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Kinematics

Kinematics

Distance-time graphs

Distance Time Graphs Core + Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. Reading a distance-time graph
  4. Two journeys on one graph
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 2.9: distance-time (travel) graphs
  • 2.9Read and draw distance-time graphs; a flat section means stationary.
  • 2.9Find speed as the gradient, in the right units.
  • 2.9Find where and when two journeys meet.

Before you start: Speed, distance and time and gradient. Next: Speed-time graphs.

Formula bank

Speed from the graphLearn itspeed = gradient = \(\dfrac{\text{change in distance}}{\text{change in time}}\)change minutes to hours for km/h.Use it when a straight sloping section.
Average speedLearn it\(\dfrac{\text{total distance}}{\text{total time}}\)include stops.Use it when the whole journey.

Reading a distance-time graph

Journey graph204060801004812t (min)d (km)
A cycle ride: out 12 km, a rest, then back home.
LineMeaning
sloping upmoving away
flatstationary (stopped)
sloping downcoming back
steeperfaster
curvedspeed changing

General methodHow to find a speed from a distance-time graph

  1. Pick the straight section; read the distance at its start and end.
  2. Read the time at its start and end; change minutes to hours (÷ 60) if you want km/h.
  3. Speed = change in distance ÷ change in time.

Example From the graph, find the speed for the first 30 minutes.

12 km in 30 min = 0.5 h.

  1. \(12\div0.5\)

Answer 24 km/h

Try it yourself

1
From the graph, the return speed (12 km in 45 min) in km/h?
Show answer

Answer 16

2
From the graph, how long was the rest, in minutes?
Show answer

Answer 15

3
10 km in 25 minutes. Speed in km/h?
Show answer

Answer 24

Two journeys on one graph

Where two lines cross, the two people are at the same place at the same time - they meet. Read the time on the horizontal axis and the distance on the vertical axis.

Two travellers204060801004812t (min)d (km)meet (22.5, 9)
Ali (blue) cycles out at 24 km/h; Ben (pink) walks back from 12 km at 8 km/h.

They meet after 22.5 minutes, 9 km from Ali's home. Without a graph: they close the 12 km gap at 24 + 8 = 32 km/h, so the time is 12 ÷ 32 = 0.375 h = 22.5 min.

Ways they test it

Type 1 Read the story

"How long did he stop for? How far from home was he at 10 30?" (1 mark each)

Spot it a distance-time graph and simple questions.

  1. Flat part = stop.
  2. Read straight off the axes.

Type 2 Speed of a section

"Find the speed between 10 15 and 10 45, in km/h." (2 marks)

Spot it 'speed' and a time interval.

  1. Distance change ÷ time change in hours.

Type 3 Complete the graph

"He returns home at 15 km/h. Complete the graph." (2 marks)

Spot it a missing section.

  1. Time = distance ÷ speed.
  2. Draw from the last point to (end time, 0).

Type 4 Meeting

"When and where do they pass each other?" (2-3 marks)

Spot it two lines on one graph.

  1. Read the crossing point, or use the closing speed.

Mistakes to avoid

  • ✗ Speed 0.4 (km per minute) given as km/h✓ × 60: 24 km/hUnits.
  • ✗ Flat line = constant speed✓ stationaryThe distance isn't changing.
  • ✗ Average speed ignoring the stop✓ include all the timeTotal time.

Summary

  • Gradient = speed; flat = stopped; down = coming back.
  • Change minutes to hours for km/h.
  • Crossing lines = the two people meet.

If you see …

  • "find the speed" from a graphdistance change ÷ time change, in hours.
  • "when do they meet?"read the crossing point.

Practice

Practice: easy, medium, hard

Easy

1
6 km in 20 minutes. Speed in km/h?
Show answer

Answer 18

2
6 km in 40 minutes. Speed in km/h?
Show answer

Answer 9

3
A graph is flat from 10 15 to 10 40. How long was the stop, in minutes?
Show answer

Answer 25

Medium

1
12 km out in 30 min, a 15 min rest, back in 45 min. Average speed for the whole trip (km/h)?
Guide me

What is it asking? average speed

What do we know? total 24 km

Plan total 90 min = 1.5 h

Show working
  1. \(24\div1.5=16\)

Answer 16

2
A runner goes 9 km in 45 minutes. Outward speed (km/h)?
Guide me

What is it asking? a speed

What do we know? 45 min = 0.75 h

Plan 9 ÷ 0.75

Show working
  1. \(9\div0.75=12\)

Answer 12

3
She then rests 15 minutes and runs the 9 km back at 12 km/h. How long does the return take, in minutes?
Guide me

What is it asking? a time

What do we know? t = d ÷ s

Plan 9 ÷ 12 h

Show working
  1. \(9\div12=0.75\) h = 45 min

Answer 45

Hard

1
Ali leaves home cycling at 24 km/h towards a village 12 km away. At the same time Ben leaves the village walking towards Ali at 8 km/h. After how many minutes do they meet?
Guide me

What is it asking? a time

What do we know? closing speed 24 + 8

Plan 12 ÷ 32 hours

Show working
  1. \(12\div32=0.375\) h
  2. = 22.5 min

Answer 22.5

PartAnswer / workingMarksGuidance
closing speed 32M1
0.375 hM1
22.5 minA1
Total: 3 marks
2
How far from Ali's home do they meet? (km)
Guide me

What is it asking? a distance

What do we know? Ali's speed × time

Plan \(24\times0.375\)

Show working
  1. \(24\times0.375=9\)

Answer 9

PartAnswer / workingMarksGuidance
9B1
Total: 1 mark
3
A runner starts at 09 00, runs 9 km in 45 min, rests 15 min, and returns at 12 km/h. At what time does she get home?
Guide me

What is it asking? a time

What do we know? 45 + 15 + 45 minutes

Plan add to 09 00

Show working
  1. 105 minutes
  2. 09 00 + 1 h 45 min = 10 45

Answer 10 45

PartAnswer / workingMarksGuidance
return 45 minM1
10 45A1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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