Modelling and vectors in mechanics

A window cleaner raises a bucket of water of mass 4.5 kg to a balcony. A rope is attached to the bucket, passes over a fixed pulley above the balcony, and the window cleaner pulls vertically downwards on the free end. Both parts of the rope are vertical. In a model, the bucket is a particle, the rope is light and inextensible, and the pulley is small and smooth. Take g = 9.8 m s⁻².

[In this question i and j are unit vectors due east and due north respectively.] A robotic lawnmower crosses a flat field in two straight stages. Its displacement in the first stage is a = (7i − 4j) m and in the second stage is b = (−2i + 9j) m.

[In this question i and j are unit vectors due east and due north respectively.] A hiker walks 3.0 km on a bearing of 040° and then 5.0 km on a bearing of 130°.

[In this question i and j are unit vectors due east and due north respectively, and position vectors are given relative to a fixed point O on the bank of a pond.] A model boat is at the point with position vector (−2i + 5j) m when t = 0, and moves with constant velocity (1.5i − 0.8j) m s⁻¹.

Figure 1 shows a child sitting on a sledge on horizontal snow. A parent pulls the sledge with a rope. The tension in the rope is 60 N and the rope makes the angle shown in Figure 1 with the horizontal. The child and sledge have a total mass of 18 kg and are modelled as a single particle; the rope is modelled as light and the snow as smooth.

Figure 1 shows two forces P and Q acting at a point O, drawn on a square grid in which one square represents 1 N. The unit vectors i and j are parallel to the gridlines, as shown.

[In this question i and j are unit vectors due east and due north respectively.] Figure 1 shows the route of a kayak. It leaves a harbour H and paddles in a straight line to a rock A, then in a straight line to a beach B. The distances and bearings of the two stages are shown in Figure 1.

[In this question i and j are unit vectors due east and due north respectively, and position vectors are given relative to a fixed point O.] A ball rolls across a smooth horizontal floor with constant velocity and is modelled as a particle. Figure 1 shows its position A at time t = 0 and its position B at time t = 4 s, on a grid of 1 m squares.

A small ring rests on a smooth horizontal table. Three horizontal forces act on it: F₁ = (7i − 2j) N, F₂ = (−4i + 6j) N and F₃. The ring is in equilibrium. The ring has mass 0.4 kg and is modelled as a particle.

Two forces F and G act on a particle. F = (4i − 3j) N. G has magnitude 13 N and acts in the direction of the vector (5i + 12j).

A model hovercraft of mass 0.5 kg moves on a smooth horizontal surface. It is modelled as a particle acted on by two constant horizontal forces, (2i + j) N and (−0.5i + j) N. At time t = 0 its velocity is (2i − 6j) m s⁻¹.

[In this question i and j are unit vectors due east and due north respectively, and position vectors are relative to a fixed point O.] Two small robots, A and B, move on a level floor, each with constant velocity. At time t = 0, A is at O and B is at (10i − 4j) m. The velocity of A is (2i + 3j) m s⁻¹ and the velocity of B is (−0.5i + 4j) m s⁻¹. The robots are modelled as particles.

[In this question i and j are unit vectors due east and due north respectively, and position vectors are relative to a fixed point O.] A radio-controlled toy car of mass 1.2 kg moves across a horizontal car park with constant acceleration (i − 2j) m s⁻². At time t = 0 it is at the point with position vector (−4i + 6j) m and has velocity (2i + j) m s⁻¹. The toy car is modelled as a particle.

[In this question i and j are unit vectors due east and due north respectively.] A light aircraft flies with a velocity relative to the air of 180 km h⁻¹ on a bearing of 070°. A wind blows from due north at 40 km h⁻¹, so the velocity of the air is −40j km h⁻¹. The velocity of the aircraft over the ground is the resultant of its velocity relative to the air and the velocity of the air. The aircraft is modelled as a particle and the wind velocity is assumed constant.

Figure 1 is a plan view of three horizontal forces acting on a puck at rest on smooth horizontal ice. A force of 14 N acts in the direction of −j. Forces of magnitude P N and Q N act at the angles shown, where i and j are perpendicular unit vectors in the horizontal plane. The puck has mass 2.5 kg, is modelled as a particle and is in equilibrium.

[In this question i and j are unit vectors parallel and perpendicular to the river banks, as shown.] Figure 1 shows a straight river with parallel banks 40 m apart. The water flows with a constant velocity of 0.8i m s⁻¹. A swimmer starts at S on the south bank and swims at 1.5 m s⁻¹ relative to the water, heading upstream at the angle to the bank shown. The velocity of the swimmer over the river bed is the resultant of the swimmer's velocity relative to the water and the velocity of the water. The swimmer is modelled as a particle.

[In this question i and j are unit vectors due east and due north respectively, and position vectors are given relative to a fixed point O.] A ship moves with constant velocity. Figure 1 shows the position of a fixed buoy and the positions of the ship at times t = 0, 1, 2, 3 and 4 hours, on a grid of 1 km squares. The ship and buoy are modelled as particles.

A puck of mass 0.25 kg slides on a horizontal air table under the action of a constant horizontal force, and is modelled as a particle. Figure 1 is a velocity diagram, with i and j perpendicular unit vectors in the plane of the table. It shows the velocity of the puck at time t = 0 and at time t = 5 s, drawn from O.

A bead of mass 0.2 kg is threaded on a fixed straight horizontal wire. The bead is pulled by a force of magnitude 1.5 N acting in the vertical plane containing the wire, at the angle below the horizontal shown in Figure 1. The wire is modelled as smooth. Take g = 9.8 m s⁻².

Figure 1 is a plan view of a barge in a straight canal. The barge is pulled by two horizontal ropes attached to the same point at its front. One rope goes to a horse on the north towpath and has tension 900 N; the other goes to a horse on the south towpath and has tension T N. The angles between the ropes and the direction of the canal are shown in Figure 1. The barge moves along the canal, parallel to the banks. The barge has mass 2400 kg and is modelled as a particle; the ropes are light and inextensible. The water exerts a constant resistance of 600 N opposing the motion.

[In this question i and j are unit vectors due east and due north respectively, and position vectors are relative to a fixed point O.] At 09:00 yacht A is at (−6i + 2j) km and yacht B is at (18i + 14j) km. A sails with constant velocity (8i + 6j) km h⁻¹ and B sails with constant velocity (−4i + 10j) km h⁻¹. At time t hours after 09:00 the distance between the yachts is d km.

[In this question i and j are unit vectors due east and due north respectively, and position vectors are relative to a fixed point O.] A particle P of mass 2 kg moves on a smooth horizontal plane under the action of two constant forces, F₁ = (3i − 2j) N and F₂. At time t = 0, P is at the point with position vector (i − 12j) m and has velocity (5i − 2j) m s⁻¹. When t = 4 s, the velocity of P is (−3i + 14j) m s⁻¹.

A crate of mass 6 kg rests on smooth horizontal ice. Three light horizontal ropes are attached to it. The tension in the first rope is 40 N, acting in the direction of i. The tension in the second rope is 30 N, acting at 120° to i, measured anticlockwise. The tension in the third rope has magnitude T N and acts at an angle θ° to i, measured anticlockwise, where 0 ≤ θ < 360. The crate is modelled as a particle and is in equilibrium.

[In this question i and j are unit vectors due east and due north respectively, and position vectors are relative to a fixed point O.] Two drones, P and Q, fly at the same constant height and are modelled as particles moving in a horizontal plane. At time t = 0, P is at (−10i + 4j) m and Q is at (8i − 6j) m. P flies with constant velocity (6i + 2j) m s⁻¹ and Q flies with constant velocity (−3i + 7j) m s⁻¹.

[In this question i and j are unit vectors due east and due north respectively, and position vectors are given relative to a fixed point O.] Figure 1 shows a radar station R and the starting position S of a ship at 12:00, on a grid in kilometres. The ship sails with constant velocity (3i + 4j) km h⁻¹ in the direction shown. At time t hours after 12:00 the distance between the ship and R is d km. The ship and the station are modelled as particles.

[In this question i and j are unit vectors due east and due north respectively, and position vectors are given relative to a fixed point O.] A particle P of mass 0.5 kg moves on a smooth horizontal plane under the action of a single constant force. At time t = 0, P is at the point A with position vector r₀, and it has velocity u m s⁻¹ and constant acceleration a m s⁻². Figure 1 shows the path of P, with its positions at t = 0, 1, 2, 3 and 4 s marked on a grid of 1 m squares.

[In this question i and j are unit vectors due east and due north respectively, and position vectors are given relative to a fixed point O.] A puck of mass 0.5 kg moves on smooth horizontal ice and is modelled as a particle. At time t = 0 it passes through O with velocity (4i − j) m s⁻¹. For 0 ≤ t ≤ 4 a constant horizontal force (−0.5i + 0.5j) N acts on the puck; at t = 4 the force is removed. The puck then moves with constant velocity until it reaches a straight barrier. Figure 1 shows the path of the puck and the barrier on a grid of 1 m squares.

[In this question i and j are unit vectors due east and due north respectively, and position vectors are given relative to a fixed point O.] Figure 1 shows the position S of a cargo ship at 12:00 and the position of a patrol base B, on a grid in kilometres. The ship sails with constant velocity (6i − 4j) km h⁻¹ in the direction shown. A patrol boat leaves B at 12:30 and moves with constant velocity so that it meets the ship at 14:00. Both vessels are modelled as particles. Let t be the time in hours after 12:00.