Questions M1 (2067 questions)

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CAIE M1 2003 November Q2
5 marks Easy -1.3
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
5 marks Moderate -0.8
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
6 marks Moderate -0.8
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
7 marks Moderate -0.3
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
10 marks Standard +0.3
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
13 marks Standard +0.3
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
5 marks Moderate -0.8
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
5 marks Moderate -0.8
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
5 marks Standard +0.3
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
8 marks Standard +0.3
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
8 marks Moderate -0.3
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.
CAIE M1 2004 November Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{38ece0f6-1c29-4e7a-9d66-16c3e2b695f9-3_330_572_1037_788} Two identical boxes, each of mass 400 kg , are at rest, with one on top of the other, on horizontal ground. A horizontal force of magnitude \(P\) newtons is applied to the lower box (see diagram). The coefficient of friction between the lower box and the ground is 0.75 and the coefficient of friction between the two boxes is 0.4 .
  1. Show that the boxes will remain at rest if \(P \leqslant 6000\). The boxes start to move with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Given that no sliding takes place between the boxes, show that \(a \leqslant 4\) and deduce the maximum possible value of \(P\).
CAIE M1 2004 November Q7
10 marks Standard +0.3
7 A particle starts from rest at the point \(A\) and travels in a straight line until it reaches the point \(B\). The velocity of the particle \(t\) seconds after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.009 t ^ { 2 } - 0.0001 t ^ { 3 }\). Given that the velocity of the particle when it reaches \(B\) is zero, find
  1. the time taken for the particle to travel from \(A\) to \(B\),
  2. the distance \(A B\),
  3. the maximum velocity of the particle.
CAIE M1 2005 November Q1
4 marks Moderate -0.3
1 A car travels in a straight line with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). It passes the points \(A , B\) and \(C\), in this order, with speeds \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 } , 7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The distances \(A B\) and \(B C\) are \(d _ { 1 } \mathrm {~m}\) and \(d _ { 2 } \mathrm {~m}\) respectively.
  1. Write down an equation connecting
    1. \(d _ { 1 }\) and \(a\),
    2. \(d _ { 2 }\) and \(a\).
    3. Hence find \(d _ { 1 }\) in terms of \(d _ { 2 }\).
CAIE M1 2005 November Q2
5 marks Moderate -0.3
2 A crate of mass 50 kg is dragged along a horizontal floor by a constant force of magnitude 400 N acting at an angle \(\alpha ^ { \circ }\) upwards from the horizontal. The total resistance to motion of the crate has constant magnitude 250 N . The crate starts from rest at the point \(O\) and passes the point \(P\) with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(O P\) is 20 m . For the crate's motion from \(O\) to \(P\), find
  1. the increase in kinetic energy of the crate,
  2. the work done against the resistance to the motion of the crate,
  3. the value of \(\alpha\).
CAIE M1 2005 November Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-2_479_771_1356_687} Each of three light strings has a particle attached to one of its ends. The other ends of the strings are tied together at a point \(A\). The strings are in equilibrium with two of them passing over fixed smooth horizontal pegs, and with the particles hanging freely. The weights of the particles, and the angles between the sloping parts of the strings and the vertical, are as shown in the diagram. Find the values of \(W _ { 1 }\) and \(W _ { 2 }\).
[0pt] [6]
CAIE M1 2005 November Q4
6 marks Moderate -0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-3_276_570_264_790} A stone slab of mass 320 kg rests in equilibrium on rough horizontal ground. A force of magnitude \(X \mathrm {~N}\) acts upwards on the slab at an angle of \(\theta\) to the vertical, where \(\tan \theta = \frac { 7 } { 24 }\) (see diagram).
  1. Find, in terms of \(X\), the normal component of the force exerted on the slab by the ground.
  2. Given that the coefficient of friction between the slab and the ground is \(\frac { 3 } { 8 }\), find the value of \(X\) for which the slab is about to slip.
CAIE M1 2005 November Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-3_917_1451_1059_347} The diagram shows the displacement-time graph for a car's journey. The graph consists of two curved parts \(A B\) and \(C D\), and a straight line \(B C\). The line \(B C\) is a tangent to the curve \(A B\) at \(B\) and a tangent to the curve \(C D\) at \(C\). The gradient of the curves at \(t = 0\) and \(t = 600\) is zero, and the acceleration of the car is constant for \(0 < t < 80\) and for \(560 < t < 600\). The displacement of the car is 400 m when \(t = 80\).
  1. Sketch the velocity-time graph for the journey.
  2. Find the velocity at \(t = 80\).
  3. Find the total distance for the journey.
  4. Find the acceleration of the car for \(0 < t < 80\).
CAIE M1 2005 November Q6
10 marks Standard +0.3
6 A particle \(P\) starts from rest at \(O\) and travels in a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by \(v = 8 t - 2 t ^ { 2 }\) for \(0 \leqslant t \leqslant 3\), and \(v = \frac { 54 } { t ^ { 2 } }\) for \(t > 3\). Find
  1. the distance travelled by \(P\) in the first 3 seconds,
  2. an expression in terms of \(t\) for the displacement of \(P\) from \(O\), valid for \(t > 3\),
  3. the value of \(v\) when the displacement of \(P\) from \(O\) is 27 m .
CAIE M1 2005 November Q7
10 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-4_601_515_699_815} Two particles \(A\) and \(B\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. Particle \(B\) is held on the horizontal floor and particle \(A\) hangs in equilibrium. Particle \(B\) is released and each particle starts to move vertically with constant acceleration of magnitude \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the value of \(a\). Particle \(A\) hits the floor 1.2 s after it starts to move, and does not rebound upwards.
  2. Show that \(A\) hits the floor with a speed of \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find the gain in gravitational potential energy by \(B\), from leaving the floor until reaching its greatest height.
CAIE M1 2006 November Q1
5 marks Moderate -0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{d0fa61eb-f320-427e-8883-de224d293933-2_421_1223_267_461} A box of mass 8 kg is pulled, at constant speed, up a straight path which is inclined at an angle of \(15 ^ { \circ }\) to the horizontal. The pulling force is constant, of magnitude 30 N , and acts upwards at an angle of \(10 ^ { \circ }\) from the path (see diagram). The box passes through the points \(A\) and \(B\), where \(A B = 20 \mathrm {~m}\) and \(B\) is above the level of \(A\). For the motion from \(A\) to \(B\), find
  1. the work done by the pulling force,
  2. the gain in potential energy of the box,
  3. the work done against the resistance to motion of the box.
CAIE M1 2006 November Q2
5 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{d0fa61eb-f320-427e-8883-de224d293933-2_701_323_1244_913} A small ring of mass 0.6 kg is threaded on a rough rod which is fixed vertically. The ring is in equilibrium, acted on by a force of magnitude 5 N pulling upwards at \(30 ^ { \circ }\) to the vertical (see diagram).
  1. Show that the frictional force acting on the ring has magnitude 1.67 N , correct to 3 significant figures.
  2. The ring is on the point of sliding down the rod. Find the coefficient of friction between the ring and the rod.
CAIE M1 2006 November Q3
5 marks Moderate -0.3
3 A cyclist travels along a straight road working at a constant rate of 420 W . The total mass of the cyclist and her cycle is 75 kg . Ignoring any resistance to motion, find the acceleration of the cyclist at an instant when she is travelling at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  1. given that the road is horizontal,
  2. given instead that the road is inclined at \(1.5 ^ { \circ }\) to the horizontal and the cyclist is travelling up the slope.
CAIE M1 2006 November Q4
7 marks Moderate -0.8
4 The velocity of a particle \(t \mathrm {~s}\) after it starts from rest is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 1.25 t - 0.05 t ^ { 2 }\). Find
  1. the initial acceleration of the particle,
  2. the displacement of the particle from its starting point at the instant when its acceleration is \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
CAIE M1 2006 November Q5
7 marks Moderate -0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{d0fa61eb-f320-427e-8883-de224d293933-3_515_789_995_676} The diagram shows the vertical cross-section \(L M N\) of a fixed smooth surface. \(M\) is the lowest point of the cross-section. \(L\) is 2.45 m above the level of \(M\), and \(N\) is 1.2 m above the level of \(M\). A particle of mass 0.5 kg is released from rest at \(L\) and moves on the surface until it leaves it at \(N\). Find
  1. the greatest speed of the particle,
  2. the kinetic energy of the particle at \(N\). The particle is now projected from \(N\), with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), along the surface towards \(M\).
  3. Find the least value of \(v\) for which the particle will reach \(L\).