Questions M1 (2067 questions)

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CAIE M1 2006 November Q6
10 marks Moderate -0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{d0fa61eb-f320-427e-8883-de224d293933-4_474_831_269_657} Forces of magnitudes \(P \mathrm {~N}\) and 25 N act at right angles to each other. The resultant of the two forces has magnitude \(R \mathrm {~N}\) and makes an angle of \(\theta ^ { \circ }\) with the \(x\)-axis (see diagram). The force of magnitude \(P \mathrm {~N}\) has components - 2.8 N and 9.6 N in the \(x\)-direction and the \(y\)-direction respectively, and makes an angle of \(\alpha ^ { \circ }\) with the negative \(x\)-axis.
  1. Find the values of \(P\) and \(R\).
  2. Find the value of \(\alpha\), and hence find the components of the force of magnitude 25 N in
    1. the \(x\)-direction,
    2. the \(y\)-direction.
    3. Find the value of \(\theta\).
CAIE M1 2006 November Q7
11 marks Standard +0.3
7 A particle of mass \(m \mathrm {~kg}\) moves up a line of greatest slope of a rough plane inclined at \(21 ^ { \circ }\) to the horizontal. The frictional and normal components of the contact force on the particle have magnitudes \(F \mathrm {~N}\) and \(R \mathrm {~N}\) respectively. The particle passes through the point \(P\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and 2 s later it reaches its highest point on the plane.
  1. Show that \(R = 9.336 m\) and \(F = 1.416 m\), each correct to 4 significant figures.
  2. Find the coefficient of friction between the particle and the plane. After the particle reaches its highest point it starts to move down the plane.
  3. Find the speed with which the particle returns to \(P\).
CAIE M1 2008 November Q1
5 marks Moderate -0.5
1 \includegraphics[max width=\textwidth, alt={}, center]{a4cb105b-55d2-4793-95d2-3d791990a1f6-2_341_929_269_609} Forces of magnitudes 10 N and 8 N act in directions as shown in the diagram.
  1. Write down in terms of \(\theta\) the component of the resultant of the two forces
    1. parallel to the force of magnitude 10 N ,
    2. perpendicular to the force of magnitude 10 N .
    3. The resultant of the two forces has magnitude 8 N . Show that \(\cos \theta = \frac { 5 } { 8 }\).
CAIE M1 2008 November Q2
6 marks Standard +0.3
2 A block of mass 20 kg is at rest on a plane inclined at \(10 ^ { \circ }\) to the horizontal. A force acts on the block parallel to a line of greatest slope of the plane. The coefficient of friction between the block and the plane is 0.32 . Find the least magnitude of the force necessary to move the block,
  1. given that the force acts up the plane,
  2. given instead that the force acts down the plane.
CAIE M1 2008 November Q3
6 marks Standard +0.3
3 A car of mass 1200 kg is travelling on a horizontal straight road and passes through a point \(A\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The power of the car's engine is 18 kW and the resistance to the car's motion is 900 N .
  1. Find the deceleration of the car at \(A\).
  2. Show that the speed of the car does not fall below \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while the car continues to move with the engine exerting a constant power of 18 kW .
CAIE M1 2008 November Q4
6 marks Moderate -0.8
4 A load of mass 160 kg is lifted vertically by a crane, with constant acceleration. The load starts from rest at the point \(O\). After 7 s , it passes through the point \(A\) with speed \(0.5 \mathrm {~ms} ^ { - 1 }\). By considering energy, find the work done by the crane in moving the load from \(O\) to \(A\).
CAIE M1 2008 November Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{a4cb105b-55d2-4793-95d2-3d791990a1f6-3_643_481_274_831} Particles \(A\) and \(B\), of masses 0.5 kg and \(m \mathrm {~kg}\) respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. Particle \(B\) is held at rest on the horizontal floor and particle \(A\) hangs in equilibrium (see diagram). \(B\) is released and each particle starts to move vertically. \(A\) hits the floor 2 s after \(B\) is released. The speed of each particle when \(A\) hits the floor is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. For the motion while \(A\) is moving downwards, find
    1. the acceleration of \(A\),
    2. the tension in the string.
    3. Find the value of \(m\).
CAIE M1 2008 November Q6
9 marks Standard +0.3
6 A train travels from \(A\) to \(B\), a distance of 20000 m , taking 1000 s . The journey has three stages. In the first stage the train starts from rest at \(A\) and accelerates uniformly until its speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the second stage the train travels at constant speed \(V _ { \mathrm { m } } { } ^ { - 1 }\) for 600 s . During the third stage of the journey the train decelerates uniformly, coming to rest at \(B\).
  1. Sketch the velocity-time graph for the train's journey.
  2. Find the value of \(V\).
  3. Given that the acceleration of the train during the first stage of the journey is \(0.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the distance travelled by the train during the third stage of the journey. \(7 \quad\) A particle \(P\) is held at rest at a fixed point \(O\) and then released. \(P\) falls freely under gravity until it reaches the point \(A\) which is 1.25 m below \(O\).
  4. Find the speed of \(P\) at \(A\) and the time taken for \(P\) to reach \(A\). The particle continues to fall, but now its downward acceleration \(t\) seconds after passing through \(A\) is \(( 10 - 0.3 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  5. Find the total distance \(P\) has fallen, 3 s after being released from \(O\).
CAIE M1 2009 November Q1
4 marks Moderate -0.8
1 A car of mass 1000 kg moves along a horizontal straight road, passing through points \(A\) and \(B\). The power of its engine is constant and equal to 15000 W . The driving force exerted by the engine is 750 N at \(A\) and 500 N at \(B\). Find the speed of the car at \(A\) and at \(B\), and hence find the increase in the car's kinetic energy as it moves from \(A\) to \(B\).
CAIE M1 2009 November Q2
4 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-2_609_967_536_589} A smooth narrow tube \(A E\) has two straight parts, \(A B\) and \(D E\), and a curved part \(B C D\). The part \(A B\) is vertical with \(A\) above \(B\), and \(D E\) is horizontal. \(C\) is the lowest point of the tube and is 0.65 m below the level of \(D E\). A particle is released from rest at \(A\) and travels through the tube, leaving it at \(E\) with speed \(6 \mathrm {~ms} ^ { - 1 }\) (see diagram). Find
  1. the height of \(A\) above the level of \(D E\),
  2. the maximum speed of the particle.
CAIE M1 2009 November Q3
4 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-2_462_721_1672_712} Two forces have magnitudes \(P \mathrm {~N}\) and \(Q \mathrm {~N}\). The resultant of the two forces has magnitude 12 N and acts in a direction \(40 ^ { \circ }\) clockwise from the force of magnitude \(P \mathrm {~N}\) and \(80 ^ { \circ }\) anticlockwise from the force of magnitude \(Q \mathrm {~N}\) (see diagram). Find the value of \(Q\).
CAIE M1 2009 November Q4
8 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-3_335_751_264_696} A particle \(P\) of weight 5 N is attached to one end of each of two light inextensible strings of lengths 30 cm and 40 cm . The other end of the shorter string is attached to a fixed point \(A\) of a rough rod which is fixed horizontally. A small ring \(S\) of weight \(W \mathrm {~N}\) is attached to the other end of the longer string and is threaded on to the rod. The system is in equilibrium with the strings taut and \(A S = 50 \mathrm {~cm}\) (see diagram).
  1. By resolving the forces acting on \(P\) in the direction of \(P S\), or otherwise, find the tension in the longer string.
  2. Find the magnitude of the frictional force acting on \(S\).
  3. Given that the coefficient of friction between \(S\) and the rod is 0.75 , and that \(S\) is in limiting equilibrium, find the value of \(W\).
CAIE M1 2009 November Q5
8 marks Standard +0.3
5 A particle \(P\) of mass 0.6 kg moves upwards along a line of greatest slope of a plane inclined at \(18 ^ { \circ }\) to the horizontal. The deceleration of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the frictional and normal components of the force exerted on \(P\) by the plane. Hence find the coefficient of friction between \(P\) and the plane, correct to 2 significant figures. After \(P\) comes to instantaneous rest it starts to move down the plane with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the value of \(a\).
CAIE M1 2009 November Q6
10 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-4_712_529_264_810} Particles \(P\) and \(Q\), of masses 0.55 kg and 0.45 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. The particles are held at rest with the string taut and its straight parts vertical. Both particles are at a height of 5 m above the ground (see diagram). The system is released.
  1. Find the acceleration with which \(P\) starts to move. The string breaks after 2 s and in the subsequent motion \(P\) and \(Q\) move vertically under gravity.
  2. At the instant that the string breaks, find
    1. the height above the ground of \(P\) and of \(Q\),
    2. the speed of the particles.
    3. Show that \(Q\) reaches the ground 0.8 s later than \(P\). \(7 \quad\) A particle \(P\) starts from rest at the point \(A\) at time \(t = 0\), where \(t\) is in seconds, and moves in a straight line with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 10 s . For \(10 \leqslant t \leqslant 20 , P\) continues to move along the line with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \frac { 800 } { t ^ { 2 } } - 2\). Find
      1. the speed of \(P\) when \(t = 10\), and the value of \(a\),
      2. the value of \(t\) for which the acceleration of \(P\) is \(- a \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
      3. the displacement of \(P\) from \(A\) when \(t = 20\).
CAIE M1 2009 November Q1
4 marks Moderate -0.3
1 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{efa7175f-832b-4cd3-82ab-52e402115081-2_458_472_267_493} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{efa7175f-832b-4cd3-82ab-52e402115081-2_351_435_365_1217} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A small block of weight 12 N is at rest on a smooth plane inclined at \(40 ^ { \circ }\) to the horizontal. The block is held in equilibrium by a force of magnitude \(P \mathrm {~N}\). Find the value of \(P\) when
  1. the force is parallel to the plane as in Fig. 1,
  2. the force is horizontal as in Fig. 2.
CAIE M1 2009 November Q2
5 marks Moderate -0.3
2 A lorry of mass 15000 kg moves with constant speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the top to the bottom of a straight hill of length 900 m . The top of the hill is 18 m above the level of the bottom of the hill. The total work done by the resistive forces acting on the lorry, including the braking force, is \(4.8 \times 10 ^ { 6 } \mathrm {~J}\). Find
  1. the loss in gravitational potential energy of the lorry,
  2. the work done by the driving force. On reaching the bottom of the hill the lorry continues along a straight horizontal road against a constant resistance of 1600 N . There is no braking force acting. The speed of the lorry increases from \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the hill to \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(X\), where \(X\) is 2500 m from the bottom of the hill.
  3. By considering energy, find the work done by the driving force of the lorry while it travels from the bottom of the hill to \(X\).
CAIE M1 2009 November Q3
6 marks Standard +0.3
3 A car of mass 1250 kg travels along a horizontal straight road with increasing speed. The power provided by the car's engine is constant and equal to 24 kW . The resistance to the car's motion is constant and equal to 600 N .
  1. Show that the speed of the car cannot exceed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the acceleration of the car at an instant when its speed is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2009 November Q4
7 marks Standard +0.3
4 A particle moves up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\cos \alpha = 0.96\) and \(\sin \alpha = 0.28\).
  1. Given that the normal component of the contact force acting on the particle has magnitude 1.2 N , find the mass of the particle.
  2. Given also that the frictional component of the contact force acting on the particle has magnitude 0.4 N , find the deceleration of the particle. The particle comes to rest on reaching the point \(X\).
  3. Determine whether the particle remains at \(X\) or whether it starts to move down the plane.
CAIE M1 2009 November Q5
9 marks Moderate -0.3
5 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{efa7175f-832b-4cd3-82ab-52e402115081-3_317_517_922_468} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{efa7175f-832b-4cd3-82ab-52e402115081-3_317_522_922_1155} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A small ring of weight 12 N is threaded on a fixed rough horizontal rod. A light string is attached to the ring and the string is pulled with a force of 15 N at an angle of \(30 ^ { \circ }\) to the horizontal.
  1. When the angle of \(30 ^ { \circ }\) is below the horizontal (see Fig. 1), the ring is in limiting equilibrium. Show that the coefficient of friction between the ring and the rod is 0.666 , correct to 3 significant figures.
  2. When the angle of \(30 ^ { \circ }\) is above the horizontal (see Fig. 2), the ring is moving with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the value of \(a\).
CAIE M1 2009 November Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{efa7175f-832b-4cd3-82ab-52e402115081-4_686_511_269_817} Particles \(A\) and \(B\), of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. Particle \(A\) is held on the horizontal floor and particle \(B\) hangs in equilibrium. Particle \(A\) is released and both particles start to move vertically.
  1. Find the acceleration of the particles. The speed of the particles immediately before \(B\) hits the floor is \(1.6 \mathrm {~ms} ^ { - 1 }\). Given that \(B\) does not rebound upwards, find
  2. the maximum height above the floor reached by \(A\),
  3. the time taken by \(A\), from leaving the floor, to reach this maximum height.
CAIE M1 2009 November Q7
10 marks Moderate -0.3
7 A motorcyclist starts from rest at \(A\) and travels in a straight line. For the first part of the motion, the motorcyclist's displacement \(x\) metres from \(A\) after \(t\) seconds is given by \(x = 0.6 t ^ { 2 } - 0.004 t ^ { 3 }\).
  1. Show that the motorcyclist's acceleration is zero when \(t = 50\) and find the speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at this time. For \(t \geqslant 50\), the motorcyclist travels at constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the value of \(t\) for which the motorcyclist's average speed is \(27.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2010 November Q1
5 marks Moderate -0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-2_608_831_258_657} Two particles \(P\) and \(Q\) move vertically under gravity. The graphs show the upward velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particles at time \(t \mathrm {~s}\), for \(0 \leqslant t \leqslant 4 . P\) starts with velocity \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) starts from rest.
  1. Find the value of \(V\). Given that \(Q\) reaches the horizontal ground when \(t = 4\), find
  2. the speed with which \(Q\) reaches the ground,
  3. the height of \(Q\) above the ground when \(t = 0\).
CAIE M1 2010 November Q2
5 marks Standard +0.3
2 A car of mass 600 kg travels along a horizontal straight road, with its engine working at a rate of 40 kW . The resistance to motion of the car is constant and equal to 800 N . The car passes through the point \(A\) on the road with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's acceleration at the point \(B\) on the road is half its acceleration at \(A\). Find the speed of the car at \(B\).
CAIE M1 2010 November Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-2_606_843_1731_651} The diagram shows three particles \(A , B\) and \(C\) hanging freely in equilibrium, each being attached to the end of a string. The other ends of the three strings are tied together and are at the point \(X\). The strings carrying \(A\) and \(C\) pass over smooth fixed horizontal pegs \(P _ { 1 }\) and \(P _ { 2 }\) respectively. The weights of \(A , B\) and \(C\) are \(5.5 \mathrm {~N} , 7.3 \mathrm {~N}\) and \(W \mathrm {~N}\) respectively, and the angle \(P _ { 1 } X P _ { 2 }\) is a right angle. Find the angle \(A P _ { 1 } X\) and the value of \(W\).
CAIE M1 2010 November Q4
7 marks Standard +0.3
4 A particle \(P\) starts from a fixed point \(O\) at time \(t = 0\), where \(t\) is in seconds, and moves with constant acceleration in a straight line. The initial velocity of \(P\) is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its velocity when \(t = 10\) is \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the displacement of \(P\) from \(O\) when \(t = 10\). Another particle \(Q\) also starts from \(O\) when \(t = 0\) and moves along the same straight line as \(P\). The acceleration of \(Q\) at time \(t\) is \(0.03 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Given that \(Q\) has the same velocity as \(P\) when \(t = 10\), show that it also has the same displacement from \(O\) as \(P\) when \(t = 10\).