 3 } - 6 t ^ { 2 } + p t + q\), where \(p\) and \(q\) are constants to be found.
  2. Find the values of \(t\) when \(P\) is at instantaneous rest.
  3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\).
    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 2016 March Q1
3 marks Standard +0.3
1 A cyclist has mass 85 kg and rides a bicycle of mass 20 kg . The cyclist rides along a horizontal road against a total resistance force of 40 N . Find the total work done by the cyclist in increasing his speed from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while travelling a distance of 50 m .
CAIE M1 2016 March Q2
5 marks Moderate -0.5
2 A constant resistance of magnitude 1350 N acts on a car of mass 1200 kg .
  1. The car is moving along a straight level road at a constant speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find, in kW , the rate at which the engine of the car is working.
  2. The car travels at a constant speed up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.1\), with the engine working at 76.5 kW . Find this speed.
CAIE M1 2016 March Q3
7 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{9a99a969-db40-4d29-bb37-ea7ac15cdc2d-2_476_659_897_742} Coplanar forces of magnitudes \(50 \mathrm {~N} , 40 \mathrm {~N}\) and 30 N act at a point \(O\) in the directions shown in the diagram, where \(\tan \alpha = \frac { 7 } { 24 }\).
  1. Find the magnitude and direction of the resultant of the three forces.
  2. The force of magnitude 50 N is replaced by a force of magnitude \(P \mathrm {~N}\) acting in the same direction. The resultant of the three forces now acts in the positive \(x\)-direction. Find the value of \(P\).
CAIE M1 2016 March Q4
7 marks Standard +0.3
4 A particle \(P\) of mass 0.8 kg is placed on a rough horizontal table. The coefficient of friction between \(P\) and the table is \(\mu\). A force of magnitude 5 N , acting upwards at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\), is applied to \(P\). The particle is on the point of sliding on the table.
  1. Find the value of \(\mu\).
  2. The magnitude of the force acting on \(P\) is increased to 10 N , with the direction of the force remaining the same. Find the acceleration of \(P\).
CAIE M1 2016 March Q5
7 marks Standard +0.3
5 A car of mass 1200 kg is pulling a trailer of mass 800 kg up a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.1\). The system of the car and the trailer is modelled as two particles connected by a light inextensible cable. The driving force of the car's engine is 2500 N and the resistances to the car and trailer are 100 N and 150 N respectively.
  1. Find the acceleration of the system and the tension in the cable.
  2. When the car and trailer are travelling at a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the driving force becomes zero. The cable remains taut. Find the time, in seconds, before the system comes to rest.
CAIE M1 2016 March Q6
10 marks Standard +0.3
6 Two particles \(A\) and \(B\), of masses 0.8 kg and 0.2 kg respectively, are connected by a light inextensible string. Particle \(A\) is placed on a horizontal surface. The string passes over a small smooth pulley \(P\) fixed at the edge of the surface, and \(B\) hangs freely. The horizontal section of the string, \(A P\), is of length 2.5 m . The particles are released from rest with both sections of the string taut.
  1. Given that the surface is smooth, find the time taken for \(A\) to reach the pulley.
  2. Given instead that the surface is rough and the coefficient of friction between \(A\) and the surface is 0.1 , find the speed of \(A\) immediately before it reaches the pulley. \(7 \quad\) A particle \(P\) moves in a straight line. The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by $$\begin{array} { l l } v = 5 t ( t - 2 ) & \text { for } 0 \leqslant t \leqslant 4 \\ v = k & \text { for } 4 \leqslant t \leqslant 14 \\ v = 68 - 2 t & \text { for } 14 \leqslant t \leqslant 20 \end{array}$$ where \(k\) is a constant.
  3. Find \(k\).
  4. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 20\).
  5. Find the set of values of \(t\) for which the acceleration of \(P\) is positive.
  6. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 20\).
CAIE M1 2002 November Q1
3 marks Moderate -0.3
1 A car of mass 1000 kg travels along a horizontal straight road with its engine working at a constant rate of 20 kW . The resistance to motion of the car is 600 N . Find the acceleration of the car at an instant when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2002 November Q2
4 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{fcd2b219-d9b4-4972-b8fe-25cf543b9054-2_649_1244_482_452} A man runs in a straight line. He passes through a fixed point \(A\) with constant velocity \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t = 0\). At time \(t \mathrm {~s}\) his velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The diagram shows the graph of \(v\) against \(t\) for the period \(0 \leqslant t \leqslant 40\).
  1. Show that the man runs more than 154 m in the first 24 s .
  2. Given that the man runs 20 m in the interval \(20 \leqslant t \leqslant 24\), find how far he is from \(A\) when \(t = 40\).
CAIE M1 2002 November Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{fcd2b219-d9b4-4972-b8fe-25cf543b9054-2_438_621_1676_762} A light inextensible string has its ends attached to two fixed points \(A\) and \(B\), with \(A\) vertically above \(B\). A smooth ring \(R\), of mass 0.8 kg , is threaded on the string and is pulled by a horizontal force of magnitude \(X\) newtons. The sections \(A R\) and \(B R\) of the string make angles of \(50 ^ { \circ }\) and \(20 ^ { \circ }\) respectively with the horizontal, as shown in the diagram. The ring rests in equilibrium with the string taut. Find
  1. the tension in the string,
  2. the value of \(X\).
CAIE M1 2002 November Q4
8 marks Standard +0.3
4 Two particles \(A\) and \(B\) are projected vertically upwards from horizontal ground at the same instant. The speeds of projection of \(A\) and \(B\) are \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Find
  1. the difference in the heights of \(A\) and \(B\) when \(A\) is at its maximum height,
  2. the height of \(A\) above the ground when \(B\) is 0.9 m above \(A\).
CAIE M1 2002 November Q5
8 marks Standard +0.3
5 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fcd2b219-d9b4-4972-b8fe-25cf543b9054-3_245_335_580_906} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A force, whose direction is upwards parallel to a line of greatest slope of a plane inclined at \(35 ^ { \circ }\) to the horizontal, acts on a box of mass 15 kg which is at rest on the plane. The normal component of the contact force on the box has magnitude \(R\) newtons (see Fig. 1).
  1. Show that \(R = 123\), correct to 3 significant figures. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fcd2b219-d9b4-4972-b8fe-25cf543b9054-3_369_1045_1247_555} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} When the force parallel to the plane acting on the box has magnitude \(X\) newtons the box is about to move down the plane, and when this force has magnitude \(5 X\) newtons the box is about to move up the plane (see Fig. 2).
  2. Find the value of \(X\) and the coefficient of friction between the box and the plane.
  3. A particle \(P\) of mass 1.2 kg is released from rest at the top of a slope and starts to move. The slope has length 4 m and is inclined at \(25 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the slope is \(\frac { 1 } { 4 }\). Find
    1. the frictional component of the contact force on \(P\),
    2. the acceleration of \(P\),
    3. the speed with which \(P\) reaches the bottom of the slope.
    4. After reaching the bottom of the slope, \(P\) moves freely under gravity and subsequently hits a horizontal floor which is 3 m below the bottom of the slope.
      (a) Find the loss in gravitational potential energy of \(P\) during its motion from the bottom of the slope until it hits the floor.
      (b) Find the speed with which \(P\) hits the floor.
      [0pt] [1]
      [0pt] [3] \(7 \quad\) A particle \(P\) starts to move from a point \(O\) and travels in a straight line. At time \(t\) s after \(P\) starts to move its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.12 t - 0.0006 t ^ { 2 }\).
      1. Verify that \(P\) comes to instantaneous rest when \(t = 200\), and find the acceleration with which it starts to return towards \(O\).
      2. Find the maximum speed of \(P\) for \(0 \leqslant t \leqslant 200\).
      3. Find the displacement of \(P\) from \(O\) when \(t = 200\).
      4. Find the value of \(t\) when \(P\) reaches \(O\) again.
CAIE M1 2003 November Q1
4 marks Moderate -0.5
1 A motorcycle of mass 100 kg is travelling on a horizontal straight road. Its engine is working at a rate of 8 kW . At an instant when the speed of the motorcycle is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find, at this instant,
  1. the force produced by the engine,
  2. the resistance to motion of the motorcycle.