© Nuffield Foundation 2011

Nuffield Free-Standing Mathematics Activity

Model the motion

© Nuffield Foundation 2011

Model the motion

What would a velocity–time graph look like for each of these?

How can we model motion?

Are there any connections between the graphs?

What would a displacement–time graph look like?

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t (s)

x (m)

0 10 20 30

60

120

180

30

90

150

Sprinter runs at 9 ms-1 for 10 seconds, stops for 10 seconds, then runs at 9 ms-1 for another 10 seconds in the same direction.

0 10 20

3

6

9

30

v (ms-1)

t (s)

Gradient = tx

x

t

x

t

1090 = 9

The gradient of the displacement–time graph gives the velocity

Area under the velocity–time graph gives the displacement

Area = 10 9 = 90

Think about: What would a velocity–time graph look like?

Think about: What would a displacement–time graph look like?

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Sprinter runs at 9 ms-1 for 10 seconds, stops for 10 seconds, then takes 10 seconds to run back to his starting point.

010 20

– 3

3

9

30

v (ms-1)

6

– 6

– 9

t (s)

Gradient = 9 ms-1

The gradient of the displacement–time graph gives the velocity

Area under the velocity–time graph gives the displacement

Area = 90 m Area = –90 m

Gradient 1090- = –9 ms-1

t (s)

x (m)

0 10 20

30

60

90

30

x

t

x

t

Total = 0

Think about: What would a velocity-time graph and a displacement-time graph look like?

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t (s)

x (m)

0 4 8 122 6 10

200

100

A mechanic drives a car at a steady speed of 20 ms-1 for 8 seconds then puts on the brakes to come to a halt 4 seconds later.

The gradient of the displacement–time graph gives the velocity

Area under the velocity–time graph gives the displacement

10

20v (ms-1)

t (s)0 4 8 122 6 10

= 40 m 2204 8

160 = 20 ms-1 Area A = 160 m

A

Area B

B

Gradient B varies from 20 to 0 ms-1

B

Gradient A

A

Think about: What would a velocity–time graph and a displacement–time graph look like this time?

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A driver drives her car 1.2 km to a garage, spends 8 minutes checking her tyres then drives the car back home again.

0

10

v (ms-1)

5

– 5

– 10

t (min)4 8 122 6 10

= 10 ms-1

Gradient gives the velocity Area gives the displacement

Area = 10 120 Area = – 10 120

Gradient 1201200- = – 10 ms-1

x (m)

0 2 10 124 6 8

400

800

1200

200

600

1000

t (min)

x

t

x

t

Gradient 1201200 = 1200 m

= – 1200 m

Think about: Can you sketch a displacement-time graph?

Think about: Can you sketch a velocity-time graph?

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The distance between two bus stops is 60 metres. A bus sets off from the first bus stop and reaches a speed of 10 ms-1 before braking to stop at the second bus stop.

Gradient gives the velocityArea gives the displacement

t (s)

v (ms-1)

0 4 8 122 6 10

10

5

t (s)

x (m)

20

40

60

10

30

50

0 4 8 122 6 10

= 40 m

= 20 m

2108

2104

Area A

A

Area B

B

Gradient A varies from 0 to 10 ms-1

A

Gradient B varies from 10 to 0 ms-1

B

Think about: Can you sketch a velocity-time graph?

Think about: What is the displacement in each part?

Think about: What does the displacement-time graph look like?

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A remote controlled car reverses 10 metres then travels 30 metres forward at the same speed. This takes a total of 20 seconds.

Gradient gives the velocity Area gives the displacement

x (m)

0

10

20

– 10

15 205t (s)

10

0

1

2

v (ms-1)

– 1

– 2

t (s)15 205 10

= 2 ms-1 Area = 2 15

Area = – 2 5

Gradient 510- = –2 ms-1

x

t

x

t

Gradient 1020 = 30 m

= –10 m

Total = 20 m

Think about: What would a displacement-time graph and a velocity-time graph look like this time?

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Model the motionReflect on your work

Is speed the same as velocity?

What is the difference between distance and displacement?

How realistic are the graphs as models of the motion described?

How are velocity–time graphs and displacement–time graphs related?

What would more realistic graphs look like?