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AS Mechanics · Edexcel 8MA0 · free A-level maths worksheet

Modelling, units and vectors in mechanics

SI units, modelling assumptions (particle, light, smooth, inextensible), displacement, velocity and acceleration as vectors.

12 original questions, 65 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Write each quantity in SI units.

  1. A speed of 90 km h−1
  2. A mass of 1.2 tonnes
  3. A speed of 45 cm per minute
3 marks
Show answer
a) 25 m s-1 b) 1200 kg c) 0.0075 m s-1
2

A parcel is pulled across a floor by a rope. In a model of the motion the parcel is modelled as a particle, the rope is modelled as light and inextensible, and the floor is modelled as smooth.

State what each of these words means for the model.

  1. particle
  2. light
  3. inextensible
  4. smooth
4 marks
Show answer
a) the size of the parcel is ignored, so its mass acts at a single point and it does not rotate b) the mass of the rope is ignored, so the tension is the same all along it c) the rope does not stretch, so the parcel moves with the same speed and acceleration as the end of the rope d) there is no friction between the parcel and the floor
3

Amara walks 300 m due east and then 400 m due north. The whole walk takes 500 s.

  1. Find the total distance Amara walks.
  2. Find the magnitude of her displacement from her starting point, and its bearing.
  3. Find her average speed and the magnitude of her average velocity.
5 marks
Show answer
a) 700 m b) 500 m on a bearing of 037° (036.9°) c) average speed 1.4 m s-1, average velocity magnitude 1 m s-1
4

In this question i and j are unit vectors due east and due north.

A boat moves with constant velocity (5i − 12j) m s−1.

  1. Find the speed of the boat.
  2. Find the direction in which the boat is moving, as a bearing to the nearest degree.
4 marks
Show answer
a) 13 m s-1 b) 157°
5

A particle P moves with constant velocity (4i − j) m s−1. At time t = 0, P is at the point with position vector (2i + 3j) m relative to a fixed origin O.

  1. Find the position vector of P when t = 5
  2. Find the value of t when P is due east of O.
  3. Find the distance OP at this time.
5 marks
Show answer
a) (22i - 2j) m b) t = 3 c) 14 m
6

In this question i and j are unit vectors due east and due north, and position vectors are given relative to a fixed harbour H.

A ship moves with constant velocity. At 09:00 it is at the point with position vector (3i + 7j) km. At 11:00 it is at the point with position vector (15i − 2j) km.

  1. Find the velocity of the ship in km h−1.
  2. Find the speed of the ship.
  3. Find the bearing on which the ship is moving.
6 marks
Show answer
a) (6i - 4.5j) km h-1 b) 7.5 km h-1 c) 127°
7

A particle moves with velocity (3ki + (k − 5)j) m s−1, where k is a constant.

Given that the particle is moving parallel to the vector (i − j),

  1. find the value of k,
  2. find the speed of the particle, giving your answer to 3 significant figures.
5 marks
Show answer
a) k = 1.25 b) 5.30 m s-1
8

Two forces F1 = (3i − 2j) N and F2 = (pi + qj) N are the only forces acting on a particle of mass 2 kg. The acceleration of the particle is (4i + 5j) m s−2.

  1. Find the values of p and q.
  2. Find the magnitude of the acceleration.
  3. Find the angle between the acceleration and the vector i.
6 marks
Show answer
a) p = 5, q = 12 b) 6.40 m s-2 c) 51.3°
9

A sledge is pulled across a frozen lake by a rope held by a child. In a simple model the sledge is a particle, the ice is smooth and the rope is light and inextensible. The model is used to predict the acceleration of the sledge.

  1. Explain why the acceleration predicted by the model is likely to be greater than the actual acceleration.
  2. Give one other limitation of modelling the sledge as a particle.
  3. Suggest one refinement to the model that would make the prediction more accurate.
4 marks
Show answer
a) the model ignores friction from the ice (and air resistance), so the resultant force in the model is larger than in reality b) it ignores the size and shape of the sledge, so air resistance and any turning of the sledge are not included c) include a friction force between the sledge and the ice (or include air resistance)
10

In this question i and j are unit vectors due east and due north, and position vectors are relative to a fixed point O.

At noon boat A is at O and boat B is at the point with position vector (10i + 2j) km. Boat A moves with constant velocity (3i + 4j) km h−1 and boat B moves with constant velocity (−2i + 5j) km h−1.

  1. Find the distance between the boats at noon.
  2. Show that, t hours after noon, the position vector of B relative to A is ((10 − 5t)i + (2 + t)j) km.
  3. Find the time at which B is due north of A, and the distance between the boats at that time.
7 marks
Show answer
a) 10.2 km b) b - a = (10 - 5t)i + (2 + t)j shown c) 14:00, 4 km apart
11

A drone flies in a straight line with constant velocity. At time t = 0 it is at the point with position vector (−4i + 6j) m relative to its controller at O. At t = 5 s it is at the point with position vector (16i − 9j) m.

  1. Find the velocity of the drone and its speed.
  2. Show that the square of the distance of the drone from O at time t seconds is 25t2 − 68t + 52
  3. By completing the square, find the shortest distance between the drone and O, and the time at which it happens.
8 marks
Show answer
a) velocity (4i - 3j) m s-1, speed 5 m s-1 b) |r|2 = 25t2 - 68t + 52 shown c) 2.4 m when t = 1.36 s
12

In this question i and j are unit vectors due east and due north.

On a training run, Kwame runs 120 m on a bearing of 060° in 30 s, and then 80 m due south in 20 s.

  1. Find his displacement from his starting point as a vector in terms of i and j, with numbers to 3 significant figures.
  2. Find the distance and bearing of his finishing point from his starting point.
  3. Find his average speed and the magnitude of his average velocity for the whole run.
  4. Explain why these two answers are different.
8 marks
Show answer
a) (104i - 20j) m b) 106 m on a bearing of 101° c) average speed 4 m s-1, average velocity magnitude 2.12 m s-1 d) speed uses the total distance (200 m) but velocity uses the straight-line displacement (106 m), and he changed direction

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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