AQA A-Level Further Maths 2027 Predicted Practice Paper — Paper 3 Mechanics

Answer ALL questions. Write your answers in the spaces provided. You must write down all the stages in your working.

Section A

Short and medium-length questions. Answer ALL questions.

• 1

  Question SECTION-A1 (8 marks)

  A stone is projected from a point on horizontal ground with speed 28 m s^-1 at an angle of 35 degrees above the horizontal. Model the stone as a particle moving freely under gravity, taking g = 9.8 m s^-2 and neglecting air resistance. (a) Find the greatest height above the ground reached by the stone. [3 marks] (b) Find the horizontal range of the stone (the horizontal distance from the point of projection to the point where it returns to ground level). [4 marks] (c) State one physical assumption you have made in modelling the stone as a particle moving freely under gravity, and explain briefly how relaxing it would affect the range. [1 mark]

  (Total for Question SECTION-A1 is 8 marks)
• 2

  Question SECTION-A2 (7 marks)

  Two smooth spheres A and B of equal radius move towards each other along the same straight line on a smooth horizontal surface. Sphere A has mass 3 kg and speed 6 m s^-1; sphere B has mass 2 kg and speed 1 m s^-1 in the opposite direction. The coefficient of restitution between the spheres is 0.5. (a) Find the speed and direction of each sphere immediately after the collision. [5 marks] (b) Find the magnitude of the impulse exerted on B during the collision. [2 marks]

  (Total for Question SECTION-A2 is 7 marks)
• 3

  Question SECTION-A3 (6 marks)

  A particle moves with simple harmonic motion in a straight line. The motion has amplitude 0.4 m and period 2 s. The particle is at the centre O of the motion at time t = 0. (a) Find the maximum speed of the particle. [3 marks] (b) Find the speed of the particle when it is 0.2 m from O. [2 marks] (c) Find the magnitude of the maximum acceleration of the particle. [1 mark]

  (Total for Question SECTION-A3 is 6 marks)
• 4

  Question SECTION-A4 (8 marks)

  A box of mass 8 kg is pulled a distance of 5 m up a line of greatest slope of a rough plane inclined at 25 degrees to the horizontal. The pulling force has magnitude 60 N and acts up the plane, parallel to the line of greatest slope. The coefficient of friction between the box and the plane is 0.2. The box starts from rest. Take g = 9.8 m s^-2. (a) Find the normal reaction between the box and the plane. [2 marks] (b) Using the work-energy principle, find the speed of the box after it has moved 5 m up the plane. [6 marks]

  (Total for Question SECTION-A4 is 8 marks)
• 5

  Question SECTION-A5 (7 marks)

  A particle moves in a straight line. Its velocity v m s^-1 at time t seconds (t >= 0) is given by v = 3t^2 - 12t + 9. (a) Find the magnitude of the acceleration of the particle at t = 0. [2 marks] (b) Find the times at which the particle is instantaneously at rest. [2 marks] (c) Find the total distance travelled by the particle in the interval from t = 0 to t = 3. [3 marks]

  (Total for Question SECTION-A5 is 7 marks)
• 6

  Question SECTION-A6 (6 marks)

  A particle P of mass 0.5 kg is attached to one end of a light inextensible string of length 0.8 m. The other end of the string is fixed at a point O. The particle moves in a horizontal circle with the string making a constant angle of 30 degrees with the vertical, so that P describes a conical pendulum. Take g = 9.8 m s^-2. (a) Find the tension in the string. [3 marks] (b) Find the speed of P. [3 marks]

  (Total for Question SECTION-A6 is 6 marks)
• 7

  Question SECTION-A7 (8 marks)

  A uniform ladder AB has mass 20 kg and length 5 m. It rests in a vertical plane with end A on rough horizontal ground and end B against a smooth vertical wall. The ladder is inclined at 60 degrees to the horizontal. The coefficient of friction between the ladder and the ground is 0.3. Take g = 9.8 m s^-2. (a) Find the magnitude of the frictional force acting on the ladder at A when it is in equilibrium in this position. [5 marks] (b) By finding the least angle to the horizontal at which a ladder of this type could rest in equilibrium, determine whether the ladder in this position is on the point of slipping, and comment. [3 marks]

  (Total for Question SECTION-A7 is 8 marks)
• 8

  Question SECTION-A8 (7 marks)

  A ball of mass 0.2 kg is moving on a smooth horizontal floor. It strikes a smooth vertical wall which lies in the vertical plane containing the j direction, where i and j are perpendicular horizontal unit vectors and i is directed into the wall. Immediately before impact the velocity of the ball is (5i - 3j) m s^-1, and immediately after impact its velocity is (-2i - 3j) m s^-1. (a) Find the coefficient of restitution between the ball and the wall. [2 marks] (b) Find the magnitude of the impulse exerted by the wall on the ball. [3 marks] (c) State, with a reason, why the j component of the velocity is unchanged. [2 marks]

  (Total for Question SECTION-A8 is 7 marks)

Section B

Extended multi-part questions. Answer ALL questions.

• 1

  Question SECTION-B1 (15 marks)

  A ball is projected from a point A at the top edge of a vertical cliff. The point A is 45 m vertically above a point O at the base of the cliff on horizontal ground. The ball is projected with speed 30 m s^-1 at an angle of 20 degrees above the horizontal, directed out to sea away from the cliff. Model the ball as a particle moving freely under gravity, taking g = 9.8 m s^-2. The ball lands on the horizontal sea surface, which is at the same level as O. (a) Show that the horizontal and vertical components of the initial velocity are 28.19 m s^-1 and 10.26 m s^-1 respectively (each to 2 decimal places). [2 marks] (b) Find the greatest height of the ball above the level of A. [3 marks] (c) Find the time taken for the ball to reach the sea. [4 marks] (d) Find the horizontal distance from the base of the cliff O to the point where the ball hits the sea. [2 marks] (e) Find the speed of the ball and the angle its velocity makes below the horizontal as it hits the sea. [4 marks]

  (Total for Question SECTION-B1 is 15 marks)
• 2

  Question SECTION-B2 (14 marks)

  Two particles A and B are connected by a light inextensible string that passes over a small smooth pulley fixed at the edge of a rough horizontal table. Particle A has mass 5 kg and rests on the table; particle B has mass 4 kg and hangs freely below the pulley. The coefficient of friction between A and the table is 0.25. The section of string between A and the pulley is horizontal. The system is released from rest with the string taut. Take g = 9.8 m s^-2. (a) Show that while both particles are moving, the acceleration of the system is approximately 2.99 m s^-2. [5 marks] (b) Find the tension in the string during this motion. [3 marks] (c) Particle B descends a distance of 0.8 m before striking the floor, at which instant B stops and the string becomes slack. Find the speed of the particles at the moment B strikes the floor. [3 marks] (d) After B strikes the floor, A continues to move. Assuming A does not reach the pulley, find the further distance A travels before it comes to rest. [3 marks]

  (Total for Question SECTION-B2 is 14 marks)
• 3

  Question SECTION-B3 (14 marks)

  A small bead P of mass 0.3 kg is threaded on a smooth circular wire of radius 0.5 m that is fixed in a vertical plane. The bead is projected from the lowest point of the wire with speed 5 m s^-1. Take g = 9.8 m s^-2. (a) Explain why the bead is able to make complete revolutions of the wire, referring to the fact that it is threaded on the wire. [2 marks] (b) Find the speed of the bead at the highest point of the wire. [4 marks] (c) Find the magnitude and direction of the force exerted by the wire on the bead at the highest point. [4 marks] (d) Find the magnitude of the force exerted by the wire on the bead at the lowest point. [4 marks]

  (Total for Question SECTION-B3 is 14 marks)