Questions — CAIE M1 (786 questions)

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CAIE M1 2017 November Q2
6 marks Moderate -0.3
\includegraphics{figure_2} A block of mass 15 kg hangs in equilibrium below a horizontal ceiling attached to two strings as shown in the diagram. One of the strings is inclined at \(45°\) to the horizontal and the tension in this string is 120 N. The other string is inclined at \(θ°\) to the horizontal and the tension in this string is \(T\) N. Find the values of \(T\) and \(θ\). [6]
CAIE M1 2017 November Q3
6 marks Standard +0.3
A car travels along a straight road with constant acceleration. It passes through points \(A\), \(B\) and \(C\). The car passes point \(A\) with velocity 14 m s\(^{-1}\). The two sections \(AB\) and \(BC\) are of equal length. The times taken to travel along \(AB\) and \(BC\) are 5 s and 3 s respectively.
  1. Write down an expression for the distance \(AB\) in terms of the acceleration of the car. Write down a similar expression for the distance \(AC\). Hence show that the acceleration of the car is 4 m s\(^{-2}\). [4]
  2. Find the speed of the car as it passes point \(C\). [2]
CAIE M1 2017 November Q4
6 marks Standard +0.3
A particle \(P\) is projected vertically upwards from horizontal ground with speed 12 m s\(^{-1}\).
  1. Find the time taken for \(P\) to return to the ground. [2]
The time in seconds after \(P\) is projected is denoted by \(t\). When \(t = 1\), a second particle \(Q\) is projected vertically upwards with speed 10 m s\(^{-1}\) from a point which is 5 m above the ground. Particles \(P\) and \(Q\) move in different vertical lines.
  1. Find the set of values of \(t\) for which the two particles are moving in the same direction. [4]
CAIE M1 2017 November Q5
8 marks Standard +0.3
A cyclist is riding up a straight hill inclined at an angle \(α\) to the horizontal, where \(\sin α = 0.04\). The total mass of the bicycle and rider is 80 kg. The cyclist is riding at a constant speed of 4 m s\(^{-1}\). There is a force resisting the motion. The work done by the cyclist against this resistance force over a distance of 25 m is 600 J.
  1. Find the power output of the cyclist. [4]
The cyclist reaches the top of the hill, where the road becomes horizontal, with speed 4 m s\(^{-1}\). The cyclist continues to work at the same rate on the horizontal part of the road.
  1. Find the speed of the cyclist 10 seconds after reaching the top of the hill, given that the work done by the cyclist during this period against the resistance force is 1200 J. [4]
CAIE M1 2017 November Q6
10 marks Standard +0.3
\includegraphics{figure_6} Two particles \(P\) and \(Q\), each of mass \(m\) kg, 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 rough plane. The plane is inclined at an angle \(α\) to the horizontal, where \(\tan α = \frac{4}{3}\). Particle \(P\) rests on the plane and particle \(Q\) hangs vertically, as shown in the diagram. The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane. The system is in limiting equilibrium.
  1. Show that the coefficient of friction between \(P\) and the plane is \(\frac{4}{3}\). [5]
A force of magnitude 10 N is applied to \(P\), acting up a line of greatest slope of the plane, and \(P\) accelerates at 2.5 m s\(^{-2}\).
  1. Find the value of \(m\). [5]
CAIE M1 2017 November Q7
9 marks Standard +0.3
A particle starts from rest and moves in a straight line. The velocity of the particle at time \(t\) s after the start is \(v\) m s\(^{-1}\), where $$v = -0.01t^3 + 0.22t^2 - 0.4t.$$
  1. Find the two positive values of \(t\) for which the particle is instantaneously at rest. [2]
  2. Find the time at which the acceleration of the particle is greatest. [3]
  3. Find the distance travelled by the particle while its velocity is positive. [4]
CAIE M1 2018 November Q1
4 marks Standard +0.3
A smooth ring \(R\) of mass \(m\) kg is threaded on a light inextensible string \(ARB\). The ends of the string are attached to fixed points \(A\) and \(B\) with \(A\) vertically above \(B\). The string is taut and angle \(ARB = 90°\). The angle between the part \(AR\) of the string and the vertical is \(45°\). The ring is held in equilibrium in this position by a force of magnitude \(2.5\) N, acting on the ring in the direction \(BR\) (see diagram). Calculate the tension in the string and the mass of the ring. [4] \includegraphics{figure_1}
CAIE M1 2018 November Q2
4 marks Moderate -0.8
A block of mass \(5\) kg is being pulled by a rope up a rough plane inclined at \(6°\) to the horizontal. The rope is parallel to a line of greatest slope of the plane and the block is moving at constant speed. The coefficient of friction between the block and the plane is \(0.3\). Find the tension in the rope. [4]
CAIE M1 2018 November Q3
7 marks Moderate -0.8
\includegraphics{figure_3} The velocity of a particle moving in a straight line is \(v\) m s\(^{-1}\) at time \(t\) seconds. The diagram shows a velocity-time graph which models the motion of the particle from \(t = 0\) to \(t = T\). The graph consists of four straight line segments. The particle reaches its maximum velocity \(V\) m s\(^{-1}\) at \(t = 10\).
  1. Find the acceleration of the particle during the first \(2\) seconds. [1]
  2. Find the value of \(V\). [2]
At \(t = 6\), the particle is instantaneously at rest at the point \(A\). At \(t = T\), the particle comes to rest at the point \(B\). At \(t = 0\) the particle starts from rest at a point one third of the way from \(A\) to \(B\).
  1. Find the distance \(AB\) and hence find the value of \(T\). [4]
CAIE M1 2018 November Q4
8 marks Standard +0.3
\includegraphics{figure_4} Two particles \(P\) and \(Q\), of masses \(0.4\) kg and \(0.7\) 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 rough plane. The coefficient of friction between \(P\) and the plane is \(0.5\). The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\). Particle \(P\) lies on the plane and particle \(Q\) hangs vertically. The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). A force of magnitude \(X\) N, acting directly down the plane, is applied to \(P\).
  1. Show that the greatest value of \(X\) for which \(P\) remains stationary is \(6.2\). [4]
  2. Given instead that \(X = 0.8\), find the acceleration of \(P\). [4]
CAIE M1 2018 November Q5
8 marks Standard +0.3
A particle moves in a straight line starting from a point \(O\) with initial velocity \(1\) m s\(^{-1}\). The acceleration of the particle at time \(t\) s after leaving \(O\) is \(a\) m s\(^{-2}\), where $$a = 1.2t^{\frac{1}{2}} - 0.6t.$$
  1. At time \(T\) s after leaving \(O\) the particle reaches its maximum velocity. Find the value of \(T\). [2]
  2. Find the velocity of the particle when its acceleration is maximum (you do not need to verify that the acceleration is a maximum rather than a minimum). [6]
CAIE M1 2018 November Q6
8 marks Moderate -0.3
A car of mass \(1200\) kg is driving along a straight horizontal road at a constant speed of \(15\) m s\(^{-1}\). There is a constant resistance to motion of \(350\) N.
  1. Find the power of the car's engine. [1]
The car comes to a hill inclined at \(1°\) to the horizontal, still travelling at \(15\) m s\(^{-1}\).
  1. The car starts to descend the hill with reduced power and with an acceleration of \(0.12\) m s\(^{-2}\). Given that there is no change in the resistance force, find the new power of the car's engine at the instant when it starts to descend the hill. [3]
  2. When the car is travelling at \(20\) m s\(^{-1}\) down the hill, the power is cut off and the car gradually slows down. Assuming that the resistance force remains \(350\) N, find the distance travelled from the moment when the power is cut off until the speed of the car is reduced to \(18\) m s\(^{-1}\). [4]
CAIE M1 2018 November Q7
11 marks Standard +0.3
A particle of mass \(0.3\) kg is released from rest above a tank containing water. The particle falls vertically, taking \(0.8\) s to reach the water surface. There is no instantaneous change of speed when the particle enters the water. The depth of water in the tank is \(1.25\) m. The water exerts a force on the particle resisting its motion. The work done against this resistance force from the instant that the particle enters the water until it reaches the bottom of the tank is \(1.2\) J.
  1. Use an energy method to find the speed of the particle when it reaches the bottom of the tank. [4]
When the particle reaches the bottom of the tank, it bounces back vertically upwards with initial speed \(7\) m s\(^{-1}\). As the particle rises through the water, it experiences a constant resistance force of \(1.8\) N. The particle comes to instantaneous rest \(t\) seconds after it bounces on the bottom of the tank.
  1. Find the value of \(t\). [7]
CAIE M1 2019 November Q1
2 marks Easy -1.2
A crane is lifting a load of 1250 kg vertically at a constant speed \(V\) m s\(^{-1}\). Given that the power of the crane is a constant 20 kW, find the value of \(V\). [2]
CAIE M1 2019 November Q2
5 marks Standard +0.3
The total mass of a cyclist and her bicycle is 75 kg. The cyclist ascends a straight hill of length 0.7 km inclined at 1.5° to the horizontal. Her speed at the bottom of the hill is 10 m s\(^{-1}\) and at the top it is 5 m s\(^{-1}\). There is a resistance to motion, and the work done against this resistance as the cyclist ascends the hill is 2000 J. The cyclist exerts a constant force of magnitude \(F\) N in the direction of motion. Find \(F\). [5]
CAIE M1 2019 November Q3
7 marks Moderate -0.3
A block of mass 3 kg is at rest on a rough plane inclined at 60° to the horizontal. A force of magnitude 15 N acting up a line of greatest slope of the plane is just sufficient to prevent the block from sliding down the plane.
  1. Find the coefficient of friction between the block and the plane. [5]
The force of magnitude 15 N is now replaced by a force of magnitude \(X\) N acting up the line of greatest slope.
  1. Find the greatest value of \(X\) for which the block does not move. [2]
CAIE M1 2019 November Q4
7 marks Standard +0.3
\includegraphics{figure_4} Two blocks \(A\) and \(B\) of masses 4 kg and 5 kg respectively are joined by a light inextensible string. The blocks rest on a smooth plane inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac{7}{24}\). The string is parallel to a line of greatest slope of the plane with \(B\) above \(A\). A force of magnitude 36 N acts on \(B\), parallel to a line of greatest slope of the plane (see diagram).
  1. Find the acceleration of the blocks and the tension in the string. [5]
  1. At a particular instant, the speed of the blocks is 1 m s\(^{-1}\). Find the time, after this instant, that it takes for the blocks to travel 0.65 m. [2]
CAIE M1 2019 November Q5
8 marks Moderate -0.3
\includegraphics{figure_5} A small ring \(P\) is threaded on a fixed smooth horizontal rod \(AB\). Three horizontal forces of magnitudes 4.5 N, 7.5 N and \(F\) N act on \(P\) (see diagram).
  1. Given that these three forces are in equilibrium, find the values of \(F\) and \(\theta\). [6]
  1. It is given instead that the values of \(F\) and \(\theta\) are 9.5 and 30 respectively, and the acceleration of the ring is 1.5 m s\(^{-2}\). Find the mass of the ring. [2]
CAIE M1 2019 November Q6
9 marks Moderate -0.3
A particle of mass 0.4 kg is released from rest at a height of 1.8 m above the surface of the water in a tank. There is no instantaneous change of speed when the particle enters the water. The water exerts an upward force of 5.6 N on the particle when it is in the water.
  1. Find the velocity of the particle at the instant when it reaches the surface of the water. [2]
  1. Find the time that it takes from the instant when the particle enters the water until it comes to instantaneous rest in the water. You may assume that the tank is deep enough so that the particle does not reach the bottom of the tank. [4]
  1. Sketch a velocity-time graph for the motion of the particle from the instant at which it is released until it comes to instantaneous rest in the water. [3]
CAIE M1 2019 November Q7
12 marks Standard +0.3
A particle moves in a straight line, starting from rest at a point \(O\), and comes to instantaneous rest at a point \(P\). The velocity of the particle at time \(t\) s after leaving \(O\) is \(v\) m s\(^{-1}\), where $$v = 0.6t^2 - 0.12t^3.$$
  1. Show that the distance \(OP\) is 6.25 m. [5]
On another occasion, the particle also moves in the same straight line. On this occasion, the displacement of the particle at time \(t\) s after leaving \(O\) is \(s\) m, where $$s = kt^3 + ct^5.$$ It is given that the particle passes point \(P\) with velocity 1.25 m s\(^{-1}\) at time \(t = 5\).
  1. Find the values of the constants \(k\) and \(c\). [5]
  1. Find the acceleration of the particle at time \(t = 5\). [2]
CAIE M1 2019 November Q1
4 marks Moderate -0.8
A particle moves in a straight line. The displacement of the particle at time \(t\) s is \(s\) m, where $$s = t^3 - 6t^2 + 4t.$$ Find the velocity of the particle at the instant when its acceleration is zero. [4]
CAIE M1 2019 November Q2
5 marks Moderate -0.8
\includegraphics{figure_2} The diagram shows a velocity-time graph which models the motion of a tractor. The graph consists of four straight line segments. The tractor passes a point \(O\) at time \(t = 0\) with speed \(U\) m s\(^{-1}\). The tractor accelerates to a speed of \(V\) m s\(^{-1}\) over a period of 5 s, and then travels at this speed for a further 25 s. The tractor then accelerates to a speed of 12 m s\(^{-1}\) over a period of 5 s. The tractor then decelerates to rest over a period of 15 s.
  1. Given that the acceleration of the tractor between \(t = 30\) and \(t = 35\) is 0.8 m s\(^{-2}\), find the value of \(V\). [2]
  2. Given also that the total distance covered by the tractor in the 50 seconds of motion is 375 m, find the value of \(U\). [3]
CAIE M1 2019 November Q3
5 marks Moderate -0.3
\includegraphics{figure_3} A particle \(P\) of mass 0.3 kg is held in equilibrium above a horizontal plane by a force of magnitude 5 N, acting vertically upwards. The particle is attached to two strings \(PA\) and \(PB\) of lengths 0.9 m and 1.2 m respectively. The points \(A\) and \(B\) lie on the plane and angle \(APB = 90°\) (see diagram). Find the tension in each of the strings. [5]
CAIE M1 2019 November Q4
7 marks Moderate -0.3
A lorry of mass 25 000 kg travels along a straight horizontal road. There is a constant force of 3000 N resisting the motion.
  1. Find the power required to maintain a constant speed of 30 m s\(^{-1}\). [2]
The lorry comes to a straight hill inclined at 2° to the horizontal. The driver switches off the engine of the lorry at the point \(A\) which is at the foot of the hill. Point \(B\) is further up the hill. The speeds of the lorry at \(A\) and \(B\) are 30 m s\(^{-1}\) and 25 m s\(^{-1}\) respectively. The resistance force is still 3000 N.
  1. Use an energy method to find the height of \(B\) above the level of \(A\). [5]
CAIE M1 2019 November Q5
7 marks Moderate -0.3
Two particles \(A\) and \(B\) move in the same vertical line. Particle \(A\) is projected vertically upwards from the ground with speed 20 m s\(^{-1}\). One second later particle \(B\) is dropped from rest from a height of 40 m.
  1. Find the height above the ground at which the two particles collide. [4]
  2. Find the difference in the speeds of the two particles at the instant when the collision occurs. [3]