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Maths · Ratio and Proportion

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Speed, Distance, Time and Density

Using speed, distance and time, reading distance-time graphs, and calculating density and pressure.

  • 9 key terms
  • All boards

Learning Objectives

  1. 1Use speed \(= \dfrac{\text{distance}}{\text{time}}\) and rearrange it to find distance or time, converting between hours, minutes and decimals.
  2. 2Calculate average speed for a journey in more than one stage.
  3. 3Draw and interpret distance-time graphs.
  4. 4Use density \(= \dfrac{\text{mass}}{\text{volume}}\), and pressure \(= \dfrac{\text{force}}{\text{area}}\) (Higher tier).

Compound measures

A compound measure compares two different quantities, so it is written with "per" or a division sign: kilometres per hour, grams per cubic centimetre, newtons per square metre. In every case you divide one measure by the other. The exam trap is not the division but the units: a time of 2 hours 30 minutes is not 2.30 hours, and mixing the two is the most common way to lose marks in this topic.

Speed, distance and time

Speed is how far you go in one unit of time.

  • The units match

    If distance is in kilometres and time is in hours, speed is in km/h. For metres and seconds, speed is in m/s.

  • Changing minutes to hours

    30 minutes is 0.5 hours, 45 minutes is 0.75 hours, 20 minutes is \(\dfrac{1}{3}\) hour, and 1 hour 12 minutes is 1.2 hours because \(12 \div 60 = 0.2\).

  • Changing back

    2.5 hours is 2 hours 30 minutes, and 0.75 hours is 45 minutes. Multiply the decimal part by 60 to get minutes.

  • Always check it makes sense

    A car cannot travel at 600 km/h. If your speed looks silly, a unit is wrong.

Finding speed

A car travels 150 km in 2 hours 30 minutes. Work out its average speed in km/h.

Show the solutionHide the solution
  1. 1 Convert the time to hours 2 hours 30 minutes \(= 2.5\) hours.
  2. 2 Use speed \(=\) distance \(\div\) time \(150 \div 2.5\).
  3. 3 Calculate \(150 \div 2.5 = 60\).
  4. 4 Give the units 60 km/h.

Answer60 km/h

Finding time

A train travels 90 km at an average speed of 60 km/h. How long does the journey take? Give your answer in hours and minutes.

Show the solutionHide the solution
  1. 1 Use time \(=\) distance \(\div\) speed \(90 \div 60 = 1.5\) hours.
  2. 2 Convert the decimal \(0.5\) hours \(= 30\) minutes.
  3. 3 Write the answer 1 hour 30 minutes.

Answer1 hour 30 minutes

Average speed

Average speed for a whole journey is the total distance divided by the total time. It is not the average of the speeds.

  • Total distance over total time

    Add up all the distances and all the times first.

  • A common mistake

    Averaging two speeds, such as \(\dfrac{30 + 60}{2} = 45\), is wrong when the times are different.

  • Example

    60 km at 30 km/h takes 2 hours, and 60 km at 60 km/h takes 1 hour. The total is 120 km in 3 hours, so the average speed is 40 km/h.

  • Why

    More time is spent going slowly, so the average is pulled towards the lower speed.

Speeds from a distance-time graph

Using the graph above, work out the speed during the stage from 1.5 hours to 2.5 hours.

Show the solutionHide the solution
  1. 1 Read two points At 1.5 hours the distance is 40 km, and at 2.5 hours it is 100 km.
  2. 2 Distance travelled \(100 - 40 = 60\) km.
  3. 3 Time taken \(2.5 - 1.5 = 1\) hour.
  4. 4 Speed \(60 \div 1 = 60\) km/h.

Answer60 km/h

Density

Density tells you how much mass is packed into each unit of volume.

  • The formula

    Density \(= \dfrac{\text{mass}}{\text{volume}}\), usually in g/cm\(^3\) or kg/m\(^3\).

  • Example

    A metal block has mass 240 g and volume 30 cm\(^3\), so its density is \(240 \div 30 = 8\) g/cm\(^3\).

  • Rearranged

    Mass \(=\) density \(\times\) volume, and volume \(=\) mass \(\div\) density.

  • Another triangle

    Use the same layout as the speed triangle, with mass on top and density and volume underneath.

Using density

The density of a metal is 8 g/cm\(^3\). Work out the mass of 25 cm\(^3\) of the metal.

Show the solutionHide the solution
  1. 1 Choose the formula Mass \(=\) density \(\times\) volume.
  2. 2 Substitute \(8 \times 25\).
  3. 3 Calculate \(8 \times 25 = 200\).
  4. 4 Give the units 200 g.

Answer200 g

Pressure and changing units (Higher tier)

Pressure is another compound measure, and speeds can be given in different units.

  • Pressure

    Pressure \(= \dfrac{\text{force}}{\text{area}}\), in newtons per square metre (N/m\(^2\)). A force of 600 N on an area of 0.5 m\(^2\) gives a pressure of \(600 \div 0.5 = 1200\) N/m\(^2\).

  • Changing km/h to m/s

    Multiply by 1000 to get metres, and divide by 3600 for seconds. \(72\text{ km/h} = \dfrac{72 \times 1000}{3600} = 20\) m/s.

  • Changing m/s to km/h

    Multiply by 3600 and divide by 1000, which is multiplying by 3.6. 15 m/s is \(15 \times 3.6 = 54\) km/h.

  • Check the units

    Write the units on every line, and they tell you whether to multiply or divide.

Test yourself

  1. 1

    What is the formula for speed?

    Show answerHide answer

    Speed \(=\) distance \(\div\) time.

  2. 2

    What is 45 minutes as a decimal of an hour?

    Show answerHide answer

    0.75 hours.

  3. 3

    How long does 90 km take at 60 km/h?

    Show answerHide answer

    1.5 hours, which is 1 hour 30 minutes.

  4. 4

    What does a horizontal line on a distance-time graph mean?

    Show answerHide answer

    The object is stationary.

  5. 5

    What is the formula for density?

    Show answerHide answer

    Density \(=\) mass \(\div\) volume.

Exam technique: compound measures

These questions are won by careful units and a clear formula.

  • Write the formula first

    "Speed = distance ÷ time" shows the method before you substitute.

  • Convert time to hours

    Do it as a separate line so you can check it.

  • Include the units in the answer

    km/h, m/s or g/cm\(^3\), not just a number.

  • For average speed, use totals

    Find the whole distance and the whole time before dividing.

Summary and exam focus

  • Speed is distance divided by time, and the formula triangle gives distance and time too.
  • Convert minutes to a decimal of an hour before you divide: 30 minutes is 0.5 hours.
  • Average speed is total distance divided by total time.
  • On a distance-time graph the gradient is the speed and a flat line means stopped.
  • Density is mass divided by volume, and pressure is force divided by area (Higher tier).

Exam focus

Hira cycles 12 km in 45 minutes. Work out her average speed in km/h. (3 marks) (3 marks)

Change 45 minutes to 0.75 hours on a line of its own, then show \(12 \div 0.75 = 16\). The answer needs its unit, km/h. If you divide 12 by 45 you get 0.27, and a speed of 0.27 km/h should alert you that the units are wrong.

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Compound measure
A measure made by comparing two different quantities, such as speed (distance per time).
Speed
How far something travels in a unit of time, found by dividing distance by time.
Average speed
Total distance travelled divided by total time taken.
Distance-time graph
A graph showing the distance from the start against time, where the gradient is the speed.
Gradient
A measure of how steep a line is, found by dividing the change up by the change across.
Density
The mass of a material in each unit of volume, found by dividing mass by volume.
Mass
The amount of matter in an object, measured in grams or kilograms.
Volume
The amount of space an object takes up, measured in cubic units such as cm\(^3\).
Pressure
The force on each unit of area, found by dividing force by area.

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