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P8.1 Vectors and Scalars

GCSE Combined Science · AQA · Chapter P8: Forces in Balance

A free GCSE Combined Science: Trilogy lesson for AQA (8464) from OBA School: distance and displacement, speed and velocity, vector and scalar quantities, and drawing vectors to scale. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with every video, practice quizzes and printable worksheets with mark schemes, in our free online course.

What you will learn

  • Explain the difference between distance and displacement, and between speed and velocity
  • Define vector and scalar quantities and sort quantities into each group
  • Represent a vector with an arrow whose length shows its magnitude and whose direction shows its direction
  • Draw and use scale diagrams, always stating the scale

Key words

  • Magnitude: The size of a quantity, e.g. 40 N or 250 m. It does not include a direction.
  • Scalar: A quantity that has magnitude only, e.g. mass, distance, speed, time, energy, temperature.
  • Vector: A quantity that has magnitude and direction, e.g. force, displacement, velocity, acceleration.
  • Distance: How far something travels along its path. A scalar.
  • Displacement: The straight-line distance from start to finish in a stated direction. A vector.
  • Velocity: Speed in a stated direction. A vector.
  • Scale diagram: An accurate drawing in which a stated length represents a stated amount, e.g. 1 cm ≡ 5 N.

Distance and displacement

If you walk along a winding path, the distance you travel is the total length of the path. Your displacement is how far you end up from where you started, measured in a straight line, together with the direction. Displacement is distance in a stated direction.

Worked example: a walk to the library (shown in the figure (drawn in the printable lesson notes))

A student walks 500 m due north from a bus stop, then 1200 m due east to the library.
Distance = 500 m + 1200 m = 1700 m (no direction needed).
The displacement is the straight arrow from start to finish. Drawn on the grid (1 square ≡ 100 m) it measures 13 squares, so its magnitude is 1300 m.
Because the two legs are at right angles you can check with Pythagoras: √(500² + 1200²) = √1 690 000 = 1300 m.

Answer: distance 1700 m; displacement 1300 m at about 67° east of north

Key fact: If you finish where you started, your displacement is zero, however far you travelled. A swimmer who swims to the end of a pool and back has a displacement of 0 m.

Vectors and scalars

Every physical quantity has a magnitude (size). Some quantities also have a direction, and the direction matters: a 30 N push to the left has a very different effect from a 30 N push to the right.

  • displacement; Scalars (magnitude only) distance
  • velocity; Scalars (magnitude only) speed
  • force (including weight, friction, tension); Scalars (magnitude only) mass
  • acceleration; Scalars (magnitude only) time
  • momentum; Scalars (magnitude only) energy
  • gravitational field strength; Scalars (magnitude only) temperature, power

Some quantities come in pairs. Speed is a scalar: “the train travels at 40 m/s” only says how fast. Velocity is speed in a given direction: “the plane flies at 220 m/s due west” is a velocity. Mass (in kg) is a scalar, but weight is a force (in N) that acts downwards, so weight is a vector.

Exam tip: Ask yourself: “Does it make sense to say which way it points?” If yes, it is a vector. A force can push left or right, but a temperature or an energy does not point anywhere.

Representing a vector with an arrow

A vector can be shown as an arrow:

  • the direction of the arrow shows the direction of the vector
  • the length of the arrow represents the magnitude of the vector – a bigger force gets a longer arrow.

On a force diagram, label every arrow with the name of the force and its size. If the diagram is drawn to scale, write the scale on it, e.g. “1 cm ≡ 5 N” (≡ means “represents”).

Worked example: arrow lengths

Scale: 1 cm ≡ 5 N.
Pull of 35 N: length = 35 ÷ 5 = 7.0 cm. Friction of 10 N: length = 10 ÷ 5 = 2.0 cm.
Reading back: an arrow 4.6 cm long represents 4.6 × 5 = 23 N.

Answer: length (cm) = force ÷ (newtons per cm); force = length × (newtons per cm)

Scale diagrams of two vectors

To show two vectors on one scale diagram, choose a scale that makes the drawing a sensible size (the longest arrow roughly 5–15 cm). Draw the first arrow, then draw the second one at the correct angle using a protractor. Use a sharp pencil and a ruler.

Worked example: drawing the walk to scale

Choose 1 cm ≡ 200 m. The 500 m leg is 2.5 cm long (pointing north) and the 1200 m leg is 6.0 cm long (pointing east).
Join the start to the finish with a straight arrow. It measures 6.5 cm, so the displacement is 6.5 × 200 = 1300 m.
Measure the angle at the start with a protractor: about 67° east of north.

Answer: displacement = 1300 m at 67° east of north

Key fact: A scale diagram is only useful if the reader knows the scale. Always write it on the diagram, and measure lengths to the nearest millimetre.

Common mistakes

  • Giving a displacement or a velocity with no direction. Without a direction it is just a distance or a speed.
  • Adding the legs of a journey to get the displacement. Adding gives the distance; the displacement is the straight line from start to finish.
  • Calling mass a vector. Mass is a scalar; weight is the force (a vector) caused by gravity acting on that mass.
  • Drawing force arrows that are not to scale, or forgetting to state the scale on a scale diagram.

Revision questions

Try these, then check your answers below each one.

Question 1. Which of these quantities is a scalar? (Choose from: Temperature, Displacement, Weight, Velocity.)

Answer: Temperature

Working: Temperature has a size only, so it is a scalar. Displacement, weight and velocity all have a direction, so they are vectors.

Question 2. A student uses the scale 1 cm ≡ 40 N. How long should she draw the arrow for a force of 260 N?

Answer: 6.5 cm

Working: Length = 260 N ÷ 40 N per cm = 6.5 cm.

Question 3. True or false? A runner who completes exactly one 400 m lap and stops on the start line has a displacement of 400 m.

Answer: False

Working: False. She has run a distance of 400 m, but she finishes where she started, so her displacement is zero.

Related lessons

Continue in the full course

This lesson is section P8.1 (AGCSC3701) of our free GCSE Combined Science course for AQA. In the full course you get every video for this section, a self-marking practice quiz, a printable worksheet with a mark scheme, chapter tests, and your progress saved. Go to the full GCSE Combined Science course.

 

 

 

 

 

 

 

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