Questions M1 (2067 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
CAIE M1 2020 June Q1
3 marks Easy -1.2
1 Particles \(P\) of mass \(m \mathrm {~kg}\) and \(Q\) of mass 0.2 kg are free to move on a smooth horizontal plane. \(P\) is projected at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(Q\) which is stationary. After the collision \(P\) and \(Q\) move in opposite directions with speeds of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Find \(m\).
CAIE M1 2020 June Q2
4 marks Easy -1.2
2 A minibus of mass 4000 kg is travelling along a straight horizontal road. The resistance to motion is 900 N .
  1. Find the driving force when the acceleration of the minibus is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the power required for the minibus to maintain a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2020 June Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{55090630-1413-45cd-8201-4d58662db6bd-04_586_1003_260_571} Four coplanar forces of magnitudes \(40 \mathrm {~N} , 20 \mathrm {~N} , 50 \mathrm {~N}\) and \(F \mathrm {~N}\) act at a point in the directions shown in the diagram. The four forces are in equilibrium. Find \(F\) and \(\alpha\).
CAIE M1 2020 June Q4
7 marks Moderate -0.3
4 A car starts from rest and moves in a straight line with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for a distance of 50 m . The car then travels with constant velocity for 500 m for a period of 25 s , before decelerating to rest. The magnitude of this deceleration is \(2 a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Sketch the velocity-time graph for the motion of the car. \includegraphics[max width=\textwidth, alt={}, center]{55090630-1413-45cd-8201-4d58662db6bd-05_533_1155_534_534}
  2. Find the value of \(a\).
  3. Find the total time for which the car is in motion.
CAIE M1 2020 June Q5
9 marks Moderate -0.3
5 A block \(B\) of mass 4 kg is pushed up a line of greatest slope of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal by a force applied to \(B\), acting in the direction of motion of \(B\). The block passes through points \(P\) and \(Q\) with speeds \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. \(P\) and \(Q\) are 10 m apart with \(P\) below the level of \(Q\).
  1. Find the decrease in kinetic energy of the block as it moves from \(P\) to \(Q\).
  2. Hence find the work done by the force pushing the block up the slope as the block moves from \(P\) to \(Q\).
  3. At the instant the block reaches \(Q\), the force pushing the block up the slope is removed. Find the time taken, after this instant, for the block to return to \(P\).
CAIE M1 2020 June Q6
9 marks Standard +0.3
6 A particle travels in a straight line \(P Q\). The velocity of the particle \(t \mathrm {~s}\) after leaving \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 4.5 + 4 t - 0.5 t ^ { 2 }$$
  1. Find the velocity of the particle at the instant when its acceleration is zero.
    The particle comes to instantaneous rest at \(Q\).
  2. Find the distance \(P Q\). \includegraphics[max width=\textwidth, alt={}, center]{55090630-1413-45cd-8201-4d58662db6bd-10_625_780_260_744} Two particles \(A\) and \(B\), of masses \(3 m \mathrm {~kg}\) and \(2 m \mathrm {~kg}\) respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a plane. The plane is inclined at an angle \(\theta\) to the horizontal. \(A\) lies on the plane and \(B\) hangs vertically, 0.8 m above the floor, which is horizontal. The string between \(A\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). Initially \(A\) and \(B\) are at rest.
    1. Given that the plane is smooth, find the value of \(\theta\) for which \(A\) remains at rest.
      It is given instead that the plane is rough, \(\theta = 30 ^ { \circ }\) and the acceleration of \(A\) up the plane is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    2. Show that the coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 10 } \sqrt { 3 }\).
    3. When \(B\) reaches the floor it comes to rest. Find the length of time after \(B\) reaches the floor for which \(A\) is moving up the plane. [You may assume that \(A\) does not reach the pulley.]
      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 2021 June Q1
3 marks Moderate -0.8
1 A particle of mass 0.6 kg is projected with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down a line of greatest slope of a smooth plane inclined at \(10 ^ { \circ }\) to the horizontal. Use an energy method to find the speed of the particle after it has moved 15 m down the plane.
CAIE M1 2021 June Q2
6 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{41e63d05-d109-47dc-80a6-927953e3e607-03_659_655_258_744} Coplanar forces of magnitudes \(34 \mathrm {~N} , 30 \mathrm {~N}\) and 26 N act at a point in the directions shown in the diagram. Given that \(\sin \alpha = \frac { 5 } { 13 }\) and \(\sin \theta = \frac { 8 } { 17 }\), find the magnitude and direction of the resultant of the three forces.
CAIE M1 2021 June Q3
6 marks Standard +0.3
3 A ring of mass 0.3 kg is threaded on a horizontal rough rod. The coefficient of friction between the ring and the rod is 0.8 . A force of magnitude 8 N acts on the ring. This force acts at an angle of \(10 ^ { \circ }\) above the horizontal in the vertical plane containing the rod. Find the time taken for the ring to move, from rest, 0.6 m along the rod.
CAIE M1 2021 June Q4
6 marks Standard +0.8
4 A particle of mass 12 kg is stationary on a rough plane inclined at an angle of \(25 ^ { \circ }\) to the horizontal. A pulling force of magnitude \(P \mathrm {~N}\) acts at an angle of \(8 ^ { \circ }\) above a line of greatest slope of the plane. This force is used to keep the particle in equilibrium. The coefficient of friction between the particle and the plane is 0.3 . Find the greatest possible value of \(P\).
CAIE M1 2021 June Q5
11 marks Standard +0.3
5 A car of mass 1250 kg is pulling a caravan of mass 800 kg along a straight road. The resistances to the motion of the car and caravan are 440 N and 280 N respectively. The car and caravan are connected by a light rigid tow-bar.
  1. The car and caravan move along a horizontal part of the road at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Calculate, in kW , the power developed by the engine of the car.
    2. Given that this power is suddenly decreased by 8 kW , find the instantaneous deceleration of the car and caravan and the tension in the tow-bar.
  2. The car and caravan now travel along a part of the road inclined at \(\sin ^ { - 1 } 0.06\) to the horizontal. The car and caravan travel up the incline at constant speed with the engine of the car working at 28 kW .
    1. Find this constant speed.
    2. Find the increase in the potential energy of the caravan in one minute.
CAIE M1 2021 June Q6
8 marks Challenging +1.2
6 A particle \(A\) is projected vertically upwards from level ground with an initial speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same instant a particle \(B\) is released from rest 15 m vertically above \(A\). The mass of one of the particles is twice the mass of the other particle. During the subsequent motion \(A\) and \(B\) collide and coalesce to form particle \(C\). Find the difference between the two possible times at which \(C\) hits the ground. \(7 \quad\) A particle \(P\) moving in a straight line starts from rest at a point \(O\) and comes to rest 16 s later. At time \(t \mathrm {~s}\) after leaving \(O\), the acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of \(P\) is given by $$\begin{array} { l l } a = 6 + 4 t & 0 \leqslant t < 2 , \\ a = 14 & 2 \leqslant t < 4 , \\ a = 16 - 2 t & 4 \leqslant t \leqslant 16 . \end{array}$$ There is no sudden change in velocity at any instant.
  1. Find the values of \(t\) when the velocity of \(P\) is \(55 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Complete the sketch of the velocity-time diagram. \includegraphics[max width=\textwidth, alt={}, center]{41e63d05-d109-47dc-80a6-927953e3e607-11_511_1054_351_584}
  3. Find the distance travelled by \(P\) when it is decelerating.
    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 2021 June Q1
4 marks Standard +0.3
1 Particles \(P\) of mass 0.4 kg and \(Q\) of mass 0.5 kg are free to move on a smooth horizontal plane. \(P\) and \(Q\) are moving directly towards each other with speeds \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. After \(P\) and \(Q\) collide, the speed of \(Q\) is twice the speed of \(P\). Find the two possible values of the speed of \(P\) after the collision.
CAIE M1 2021 June Q2
5 marks Standard +0.3
2 A cyclist is travelling along a straight horizontal road. She is working at a constant rate of 150 W . At an instant when her speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), her acceleration is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The resistance to motion is 20 N .
  1. Find the total mass of the cyclist and her bicycle.
    The cyclist comes to a straight hill inclined at an angle \(\theta\) above the horizontal. She ascends the hill at constant speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She continues to work at the same rate as before and the resistance force is unchanged.
  2. Find the value of \(\theta\).
CAIE M1 2021 June Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{ba29ddb2-3558-4be1-a8a8-134e27a70149-04_456_767_260_689} Four coplanar forces act at a point. The magnitudes of the forces are \(20 \mathrm {~N} , 30 \mathrm {~N} , 40 \mathrm {~N}\) and \(F \mathrm {~N}\). The directions of the forces are as shown in the diagram, where \(\sin \alpha ^ { \circ } = 0.28\) and \(\sin \beta ^ { \circ } = 0.6\). Given that the forces are in equilibrium, find \(F\) and \(\theta\).
CAIE M1 2021 June Q4
6 marks Moderate -0.3
4 A particle is projected vertically upwards with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on horizontal ground. After 2 seconds, the height of the particle above the ground is 24 m .
  1. Show that \(u = 22\).
  2. The height of the particle above the ground is more than \(h \mathrm {~m}\) for a period of 3.6 s . Find \(h\).
CAIE M1 2021 June Q5
9 marks Standard +0.3
5 A car of mass 1400 kg is towing a trailer of mass 500 kg down a straight hill inclined at an angle of \(5 ^ { \circ }\) to the horizontal. The car and trailer are connected by a light rigid tow-bar. At the top of the hill the speed of the car and trailer is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at the bottom of the hill their speed is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. It is given that as the car and trailer descend the hill, the engine of the car does 150000 J of work, and there are no resistance forces. Find the length of the hill.
  2. It is given instead that there is a resistance force of 100 N on the trailer, the length of the hill is 200 m , and the acceleration of the car and trailer is constant. Find the tension in the tow-bar between the car and trailer.
CAIE M1 2021 June Q6
10 marks Moderate -0.3
6 A particle moves in a straight line and passes through the point \(A\) at time \(t = 0\). The velocity of the particle at time \(t \mathrm {~s}\) after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 2 t ^ { 2 } - 5 t + 3$$
  1. Find the times at which the particle is instantaneously at rest. Hence or otherwise find the minimum velocity of the particle.
  2. Sketch the velocity-time graph for the first 3 seconds of motion.
  3. Find the distance travelled between the two times when the particle is instantaneously at rest.
CAIE M1 2021 June Q7
10 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{ba29ddb2-3558-4be1-a8a8-134e27a70149-10_220_609_260_769} A particle \(P\) of mass 0.3 kg rests on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 7 } { 25 }\). A horizontal force of magnitude 4 N , acting in the vertical plane containing a line of greatest slope of the plane, is applied to \(P\) (see diagram). The particle is on the point of sliding up the plane.
  1. Show that the coefficient of friction between the particle and the plane is \(\frac { 3 } { 4 }\).
    The force acting horizontally is replaced by a force of magnitude 4 N acting up the plane parallel to a line of greatest slope.
  2. Find the acceleration of \(P\).
  3. Starting with \(P\) at rest, the force of 4 N parallel to the plane acts for 3 seconds and is then removed. Find the total distance travelled until \(P\) comes 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 2022 June Q1
4 marks Moderate -0.8
1 Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are at rest on a smooth horizontal plane. \(P\) is projected at a speed of \(4 \mathrm {~ms} ^ { - 1 }\) directly towards \(Q\). After \(P\) and \(Q\) collide, \(Q\) begins to move with a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the speed of \(P\) after the collision.
    After the collision, \(Q\) moves directly towards a third particle \(R\), of mass \(m \mathrm {~kg}\), which is at rest on the plane. The two particles \(Q\) and \(R\) coalesce on impact and move with a speed of \(2 \mathrm {~ms} ^ { - 1 }\).
  2. Find \(m\).
CAIE M1 2022 June Q2
5 marks Moderate -0.8
2 A particle \(P\) is projected vertically upwards from horizontal ground. \(P\) reaches a maximum height of 45 m . After reaching the ground, \(P\) comes to rest without rebounding.
  1. Find the speed at which \(P\) was projected.
  2. Find the total time for which the speed of \(P\) is at least \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M1 2022 June Q3
5 marks Moderate -0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-04_824_1636_264_258} The displacement of a particle moving in a straight line is \(s\) metres at time \(t\) seconds after leaving a fixed point \(O\). The particle starts from rest and passes through points \(P , Q\) and \(R\), at times \(t = 5 , t = 10\) and \(t = 15\) respectively, and returns to \(O\) at time \(t = 20\). The distances \(O P , O Q\) and \(O R\) are 50 m , 150 m and 200 m respectively. The diagram shows a displacement-time graph which models the motion of the particle from \(t = 0\) to \(t = 20\). The graph consists of two curved segments \(A B\) and \(C D\) and two straight line segments \(B C\) and \(D E\).
  1. Find the speed of the particle between \(t = 5\) and \(t = 10\).
  2. Find the acceleration of the particle between \(t = 0\) and \(t = 5\), given that it is constant.
  3. Find the average speed of the particle during its motion. \includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-06_483_880_258_630} The diagram shows a block of mass 10 kg suspended below a horizontal ceiling by two strings \(A C\) and \(B C\), of lengths 0.8 m and 0.6 m respectively, attached to fixed points on the ceiling. Angle \(A C B = 90 ^ { \circ }\). There is a horizontal force of magnitude \(F \mathrm {~N}\) acting on the block. The block is in equilibrium.
CAIE M1 2022 June Q5
8 marks Moderate -0.3
5 A cyclist is riding along a straight horizontal road. The total mass of the cyclist and her bicycle is 70 kg . At an instant when the cyclist's speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), her acceleration is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is a constant resistance to motion of magnitude 30 N .
  1. Find the power developed by the cyclist.
    The cyclist comes to the top of a hill inclined at \(5 ^ { \circ }\) to the horizontal. The cyclist stops pedalling and freewheels down the hill (so that the cyclist is no longer supplying any power). The magnitude of the resistance force remains at 30 N . Over a distance of \(d \mathrm {~m}\), the speed of the cyclist increases from \(6 \mathrm {~ms} ^ { - 1 }\) to \(12 \mathrm {~ms} ^ { - 1 }\).
  2. Find the change in kinetic energy.
  3. Use an energy method to find \(d\). \includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-10_725_785_260_680} Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley at \(B\) which is attached to two inclined planes. \(P\) lies on a smooth plane \(A B\) which is inclined at \(60 ^ { \circ }\) to the horizontal. \(Q\) lies on a plane \(B C\) which is inclined at \(30 ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes (see diagram).
CAIE M1 2022 June Q7
10 marks Standard +0.3
7 A particle \(P\) moves in a straight line through a point \(O\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\), at time \(t \mathrm {~s}\) after passing \(O\), is given by $$v = \frac { 9 } { 4 } + \frac { b } { ( t + 1 ) ^ { 2 } } - c t ^ { 2 }$$ where \(b\) and \(c\) are positive constants. At \(t = 5\), the velocity of \(P\) is zero and its acceleration is \(- \frac { 13 } { 12 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Show that \(b = 9\) and find the value of \(c\).
  2. Given that the velocity of \(P\) is zero only at \(t = 5\), find the distance travelled in the first 10 seconds of motion.
    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 2023 June Q1
3 marks Moderate -0.8
1 A particle of mass 1.6 kg is dropped from a height of 9 m above horizontal ground. The speed of the particle at the instant before hitting the ground is \(12 \mathrm {~ms} ^ { - 1 }\). Find the work done against air resistance.