Questions M1 (1912 questions)

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CAIE M1 2011 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{155bc571-80e4-4c93-859f-bb150a109211-3_489_1041_258_552}
\(A B C\) is a vertical cross-section of a surface. The part of the surface containing \(A B\) is smooth and \(A\) is 4 m higher than \(B\). The part of the surface containing \(B C\) is horizontal and the distance \(B C\) is 5 m (see diagram). A particle of mass 0.8 kg is released from rest at \(A\) and slides along \(A B C\). Find the speed of the particle at \(C\) in each of the following cases.
  1. The horizontal part of the surface is smooth.
  2. The coefficient of friction between the particle and the horizontal part of the surface is 0.3 .
CAIE M1 2011 November Q5
5 A particle \(P\) moves in a straight line. It starts from rest at \(A\) and comes to rest instantaneously at \(B\). The velocity of \(P\) at time \(t\) seconds after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 6 t ^ { 2 } - k t ^ { 3 }\) and \(k\) is a constant.
  1. Find an expression for the displacement of \(P\) from \(A\) in terms of \(t\) and \(k\).
  2. Find an expression for \(t\) in terms of \(k\) when \(P\) is at \(B\). Given that the distance \(A B\) is 108 m , find
  3. the value of \(k\),
  4. the maximum value of \(v\) when the particle is moving from \(A\) towards \(B\).
CAIE M1 2011 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{155bc571-80e4-4c93-859f-bb150a109211-3_465_410_1891_865} The diagram shows a ring of mass 2 kg threaded on a fixed rough vertical rod. A light string is attached to the ring and is pulled upwards at an angle of \(30 ^ { \circ }\) to the horizontal. The tension in the string is \(T \mathrm {~N}\). The coefficient of friction between the ring and the rod is 0.24 . Find the two values of \(T\) for which the ring is in limiting equilibrium.
CAIE M1 2011 November Q7
7 A car of mass 600 kg travels along a straight horizontal road starting from a point \(A\). The resistance to motion of the car is 750 N .
  1. The car travels from \(A\) to \(B\) at constant speed in 100 s . The power supplied by the car's engine is constant and equal to 30 kW . Find the distance \(A B\).
  2. The car's engine is switched off at \(B\) and the car's speed decreases until the car reaches \(C\) with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance \(B C\).
  3. The car's engine is switched on at \(C\) and the power it supplies is constant and equal to 30 kW . The car takes 14 s to travel from \(C\) to \(D\) and reaches \(D\) with a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance \(C D\).
CAIE M1 2012 November Q1
1 An object is released from rest at a height of 125 m above horizontal ground and falls freely under gravity, hitting a moving target \(P\). The target \(P\) is moving on the ground in a straight line, with constant acceleration \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At the instant the object is released \(P\) passes through a point \(O\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance from \(O\) to the point where \(P\) is hit by the object.
CAIE M1 2012 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{2bb3c9bb-60f0-440d-a148-b4db3478ca31-2_212_625_528_761} Particles \(A\) and \(B\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. \(A\) is held at rest on a rough horizontal table with the string passing over a small smooth pulley at the edge of the table. \(B\) hangs vertically below the pulley (see diagram). The system is released and \(B\) starts to move downwards with acceleration \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find
  1. the tension in the string after the system is released,
  2. the frictional force acting on \(A\).
CAIE M1 2012 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{2bb3c9bb-60f0-440d-a148-b4db3478ca31-2_241_535_1247_806} A particle \(P\) of mass 0.5 kg rests on a rough plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\). A force of magnitude 0.6 N , acting upwards on \(P\) at angle \(\alpha\) from a line of greatest slope of the plane, is just sufficient to prevent \(P\) sliding down the plane (see diagram). Find
  1. the normal component of the contact force on \(P\),
  2. the frictional component of the contact force on \(P\),
  3. the coefficient of friction between \(P\) and the plane.
CAIE M1 2012 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{2bb3c9bb-60f0-440d-a148-b4db3478ca31-2_387_1091_2019_525} Three coplanar forces of magnitudes \(8 \mathrm {~N} , 12 \mathrm {~N}\) and 2 N act at a point. The resultant of the forces has magnitude \(R \mathrm {~N}\). The directions of the three forces and the resultant are shown in the diagram. Find \(R\) and \(\theta\).
CAIE M1 2012 November Q5
5 Particle \(P\) travels along a straight line from \(A\) to \(B\) with constant acceleration \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Its speed at \(A\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its speed at \(B\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the time taken for \(P\) to travel from \(A\) to \(B\), and find also the distance \(A B\). Particle \(Q\) also travels along the same straight line from \(A\) to \(B\), starting from rest at \(A\). At time \(t \mathrm {~s}\) after leaving \(A\), the speed of \(Q\) is \(k t ^ { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(k\) is a constant. \(Q\) takes the same time to travel from \(A\) to \(B\) as \(P\) does.
  2. Find the value of \(k\) and find \(Q\) 's speed at \(B\).
CAIE M1 2012 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{2bb3c9bb-60f0-440d-a148-b4db3478ca31-3_382_1451_797_347} The diagram shows the vertical cross-section \(A B C D\) of a surface. \(B C\) is a circular arc, and \(A B\) and \(C D\) are tangents to \(B C\) at \(B\) and \(C\) respectively. \(A\) and \(D\) are at the same horizontal level, and \(B\) and \(C\) are at heights 2.7 m and 3.0 m respectively above the level of \(A\) and \(D\). A particle \(P\) of mass 0.2 kg is given a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\), in the direction of \(A B\) (see diagram). The parts of the surface containing \(A B\) and \(B C\) are smooth.
  1. Find the decrease in the speed of \(P\) as \(P\) moves along the surface from \(B\) to \(C\). The part of the surface containing \(C D\) exerts a constant frictional force on \(P\), as it moves from \(C\) to \(D\), and \(P\) comes to rest as it reaches \(D\).
  2. Find the speed of \(P\) when it is at the mid-point of \(C D\).
CAIE M1 2012 November Q7
7 A car of mass 1200 kg moves in a straight line along horizontal ground. The resistance to motion of the car is constant and has magnitude 960 N . The car's engine works at a rate of 17280 W .
  1. Calculate the acceleration of the car at an instant when its speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car passes through the points \(A\) and \(B\). While the car is moving between \(A\) and \(B\) it has constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Show that \(V = 18\). At the instant that the car reaches \(B\) the engine is switched off and subsequently provides no energy. The car continues along the straight line until it comes to rest at the point \(C\). The time taken for the car to travel from \(A\) to \(C\) is 52.5 s .
  3. Find the distance \(A C\).
CAIE M1 2012 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{9fbb63e3-4017-461e-9110-500be2c20778-2_122_803_255_671} A block is pushed along a horizontal floor by a force of magnitude 45 N acting at an angle of \(14 ^ { \circ }\) to the horizontal (see diagram). Find the work done by the force in moving the block a distance of 25 m .
CAIE M1 2012 November Q2
2 Particles \(A\) and \(B\) of masses \(m \mathrm {~kg}\) and \(( 1 - m ) \mathrm { kg }\) respectively are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. The system is released from rest with the straight parts of the string vertical. \(A\) moves vertically downwards and 0.3 seconds later it has speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the acceleration of \(A\),
  2. the value of \(m\) and the tension in the string.
CAIE M1 2012 November Q3
6 marks
3 A car travels along a straight road with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). It passes through points \(A , B\) and \(C\); the time taken from \(A\) to \(B\) and from \(B\) to \(C\) is 5 s in each case. The speed of the car at \(A\) is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the distances \(A B\) and \(B C\) are 55 m and 65 m respectively. Find the values of \(a\) and \(u\). [6]
CAIE M1 2012 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{9fbb63e3-4017-461e-9110-500be2c20778-2_583_862_1343_644} Three coplanar forces of magnitudes \(68 \mathrm {~N} , 75 \mathrm {~N}\) and 100 N act at an origin \(O\), as shown in the diagram. The components of the three forces in the positive \(x\)-direction are \(- 60 \mathrm {~N} , 0 \mathrm {~N}\) and 96 N , respectively. Find
  1. the components of the three forces in the positive \(y\)-direction,
  2. the magnitude and direction of the resultant of the three forces.
    \(5 A , B\) and \(C\) are three points on a line of greatest slope of a plane which is inclined at \(\theta ^ { \circ }\) to the horizontal, with \(A\) higher than \(B\) and \(B\) higher than \(C\). Between \(A\) and \(B\) the plane is smooth, and between \(B\) and \(C\) the plane is rough. A particle \(P\) is released from rest on the plane at \(A\) and slides down the line \(A B C\). At time 0.8 s after leaving \(A\), the particle passes through \(B\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2012 November Q6
6 A car of mass 1250 kg moves from the bottom to the top of a straight hill of length 500 m . The top of the hill is 30 m above the level of the bottom. The power of the car's engine is constant and equal to 30000 W . The car's acceleration is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at the bottom of the hill and is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at the top. The resistance to the car's motion is 1000 N . Find
  1. the car's gain in kinetic energy,
  2. the work done by the car's engine.
CAIE M1 2012 November Q7
7 A particle \(P\) starts to move from a point \(O\) and travels in a straight line. The velocity of \(P\) is \(k \left( 60 t ^ { 2 } - t ^ { 3 } \right) \mathrm { ms } ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(O\), where \(k\) is a constant. The maximum velocity of \(P\) is \(6.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(k = 0.0002\).
    \(P\) comes to instantaneous rest at a point \(A\) on the line. Find
  2. the distance \(O A\),
  3. the magnitude of the acceleration of \(P\) at \(A\),
  4. the speed of \(P\) when it subsequently passes through \(O\).
CAIE M1 2012 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{631ddcd9-17c0-4a15-8671-40788c3a84d3-2_366_780_251_680}
\(A B C D\) is a semi-circular cross-section, in a vertical plane, of the inner surface of half a hollow cylinder of radius 2.5 m which is fixed with its axis horizontal. \(A D\) is horizontal, \(B\) is the lowest point of the cross-section and \(C\) is at a height of 1.8 m above the level of \(B\) (see diagram). A particle \(P\) of mass 0.8 kg is released from rest at \(A\) and comes to instantaneous rest at \(C\).
  1. Find the work done on \(P\) by the resistance to motion while \(P\) travels from \(A\) to \(C\). The work done on \(P\) by the resistance to motion while \(P\) travels from \(A\) to \(B\) is 0.6 times the work done while \(P\) travels from \(A\) to \(C\).
  2. Find the speed of \(P\) when it passes through \(B\).
CAIE M1 2012 November Q2
2 A particle moves in a straight line. Its velocity \(t\) seconds after leaving a fixed point \(O\) on the line is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.2 t + 0.006 t ^ { 2 }\). For the instant when the acceleration of the particle is 2.5 times its initial acceleration,
  1. show that \(t = 25\),
  2. find the displacement of the particle from \(O\).
CAIE M1 2012 November Q3
3 A particle \(P\) is projected vertically upwards, from a point \(O\), with a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The point \(A\) is the highest point reached by \(P\). Find
  1. the speed of \(P\) when it is at the mid-point of \(O A\),
  2. the time taken for \(P\) to reach the mid-point of \(O A\) while moving upwards.
CAIE M1 2012 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{631ddcd9-17c0-4a15-8671-40788c3a84d3-2_396_880_1996_630} A particle \(P\) of weight 21 N is attached to one end of each of two light inextensible strings, \(S _ { 1 }\) and \(S _ { 2 }\), of lengths 0.52 m and 0.25 m respectively. The other end of \(S _ { 1 }\) is attached to a fixed point \(A\), and the other end of \(S _ { 2 }\) is attached to a fixed point \(B\) at the same horizontal level as \(A\). The particle \(P\) hangs in equilibrium at a point 0.2 m below the level of \(A B\) with both strings taut (see diagram). Find the tension in \(S _ { 1 }\) and the tension in \(S _ { 2 }\).
CAIE M1 2012 November Q5
5 An object of mass 12 kg slides down a line of greatest slope of a smooth plane inclined at \(10 ^ { \circ }\) to the horizontal. The object passes through points \(A\) and \(B\) with speeds \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.
  1. Find the increase in kinetic energy of the object as it moves from \(A\) to \(B\).
  2. Hence find the distance \(A B\), assuming there is no resisting force acting on the object. The object is now pushed up the plane from \(B\) to \(A\), with constant speed, by a horizontal force.
  3. Find the magnitude of this force.
CAIE M1 2012 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{631ddcd9-17c0-4a15-8671-40788c3a84d3-3_255_511_794_817} The diagram shows a particle of mass 0.6 kg on a plane inclined at \(25 ^ { \circ }\) to the horizontal. The particle is acted on by a force of magnitude \(P \mathrm {~N}\) directed up the plane parallel to a line of greatest slope. The coefficient of friction between the particle and the plane is 0.36 . Given that the particle is in equilibrium, find the set of possible values of \(P\).
CAIE M1 2012 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{631ddcd9-17c0-4a15-8671-40788c3a84d3-3_565_828_1402_660} Particles \(A\) and \(B\) have masses 0.32 kg and 0.48 kg respectively. The particles are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the edge of a smooth horizontal table. Particle \(B\) is held at rest on the table at a distance of 1.4 m from the pulley. \(A\) hangs vertically below the pulley at a height of 0.98 m above the floor (see diagram). \(A , B\), the string and the pulley are all in the same vertical plane. \(B\) is released and \(A\) moves downwards.
  1. Find the acceleration of \(A\) and the tension in the string.
    \(A\) hits the floor and \(B\) continues to move towards the pulley. Find the time taken, from the instant that \(B\) is released, for
  2. \(A\) to reach the floor,
  3. \(B\) to reach the pulley.
CAIE M1 2013 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-2_291_591_255_776} A particle \(P\) of mass 0.3 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(X\). A horizontal force of magnitude \(F \mathrm {~N}\) is applied to the particle, which is in equilibrium when the string is at an angle \(\alpha\) to the vertical, where \(\tan \alpha = \frac { 8 } { 15 }\) (see diagram). Find the tension in the string and the value of \(F\).