© T Madas

© T Madas

• Marc started from his house at 9 a.m. cycling at constant speed to Andy’s house, 7 km away.

• He arrived there one hour later.• Marc stayed at Andy’s for 1½ hours.• He started cycling back home.• Half hour into his return journey he stopped

at Elli’s house that lives 4 km from his house.

• He stayed at Elli’s for half an hour.• He continued cycling home at constant

speed arriving back at 2 p.m.

Graphs which show how the distance of an object from a starting point changes with time, are called distance-time graphs or simply travel-graphs

Draw a distance-time graph for this information

© T Madas

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© T Madas

• Marc started from his house at 9 a.m. cycling at constant speed to Andy’s house, 7 km away.

• He arrived there one hour later.• Marc stayed at Andy’s for 1½ hours.• He started cycling back home.• Half hour into his return journey he stopped

at Elli’s house that lives 4 km from his house.

• He stayed at Elli’s for half an hour.• He continued cycling home at constant

speed arriving back at 2 p.m.

Graphs which show how the distance of an object from a starting point changes with time, are called distance-time graphs or simply travel-graphs

Draw a distance-time graph for this information

© T Madas

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Time always plottedon the x axis

© T Madas

• Marc started from his house at 9 a.m. cycling at constant speed to Andy’s house, 7 km away.

• He arrived there one hour later.• Marc stayed at Andy’s for 1½ hours.• He started cycling back home.• Half hour into his return journey he stopped

at Elli’s house that lives 4 km from his house.

• He stayed at Elli’s for half an hour.• He continued cycling home at constant

speed arriving back at 2 p.m.

Graphs which show how the distance of an object from a starting point changes with time, are called distance-time graphs or simply travel-graphs

Draw a distance-time graph for this information

© T Madas

9:00 10:00 11:00 12:00 13:00 14:00

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Time

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Time always plottedon the x axis

© T Madas

• Marc started from his house at 9 a.m. cycling at constant speed to Andy’s house, 7 km away.

• He arrived there one hour later.• Marc stayed at Andy’s for 1½ hours.• He started cycling back home.• Half hour into his return journey he stopped

at Elli’s house that lives 4 km from his house.

• He stayed at Elli’s for half an hour.• He continued cycling home at constant

speed arriving back at 2 p.m.

Graphs which show how the distance of an object from a starting point changes with time, are called distance-time graphs or simply travel-graphs

Draw a distance-time graph for this information

© T Madas

9:00 10:00 11:00 12:00 13:00 14:00

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Time

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Time always plottedon the x axis

© T Madas

• Marc started from his house at 9 a.m. cycling at constant speed to Andy’s house, 7 km away.

• He arrived there one hour later.• Marc stayed at Andy’s for 1½ hours.• He started cycling back home.• Half hour into his return journey he stopped

at Elli’s house that lives 4 km from his house.

• He stayed at Elli’s for an half hour.• He continued cycling home at constant

speed arriving back at 2 p.m.

Graphs which show how the distance of an object from a starting point changes with time, are called distance-time graphs or simply travel-graphs

Draw a distance-time graph for this information

© T Madas

9:00 10:00 11:00 12:00 13:00 14:00

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Time

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Time always plottedon the x axis

© T Madas

• Marc started from his house at 9 a.m. cycling at constant speed to Andy’s house, 7 km away.

• He arrived there one hour later.• Marc stayed at Andy’s for 1½ hours.• He started cycling back home.• Half hour into his return journey he stopped

at Elli’s house that lives 4 km from his house.

• He stayed at Elli’s for half an hour.• He continued cycling home at constant

speed arriving back at 2 p.m.

Graphs which show how the distance of an object from a starting point changes with time, are called distance-time graphs or simply travel-graphs

Draw a distance-time graph for this information

© T Madas

9:00 10:00 11:00 12:00 13:00 14:00

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Time

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Time always plottedon the x axis

What do the “flat” parts of the graph represent?

Flat = not travelling/a stop

© T Madas

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Time always plottedon the x axis

What do the “uphill” parts of the graph represent?

Flat = not travelling/a stop

Uphill = travelling away

© T Madas

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Time always plottedon the x axis

What do the “down hill” parts of the graph represent?

Flat = not travelling/a stop

Uphill = travelling away

Downhill = travelling back

© T Madas

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What was Marc’s speed in cycling from his house to Andy’s house?

speed =distancetime

7 kmperhour

7 km /h

in 1 hour7 km 71

= =

=

© T Madas

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What was Marc’s speed in cycling from Andy’s house to Elli’s house?

6 kmperhour

6 km /h

in ½ hour3 km 3½

= =

=

in ½ hour3 km

in 1 hour6 km

6 km /h=

speed =distancetime

© T Madas

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What was Marc’s speed in cycling from Ellis’s house back home?

2.67 kmperhour

2.67 km /h

in 1½ hour4 km 41.5

= ≈

≈

in 1½ hour4 km

in 3 hours8 km

÷8 =383

=2 23

km /h

2.67 km /h≈

speed =distancetime

© T Madas

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What was Marc’s average speed for the entire journey?

2.8 kmperhour

2.8 km /h

in 5 hours14 km 145

= =

=

2.8 km /h=

speed =distancetime

km14282.8

Hours5

101

© T Madas

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The speed for each section is also represented by the slope of each section.

1

77 k

m/h ½

3

-6 k

m/h

1.5

4

-2.67 km/h

Gradient = Speed[in a distance-time graph]

The steeper the line,the greater the speed

speed =distancetime

© T Madas

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7 k

m/h

-6 k

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-2.67 km/h

Gradient = Speed[in a distance-time graph]

The steeper the line,the greater the speed

speed =distancetime

The speed for each section is also represented by the slope of each section.

© T Madas

Graphs which show how the distance of an object from a starting point changes with time, are called distance-time graphs or simply travel-graphs

Time is always plotted on the x axis

the “flat” parts of the graph indicate stops (no motion) the “uphill” parts indicate travelling away from the starting point the “downhill” parts indicate travelling back to the starting point

To find the speed we use:

The speed in a distance-time graph is given by the slope (also called gradient) of a line

Distance-Time Graphs Summary

speed =distancetime

© T Madas

© T Madas

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A sales rep drove from London to Manchester, stopping on the way at Birmingham.After his meeting in Manchester, he drove straight to London without a break . His entire journey is shown in the graph below:

Q1 How many miles is Birmingham from London? 120

miles

© T Madas

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A sales rep drove from London to Manchester, stopping on the way at Birmingham.After his meeting in Manchester, he drove straight to London without a break . His entire journey is shown in the graph below:

Q2 How long did he stop in Birmingham for? half an

hour

© T Madas

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A sales rep drove from London to Manchester, stopping on the way at Birmingham.After his meeting in Manchester, he drove straight to London without a break . His entire journey is shown in the graph below:

Q3 How far is Manchester from Birmingham? 90

miles

© T Madas

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A sales rep drove from London to Manchester, stopping on the way at Birmingham.After his meeting in Manchester, he drove straight to London without a break . His entire journey is shown in the graph below:

Q4 How long did the entire journey take 9 hours

© T Madas

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A sales rep drove from London to Manchester, stopping on the way at Birmingham.After his meeting in Manchester, he drove straight to London without a break . His entire journey is shown in the graph below:

Q5 What was the average speed for the driving parts of journey?average speed =total distance

total time

= 4206.5

≈ 65 miles/hour

© T Madas

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A sales rep drove from London to Manchester, stopping on the way at Birmingham.After his meeting in Manchester, he drove straight to London without a break . His entire journey is shown in the graph below:

Q6 What was the speed for each part of the journey?

2

120

60 m

ph

2

90

45 m

ph

2½

210

84 m

ph

speed =distancetime

Miles21042084084

Hours2½5

101

© T Madas

© T Madas

0 1 2 3 4 5 6 7 8 9 10 11Time in seconds

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The distance-time graph below shows a 100 metre race for 3 sprinters labelled as A, B and C.1. Who won the race and what was the winning time.2. How many seconds behind the winner was the runner up?3. Explain what happened 10 seconds into the race.4. What is a possible explanation for the run of sprinter C ?

A

C

A : 100 m in 10.5 sec B : 100 m in 11 secC : did not finish

1. 10.5 sec2. Half a second3. A overtook B4. Possibly a muscle pull

B

© T Madas

© T Madas

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¼ ½ ¾ 1 1¼ 1½ 1¾ 2

The travel-graph below shows a car journey from London to Gatwick Airport .

The 1st part of the journey to Croydon took 1 hour, due to traffic1. Calculate the speed for the London-Croydon section

2. Calculate the speed for the Croydon-Gatwick section

3. Calculate the average speed for the entire journey

speed =distancetime

1

40

40 km/h

½

80

160 k

m/h

© T Madas

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¼ ½ ¾ 1 1¼ 1½ 1¾ 2

The travel-graph below shows a car journey from London to Gatwick Airport .

The 1st part of the journey to Croydon took 1 hour, due to traffic1. Calculate the speed for the London-Croydon section

2. Calculate the speed for the Croydon-Gatwick section

3. Calculate the average speed for the entire journey

1½

12080

km

/h

average speed =total distancetotal time

= 1201.5

= 80 km/h

km12024080

Hours1½31

or

© T Madas

© T Madas

The distance – time graph shown opposite describes Jon’s journey.

Tick the correct boxes of the 4 statements which describe part A, part B and part C of his journey.

time

dis

tan

ce

A

B

C

… was walking in a North-East direction

… was walking back to his starting point

… was walking at his fastest

… was walking at constant speed

… took some rest

… was walking uphill

… was walking slower and slower

… was walking faster and faster

CBAOn this part, Jon …

© T Madas

© T Madas

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The following distance-time graph shows Lee’s egg and spoon race, from the starting line to the end of the track and back to the starting line.

1. What was the total distance run? 70 metres2. How many times did Lee drop the egg? Twice3. What was Lee’s average speed? approx 1.17 m/s4. Calculate Lee’s speed at various parts of the race

35 metres to the end of the track and 35 metres back

average speed =total distancetotal time

= 7060

≈ 1.17 m/s

© T Madas

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The following distance-time graph shows Lee’s egg and spoon race, from the starting line to the end of the track and back to the starting line.

1. What was the total distance run? 70 metres2. How many times did Lee drop the egg? Twice3. What was Lee’s average speed? approx 1.17 m/s4. Calculate Lee’s speed at various parts of the race

speed =distancetime

10

20

2 m

/s

15

151 m

/s

5

25

5 m

/s

20

10½ m/s

© T Madas

© T Madas

Two joggers start from the same point and run the same distance in 21 minutes as follows:

Jogger A: He runs every 800 metres in 5 minutes followed by a 3 minute restJogger B: She runs for 21 minutes at constant speed.

1. Plot a distance-time graph

2. What distance do the joggers cover?

3. After how many minutes does jogger B overtake jogger A for the first time?

4. After how many metres does jogger A overtake jogger B ?

© T Madas

Two joggers start from the same point and run the same distance in 21 minutes as follows:

Jogger A: He runs every 800 metres in 5 minutes followed by a 3 minute restJogger B: She runs for 21 minutes at constant speed.

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0 2 4 6 8 10 12 14 16 18 20 22 24 min

metr

es

A

B

© T Madas

Two joggers start from the same point and run the same distance in 21 minutes as follows:

Jogger A: He runs every 800 metres in 5 minutes followed by a 3 minute restJogger B: She runs for 21 minutes at constant speed.

1. Plot a distance-time graph

2. What distance do the joggers cover?

3. After how many minutes does jogger B overtake jogger A for the first time?

4. After how many metres does jogger A overtake jogger B ?

© T Madas

Two joggers start from the same point and run the same distance in 21 minutes as follows:

Jogger A: He runs every 800 metres in 5 minutes followed by a 3 minute restJogger B: She runs for 21 minutes at constant speed.

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A

B

© T Madas

Two joggers start from the same point and run the same distance in 21 minutes as follows:

Jogger A: He runs every 800 metres in 5 minutes followed by a 3 minute restJogger B: She runs for 21 minutes at constant speed.

1. Plot a distance-time graph

2. What distance do the joggers cover?

3. After how many minutes does jogger B overtake jogger A for the first time?

4. After how many metres does jogger A overtake jogger B ?

© T Madas

Two joggers start from the same point and run the same distance in 21 minutes as follows:

Jogger A: He runs every 800 metres in 5 minutes followed by a 3 minute restJogger B: She runs for 21 minutes at constant speed.

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es

A

B

© T Madas

Two joggers start from the same point and run the same distance in 21 minutes as follows:

Jogger A: He runs every 800 metres in 5 minutes followed by a 3 minute restJogger B: She runs for 21 minutes at constant speed.

1. Plot a distance-time graph

2. What distance do the joggers cover?

3. After how many minutes does jogger B overtake jogger A for the first time?

4. After how many metres does jogger A overtake jogger B ?

© T Madas

Two joggers start from the same point and run the same distance in 21 minutes as follows:

Jogger A: He runs every 800 metres in 5 minutes followed by a 3 minute restJogger B: She runs for 21 minutes at constant speed.

2800

2400

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800

400

0 2 4 6 8 10 12 14 16 18 20 22 24 min

metr

es

A

B

© T Madas

Two joggers start from the same point and run the same distance in 21 minutes as follows:

Jogger A: He runs every 800 metres in 5 minutes followed by a 3 minute restJogger B: She runs for 21 minutes at constant speed.

1. Plot a distance-time graph

2. What distance do the joggers cover?

3. After how many minutes does jogger B overtake jogger A for the first time?

4. After how many metres does jogger A overtake jogger B ?

© T Madas

© T Madas

0 1 2 3 4 5 6 7 8 9 10 11Time in seconds

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A thief seeking to avoid capture runs through a park with a speed of 6 m/s

As he runs past Nick, Nick sets in pursuit of the thief, on his mountain bike.

Nick cycles at a speed of 9 m/s but took 3 seconds to react and get on his bike.

Assuming that Nick did not see the thief until he was running past him:1. Draw a distance

time graph showing this information.

2. Hence find how many seconds does Nick have to cycle until he draws level with the thief

© T Madas

A thief seeking to avoid capture runs through a park with a speed of 6 m/s

As he runs past Nick, Nick sets in pursuit of the thief, on his mountain bike.

Nick cycles at a speed of 9 m/s but took 3 seconds to react and get on his bike.

Assuming that Nick did not see the thief until he was running past him:1. Draw a distance

time graph showing this information.

2. Hence find how many seconds does Nick have to cycle until he draws level with the thief

The thief’s speed is 6 m/s

1 sec2 sec3 sec

10 secetc

6 m12 m18 m60 metc

0 1 2 3 4 5 6 7 8 9 10 11Time in seconds

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20

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Dis

tance

in m

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from

Nic

k’s

spot

© T Madas

A thief seeking to avoid capture runs through a park with a speed of 6 m/s

As he runs past Nick, Nick sets in pursuit of the thief, on his mountain bike.

Nick cycles at a speed of 9 m/s but took 3 seconds to react and get on his bike.

Assuming that Nick did not see the thief until he was running past him:1. Draw a distance

time graph showing this information.

2. Hence find how many seconds does Nick have to cycle until he draws level with the thief

1 sec2 sec3 sec4 sec

etc

9 m18 m27 m36 metc

0 1 2 3 4 5 6 7 8 9 10 11Time in seconds

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90

80

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Dis

tance

in m

etr

es

from

Nic

k’s

spot

4 sec5 sec6 sec7 sec

etc

Nick’s speed is 9 m/s

© T Madas

A thief seeking to avoid capture runs through a park with a speed of 6 m/s

As he runs past Nick, Nick sets in pursuit of the thief, on his mountain bike.

Nick cycles at a speed of 9 m/s but took 3 seconds to react and get on his bike.

Assuming that Nick did not see the thief until he was running past him:1. Draw a distance

time graph showing this information.

2. Hence find how many seconds does Nick have to cycle until he draws level with the thief

0 1 2 3 4 5 6 7 8 9 10 11Time in seconds

100

90

80

70

60

50

40

30

20

10

Dis

tance

in m

etr

es

from

Nic

k’s

spot

Nick had to cycle for 6 seconds

How far from Nick’s spot did Nick draw level?• Using the graph• Using the thief’s

speed• Using Nick’s speed

© T Madas

© T Madas

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Time in seconds

A ball is thrown vertically upwards and its height at subsequent times is recorded. The results were plotted in the graph below.

1. By drawing the tangents on this curve, estimate the speed of the ball at times t = 2 and t = 4.

2. What is the meaning of a negative gradient?

2.5

25.5

gradient =25.52.5 = 10.2

© T Madas

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Time in seconds

A ball is thrown vertically upwards and its height at subsequent times is recorded. The results were plotted in the graph below.

1. By drawing the tangents on this curve, estimate the speed of the ball at times t = 2 and t = 4.

2. What is the meaning of a negative gradient?

2.5

-23.5

gradient =25.52.5 = 10.2

gradient =-23.52.5 = -9.2

at t = 2speed ≈ 10.2 m/s

at t = 4speed ≈9.2 m/s

© T Madas

© T Madas

The graph below shows a journey on a lift, starting at the ground floor and returning to the ground floor sometime later.

•Find the time the lift spent stationary and the time it spent moving

•Show that the lift is travelling faster when going down

•Calculate the average speed for the lift when moving in m/s, if floors are 3 metres apart

Time (seconds)

Floor

Nu

mber

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

20

15

10

5

total time:

1st stop:2nd stop:3rd stop:

time stationary:

time moving:

150 s

10 s 25 s 15 s

50 s

100 s

© T Madas

The graph below shows a journey on a lift, starting at the ground floor and returning to the ground floor sometime later.

•Find the time the lift spent stationary and the time it spent moving

•Show that the lift is travelling faster when going down

•Calculate the average speed for the lift when moving in m/s, if floors are 3 metres apart

Time (seconds)

Floor

Nu

mber

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

20

15

10

5

4 floorsin 15 s

8 floorsin 30 s

4 floorsin 15 s 16 floors

in 40 s

8 floorsin 20 s

4 floorsin 10 s

© T Madas

The graph below shows a journey on a lift, starting at the ground floor and returning to the ground floor sometime later.

•Find the time the lift spent stationary and the time it spent moving

•Show that the lift is travelling faster when going down

•Calculate the average speed for the lift when moving in m/s, if floors are 3 metres apart

Time (seconds)

Floor

Nu

mber

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

20

15

10

5

time moving:

floors up:floors down:total floors:

total distance:

100 s

161632

96 m

speed =distancetime

= 96100= 0.96

m/s

© T Madas