Questions — CAIE M1 (732 questions)

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CAIE M1 2017 March Q1
1 A particle of mass 0.4 kg is projected with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal.
  1. Find the initial kinetic energy of the particle.
  2. Use an energy method to find the distance the particle moves up the plane before coming to instantaneous rest.
    \includegraphics[max width=\textwidth, alt={}, center]{f3ddfde8-b678-4c3a-a9a6-d984c2c38bd9-03_266_874_260_632} A particle \(P\) of mass 1.6 kg is suspended in equilibrium by two light inextensible strings attached to points \(A\) and \(B\). The strings make angles of \(20 ^ { \circ }\) and \(40 ^ { \circ }\) respectively with the horizontal (see diagram). Find the tensions in the two strings.
    \includegraphics[max width=\textwidth, alt={}, center]{f3ddfde8-b678-4c3a-a9a6-d984c2c38bd9-04_286_664_251_737} A particle of mass 0.6 kg is placed on a rough plane which is inclined at an angle of \(21 ^ { \circ }\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(P \mathrm {~N}\) acting parallel to a line of greatest slope of the plane, as shown in the diagram. The coefficient of friction between the particle and the plane is 0.3 . Show that the least possible value of \(P\) is 0.470 , correct to 3 significant figures, and find the greatest possible value of \(P\).
CAIE M1 2017 March Q4
4 A car of mass 900 kg is moving on a straight horizontal road \(A B C D\). There is a constant resistance of magnitude 800 N in the sections \(A B\) and \(B C\), and a constant resistance of magnitude \(R \mathrm {~N}\) in the section \(C D\). The power of the car's engine is a constant 36 kW .
  1. The car moves from \(A\) to \(B\) at a constant speed in 120 s . Find the speed of the car and the distance \(A B\).
    The car's engine is switched off at \(B\).
  2. The distance \(B C\) is 450 m . Find the speed of the car at \(C\).
  3. The car comes to rest at \(D\). The distance \(A D\) is 6637.5 m . Find the deceleration of the car and the value of \(R\).
CAIE M1 2017 March Q5
5 A particle \(P\) moves in a straight line starting from a point \(O\) and comes to rest 35 s later. At time \(t \mathrm {~s}\) after leaving \(O\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is given by $$\begin{array} { l l } v = \frac { 4 } { 5 } t ^ { 2 } & 0 \leqslant t \leqslant 5
v = 2 t + 10 & 5 \leqslant t \leqslant 15
v = a + b t ^ { 2 } & 15 \leqslant t \leqslant 35 \end{array}$$ where \(a\) and \(b\) are constants such that \(a > 0\) and \(b < 0\).
  1. Show that the values of \(a\) and \(b\) are 49 and - 0.04 respectively.
  2. Sketch the velocity-time graph.
    \includegraphics[max width=\textwidth, alt={}, center]{f3ddfde8-b678-4c3a-a9a6-d984c2c38bd9-09_689_1323_349_452}
  3. Find the total distance travelled by \(P\) during the 35 s .
    \includegraphics[max width=\textwidth, alt={}, center]{f3ddfde8-b678-4c3a-a9a6-d984c2c38bd9-10_487_506_260_817} Two particles of masses 1.2 kg and 0.8 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The particles hang vertically. The system is released from rest with both particles 0.64 m above the floor (see diagram). In the subsequent motion the 0.8 kg particle does not reach the pulley.
  4. Show that the acceleration of the particles is \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the tension in the string.
  5. Find the total distance travelled by the 0.8 kg particle during the first second after the particles are released.
CAIE M1 2019 March Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{06df8c0d-dd38-4e3b-b1a4-72120a81050e-02_415_623_255_760} A small ring \(P\) of mass 0.03 kg is threaded on a rough vertical rod. A light inextensible string is attached to the ring and is pulled upwards at an angle of \(15 ^ { \circ }\) to the horizontal. The tension in the string is 2.5 N (see diagram). The ring is in limiting equilibrium and on the point of sliding up the rod. Find the coefficient of friction between the ring and the rod.
CAIE M1 2019 March Q2
2 A particle is projected vertically upwards with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on horizontal ground.
  1. Show that the maximum height above the ground reached by the particle is 45 m .
  2. Find the time that it takes for the particle to reach a height of 33.75 m above the ground for the first time. Find also the speed of the particle at this time.
    \includegraphics[max width=\textwidth, alt={}, center]{06df8c0d-dd38-4e3b-b1a4-72120a81050e-04_645_661_260_740} Four coplanar forces of magnitudes \(F \mathrm {~N} , 5 \mathrm {~N} , 25 \mathrm {~N}\) and 15 N are acting at a point \(P\) in the directions shown in the diagram. Given that the forces are in equilibrium, find the values of \(F\) and \(\alpha\).
CAIE M1 2019 March Q4
4 A car of mass 1500 kg is pulling a trailer of mass 300 kg along a straight horizontal road at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The system of the car and trailer is modelled as two particles, connected by a light rigid horizontal rod. The power of the car's engine is 6000 W . There are constant resistances to motion of \(R \mathrm {~N}\) on the car and 80 N on the trailer.
  1. Find the value of \(R\).
    The power of the car's engine is increased to 12500 W . The resistance forces do not change.
  2. Find the acceleration of the car and trailer and the tension in the rod at an instant when the speed of the car is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2019 March Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{06df8c0d-dd38-4e3b-b1a4-72120a81050e-08_759_1447_260_349} The velocity of a particle moving in a straight line is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds after leaving a fixed point \(O\). The diagram shows a velocity-time graph which models the motion of the particle from \(t = 0\) to \(t = 16\). The graph consists of five straight line segments. The acceleration of the particle from \(t = 0\) to \(t = 3\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The velocity of the particle at \(t = 5\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it comes to instantaneous rest at \(t = 8\). The particle then comes to rest again at \(t = 16\). The minimum velocity of the particle is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the distance travelled by the particle in the first 8 s of its motion.
  2. Given that when the particle comes to rest at \(t = 16\) its displacement from \(O\) is 32 m , find the value of \(V\).
CAIE M1 2019 March Q6
6 A particle moves in a straight line. It starts from rest at a fixed point \(O\) on the line. Its acceleration at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.4 t ^ { 3 } - 4.8 t ^ { \frac { 1 } { 2 } }\).
  1. Show that, in the subsequent motion, the acceleration of the particle when it comes to instantaneous rest is \(16 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the displacement of the particle from \(O\) at \(t = 5\).
    \includegraphics[max width=\textwidth, alt={}, center]{06df8c0d-dd38-4e3b-b1a4-72120a81050e-12_554_878_260_635} The diagram shows the vertical cross-section \(P Q R\) of a slide. The part \(P Q\) is a straight line of length 8 m inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.8\). The straight part \(P Q\) is tangential to the curved part \(Q R\), and \(R\) is \(h \mathrm {~m}\) above the level of \(P\). The straight part \(P Q\) of the slide is rough and the curved part \(Q R\) is smooth. A particle of mass 0.25 kg is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(P\) towards \(Q\) and comes to rest at \(R\). The coefficient of friction between the particle and \(P Q\) is 0.5 .
  3. Find the work done by the friction force during the motion of the particle from \(P\) to \(Q\).
  4. Hence find the speed of the particle at \(Q\).
  5. Find the value of \(h\).
    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 2002 November Q1
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
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
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
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
4 marks
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
    (a) the frictional component of the contact force on \(P\),
    (b) the acceleration of \(P\),
    (c) 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 \(v = 0.12 t - 0.0006 t ^ { 2 }\).
  5. Verify that \(P\) comes to instantaneous rest when \(t = 200\), and find the acceleration with which it starts to return towards \(O\).
  6. Find the maximum speed of \(P\) for \(0 \leqslant t \leqslant 200\).
  7. Find the displacement of \(P\) from \(O\) when \(t = 200\).
  8. Find the value of \(t\) when \(P\) reaches \(O\) again.
CAIE M1 2003 November Q1
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.
CAIE M1 2003 November Q2
2 A stone is released from rest and falls freely under gravity. Find
  1. the speed of the stone after 2 s ,
  2. the time taken for the stone to fall a distance of 45 m from its initial position,
  3. the distance fallen by the stone from the instant when its speed is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to the instant when its speed is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2003 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-2_143_611_1050_769} A crate of mass 3 kg is pulled at constant speed along a horizontal floor. The pulling force has magnitude 25 N and acts at an angle of \(15 ^ { \circ }\) to the horizontal, as shown in the diagram. Find
  1. the work done by the pulling force in moving the crate a distance of 2 m ,
  2. the normal component of the contact force on the crate.
CAIE M1 2003 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-2_227_586_1631_781} The diagram shows a vertical cross-section of a surface. \(A\) and \(B\) are two points on the cross-section. A particle of mass 0.15 kg is released from rest at \(A\).
  1. Assuming that the particle reaches \(B\) with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and that there are no resistances to motion, find the height of \(A\) above \(B\).
  2. Assuming instead that the particle reaches \(B\) with a speed of \(6 \mathrm {~ms} ^ { - 1 }\) and that the height of \(A\) above \(B\) is 4 m , find the work done against the resistances to motion.
CAIE M1 2003 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-3_300_792_274_680} Particles \(A\) and \(B\), of masses 0.4 kg and 0.1 kg respectively, are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a horizontal table with the string passing over a smooth pulley at the edge of the table. Particle \(B\) hangs vertically below the pulley (see diagram). The system is released from rest. In the subsequent motion a constant frictional force of magnitude 0.6 N acts on \(A\). Find
  1. the tension in the string,
  2. the speed of \(B 1.5 \mathrm {~s}\) after it starts to move.
CAIE M1 2003 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-3_579_469_1142_840} One end of a light inextensible string is attached to a fixed point \(A\) of a fixed vertical wire. The other end of the string is attached to a small ring \(B\), of mass 0.2 kg , through which the wire passes. A horizontal force of magnitude 5 N is applied to the mid-point \(M\) of the string. The system is in equilibrium with the string taut, with \(B\) below \(A\), and with angles \(A B M\) and \(B A M\) equal to \(30 ^ { \circ }\) (see diagram).
  1. Show that the tension in \(B M\) is 5 N .
  2. The ring is on the point of sliding up the wire. Find the coefficient of friction between the ring and the wire.
  3. A particle of mass \(m \mathrm {~kg}\) is attached to the ring. The ring is now on the point of sliding down the wire. Given that the coefficient of friction between the ring and the wire is unchanged, find the value of \(m\).
CAIE M1 2003 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-4_547_1237_269_456} A tractor \(A\) starts from rest and travels along a straight road for 500 seconds. The velocity-time graph for the journey is shown above. This graph consists of three straight line segments. Find
  1. the distance travelled by \(A\),
  2. the initial acceleration of \(A\). Another tractor \(B\) starts from rest at the same instant as \(A\), and travels along the same road for 500 seconds. Its velocity \(t\) seconds after starting is \(\left( 0.06 t - 0.00012 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
  3. how much greater \(B\) 's initial acceleration is than \(A\) 's,
  4. how much further \(B\) has travelled than \(A\), at the instant when \(B\) 's velocity reaches its maximum.
CAIE M1 2004 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{38ece0f6-1c29-4e7a-9d66-16c3e2b695f9-2_200_529_269_808} Two particles \(P\) and \(Q\), of masses 1.7 kg and 0.3 kg respectively, are connected by a light inextensible string. \(P\) is held on a smooth horizontal table with the string taut and passing over a small smooth pulley fixed at the edge of the table. \(Q\) is at rest vertically below the pulley. \(P\) is released. Find the acceleration of the particles and the tension in the string.
CAIE M1 2004 November Q2
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{38ece0f6-1c29-4e7a-9d66-16c3e2b695f9-2_229_382_852_589} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{38ece0f6-1c29-4e7a-9d66-16c3e2b695f9-2_222_383_854_1178} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A small block of weight 18 N is held at rest on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal, by a force of magnitude \(P\) N. Find
  1. the value of \(P\) when the force is parallel to the plane, as in Fig. 1,
  2. the value of \(P\) when the force is horizontal, as in Fig. 2.
CAIE M1 2004 November Q3
3 A car of mass 1250 kg travels down a straight hill with the engine working at a power of 22 kW . The hill is inclined at \(3 ^ { \circ }\) to the horizontal and the resistance to motion of the car is 1130 N . Find the speed of the car at an instant when its acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
CAIE M1 2004 November Q4
4 A lorry of mass 16000 kg climbs from the bottom to the top of a straight hill of length 1000 m at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The top of the hill is 20 m above the level of the bottom of the hill. The driving force of the lorry is constant and equal to 5000 N . Find
  1. the gain in gravitational potential energy of the lorry,
  2. the work done by the driving force,
  3. the work done against the force resisting the motion of the lorry. On reaching the top of the hill the lorry continues along a straight horizontal road against a constant resistance of 1500 N . The driving force of the lorry is not now constant, and the speed of the lorry increases from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill to \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(P\). The distance of \(P\) from the top of the hill is 2000 m .
  4. Find the work done by the driving force of the lorry while the lorry travels from the top of the hill to \(P\).
CAIE M1 2004 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{38ece0f6-1c29-4e7a-9d66-16c3e2b695f9-3_240_862_274_644} Particles \(P\) and \(Q\) start from points \(A\) and \(B\) respectively, at the same instant, and move towards each other in a horizontal straight line. The initial speeds of \(P\) and \(Q\) are \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The accelerations of \(P\) and \(Q\) are constant and equal to \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) respectively (see diagram).
  1. Find the speed of \(P\) at the instant when the speed of \(P\) is 1.8 times the speed of \(Q\).
  2. Given that \(A B = 51 \mathrm {~m}\), find the time taken from the start until \(P\) and \(Q\) meet.