O LevelMathsCambridgeRevision NotesCoordinate Geometry & GraphsReal-Life GraphsDistance-Time & Speed-Time Graphs

Distance-Time & Speed-Time Graphs (Cambridge O Level Maths)

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Distance-Time Graphs

How does a distance-time graph work?

  • Distance-time graphs show distance from a fixed point at different times
    • Distance is on the vertical axis, and time is on the horizontal axis.
  •  The gradient of the graph is the speed
    • speed space equals space distance over time space equals space rise over run
  • A positive gradient represents the object (or person) moving away from the starting point
  • If the graph is a horizontal line the object is stationary (not moving)
  • A negative gradient represents the object (or person) moving towards the starting point
  • If the graph is a straight line the speed is constant
  • If the graph is a curve you can draw the tangent at a point on the graph and find its gradient
    • This will be an estimate of the speed at that point

Examiner Tip

  • It is easy to get confused between different types of graph.
    • Look at the label on the vertical axis to make sure you are looking at a DISTANCE-time graph (not speed-time)

Worked example

One afternoon Mary cycled to her grandparents' house, 8 km from her own home. 

Part of her travel graph for her journey is shown below. 

real-life-graphs-worked-example-1

Mary stayed at her grandparents' house for half an hour. 

She then cycled home at a steady speed, without stopping, arriving home at 4 pm.  

a)
Complete the travel graph for Mary's journey. 
  
Begin by checking the scale on the time axis. Note that one square is 15 minutes. 
 
Mary stays at her grandparents' house for 30 minutes, so draw a horizontal line for 2 squares to show this.
 
Her cycle home is represented by a straight line (steady speed) drawn from the end of her stay to 4pm on the time axis (where the distance from home is zero).
  
real-life-graphs-worked-example-answer-1

b)
For how long did Mary stop on the way to her grandparents' house?
 
Mary's stop on the way is the short horizontal line from 1.30 pm to 1.45 pm. The horizontal line is one square long so represents 15 minutes. 
 
Mary stopped for 15 minutes. 
 
c)
What is Mary's speed between 1.45 pm and 2.45 pm?
 
Speed can be found on a distance-time graph by finding the gradient of the line at that point.
speed space equals space fraction numerator space rise over denominator run end fraction equals fraction numerator distance space space over denominator time end fraction
 RPiptmV-_real-life-graphs-worked-example-answer-2 
speed space equals space fraction numerator 6 space km over denominator 1 space hour end fraction space equals space 6 space km divided by straight h
6 km/h

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Speed-Time Graphs

What is a speed-time graph?

  • Speed-time graphs show speed at different times
    • Speed is on the vertical axis, and time is on the horizontal axis
  • The gradient of the graph is the acceleration

    • Acceleration space equals space speed over time space equals space rise over run

  • If the graph is a curve you can draw the tangent at a point on the graph and find its gradient
    • This will be an estimate of the acceleration at that point
  • A positive gradient shows positive acceleration (speeding up)
  • A horizontal line on a speed-time graph shows constant speed (no acceleration)
  • A negative gradient shows negative acceleration, or deceleration (slowing down)
  • The distance covered can be found by finding the area under the graph

Examiner Tip

  • It is easy to get confused between different types of graph.
    • Look at the label on the vertical axis to make sure you are looking at a SPEED-time graph (not distance-time)

Worked example

The speed-time graph for a car travelling between two sets of traffic lights is shown below. 

real-life-graphs-s-t-graph-we-image

a)
Calculate the acceleration in the first 6 seconds. 
 
In a speed-time graph the acceleration is the gradient of the graph. 
acceleration space equals space fraction numerator space rise over denominator run end fraction equals space speed over time
real-life-graphs-s-t-graph-we-image-2
 
acceleration space equals space fraction numerator 9 space straight m divided by straight s over denominator 6 space straight s end fraction space equals space 1.5 space fraction numerator straight m divided by straight s over denominator straight s end fraction
Acceleration = 1.5 m/s2
 
b)
Work out the distance covered by the car. 
  
In a speed-time graph the distance travelled is equal to the area under the graph. 
 
The graph is a trapezium so use the formula   Area space equals space fraction numerator open parentheses a space plus space b close parentheses h over denominator 2 end fraction.
  
Area space equals space fraction numerator open parentheses 9 space plus thin space 20 close parentheses space cross times space 9 over denominator 2 end fraction space equals space 261 over 2 space equals space 130.5 space
  
Distance travelled = 130.5 m

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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.