Physical

10

Graphs of Physical

Phenomena

MIKE'S CAR

You really ought to get your car serviced.

ACCELERATION Start/Stop

Chapter Contents

10:01 Distance/time graphs A Review of linear graphs Investigation: Graphing coins B Non-linear graphs Challenge: Rolling down an inclined plane

10:02 Relating graphs to physical phenomena Investigation: Spreadsheet graphs Fun Spot: Make words with your calculator Challenge: Curves and stopping distances

Mathematical Terms, Diagnostic Test, Revision Assignment, Working Mathematically

Learning Outcomes

Students will be able to: ? Interpret and construct both linear and non-linear distance/time graphs. ? Relate and interpret graphs of physical phenomena.

Areas of Interaction

Approaches to Learning (Knowledge Acquisition, Problem Solving, Communication, Logical Thinking, Reflection), Human Ingenuity, Environments

263

10:01 | Distance/Time Graphs

10:01A Review of linear graphs

As covered in Book 3:

? A distance?time graph (or travel graph) can be a type of line graph used to describe one or more trips or journeys.

? The vertical axis represents distance from a certain point, while the horizontal axis represents time.

? The formulae that connect distance travelled (D), time taken (T) and average speed (S) are given below.

D = S?T

D S = -T---

T

=

D -S---

Distance (km)

Example 1

50 40 30 20 10

worked examples

The travel graph shows the journey made by a cyclist. a How many hours was the journey? b How far did the cyclist travel? c What was the cyclist's average speed? d Between what times did the cyclist stop to rest? e Between what times was the cyclist travelling

fastest?

0 8 10 12 2 4 6 8 am am noon pm pm pm pm Time of day

Solution 1

a The cyclist began at 8 am and finished at 7 pm. That is a total of 11 hours.

b He travelled a total of 45 km.

c Average speed = Total distance travelled ? the time taken

= 45 ? 11

=

4

--1--11

km/h

d Between 12 am and 1:30 pm the line is flat which means no distance was travelled during

that time. This must have been the rest period.

e When the graph is at its steepest, more distance is being covered per unit of time. So this

is when the cyclist is travelling fastest. This would be from 6 pm to 7 pm.

264 INTERNATIONAL MATHEMATICS 4

Distance from Mike's house (km)

Example 2

25 20 15 10

5 0

8 10 12 2 4 6 8 am am noon pm pm pm pm

Time of day

The graph shows the journeys of two friends, Mike and Mal.

They each leave their own house at 8 am and walk to the other's house to see who can walk the fastest. a Which graph shows Mike's journey? b When do the two friends pass one another? c Who has the longest rest? d What is the speed of each friend? e Who walks the fastest at any time in their

journey?

Solution 2

a Because the graph shows the distance from Mike's house, the person who starts 0 km from

there must be Mike. Therefore the blue graph is Mike and the red graph is Mal.

b They pass one another when they are the same distance from Mike's house. So they pass one

another where the graphs cross -- at 2:15 pm.

c Mike rests for 1 hour, Mal rests for 2 hours (where the graphs are flat) so Mal has the

longest rest. d Mike's speed = 1----7-7--?--5-

Mal's speed = 1----7-8--?--5-

= 2?5 km/h

= 2?1875 km/h

e Mal walks the fastest from 5 pm to 8 pm as the gradient is the steepest of all.

Gradient

=

4-3

.

At

no

other

time

is

the

gradient

steeper

than

this.

Exercise 10:01A

1 500

Tamara

Louise

Distance travelled (km)

400

300

200

100

0 0 2 4 6 8 10 Time taken

Two women, Tamara and Louise, are travelling along the same road. Their progress is shown on the graph. a Who started first? b Who stopped for a rest? c Who had the fastest average speed? d When did they meet on the road? e What was the fastest speed for either of them?

265 CHAPTER 10 GRAPHS OF PHYSICAL PHENOMENA

2 The graph shows the journey taken by Max as he went for a training run on his bicycle.

a When was Max travelling fastest? What was the speed at this time?

b Did he stop? If so, for how long? c How many kilometres did Max cycle? d What was his average speed for the entire trip? e Max's brother Thilo did the same trip, but

cycled at a constant speed all the way and did not stop. Show his journey on the same graph.

Distance from home (km)

100 80 60 40 20 0 0 2 4 6 8 10 Time taken

3 A family left Hamburg by car at 9 am. They drove 200 km in the first 2 hours then stopped for half an hour for lunch. Then they drove 150 km along the autobahn in the next hour. They then left the autobahn in the next hour. They then left the autobahn and drove at an average speed of 50 km/h for the last 1?5 hours of their journey. a Draw a graph showing their journey. b What was their average speed for the whole trip?

4 The graph shows the progress of a group of bushwalkers hiking in bush over a number of days. It shows their distance from the start of the hike which is at the ranger's station.

a How far did they hike? b Did they hike every day? If not, on which

day did they rest? c On which day did they hike the least distance? d On average, how far did they hike per day? e How much more than the average did they

hike on the last day?

Distance from Ranger Station (km)

25 20 15 10

5 0

01 23 4 5 Day

266 INTERNATIONAL MATHEMATICS 4

tion

5 500

The graph shows the journey taken by two motorists -- one represented in blue and the other in red.

Distance from Beijing (km)

a If the speed limit on all the roads travelled is

400

80 km/h, did either motorist break the speed

300

limit? If so which one?

b Apart from when they stopped, what was the

200

slowest speed for each motorist?

c Which motorist drove most consistently?

100

d What can be said about the average speed

of the two motorists?

0 0 1 2 3 4 5 6 7 8 9 10 Time (hours)

e If both motorists were in the same make of car, and the fuel consumption is 8 L/100 km under 70 km/h and 10 L/100 km over 70 km/h, which

motorists will use the least fuel?

Investigation 10:01 | Graphing coins

By rolling a coin along a wooden ruler it is possible to graph the position of a coloured mark as the coin moves. The mark on the coin is highest when it is at the top. The mark is actually touching the ground when it is at the bottom. ? The greatest height is equal to the diameter of the coin. ? The smallest height is zero, which occurs when the

mark is on the ground. Because the coin is rolling, the height of the mark will oscillate between these two positions as the distance rolled increases.

investiga 10:01

Height of mark

Distance rolled

1 Choose three coins of different sizes. Produce a separate graph for each, similar to the one above.

2 Superimpose the three graphs onto one, using a different colour for each graph.

267 CHAPTER 10 GRAPHS OF PHYSICAL PHENOMENA

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