354 Chapter 9

9.59.5

If you are in a car that travels 80 km along a road in one hour, we saythat you are travelling at, on average, 80 km/h. You might not havebeen travelling at the same speed all the time: you might have speededup to 100 km/h to pass other vehicles, or even stopped for gas. Butoverall, during the entire hour, your average speed was 80 km/h.

Average Speed

To find the speed of a car in units of kilometres per hour(km/h), we divide the distance (in kilometres) by the time (inhours). Therefore speed v is distance ∆d divided by the time ∆t.(Some examples of units for distance, time, and speed are givenin Table 1.) Average speed, vav, is the total distance divided bythe total time for a trip. Transformed into quantity symbols, thedefining equation looks like this.

vav = �∆∆

dt� vav is read as “average speed.”

∆ is read as “change in” (delta is the fourth capital letter in the Greek alphabet).

∆d is read as “change in distance,” “elapsed distance,” or “distance.”

∆d = d2 � d1, where d1 is one distance measurement and d2 is a later distance measurement; d1 is often zero.

∆t is read as “change in time,” “elapsed time,” “period oftime,” or “time.”

∆t = t2 � t1, where t1 is one time measurement and t2 isa later time measurement.

t1, the starting time, is often zero.

Quantity symbols (Table 1) are italic letters used to representquantities (such as distance, time, and speed) in scientific equations.These quantity symbols have general international agreement,although alternatives are sometimes suggested. The quantity symbolsare not unique because there are only 26 letters in our alphabet, whilethere are many more quantities. For example, d can representdistance, diameter, or density, t can communicate time ortemperature, and v can refer to speed or volume. Quantity symbolsare always typed in italics while unit symbols are not. For example,while m is the quantity symbol for mass, m is the unit symbol formetres. Quantity symbols are found in equations (e.g., E = mc2)while unit symbols are found with values (e.g., 3.7 m).

Relating Speed to Distance and Time

Some SI Quantities and UnitsQuantity Quantity Sample Unit

symbol unit symbol

distance d millimetre mmcentimetre cmmetre mkilometre km

time t second sminute minhour hyear a

speed v metres per m/ssecondkilometres km/hper hour

Table 1

Distance and Speed 355

Instantaneous Speed

The car in Figure 1, at the time the photo was taken,was stationary. Its speed at that moment was 0 km/h.When the car was passing a truck on the road, itsspeedometer might have shown a speed of 100 km/h.These are examples of instantaneous speeds.Instantaneous speed is the speed at which an objectis travelling at a particular instant. It is not affectedby its previous speed, or by how long it has been moving.

High instantaneous speeds contribute to theseriousness of road traffic accidents. The two mostfamiliar instruments that measure instantaneousspeed were designed to improve road safety: thespeedometer and the “radar gun.”

Constant Speed

If the instantaneous speed of a car remains the same over a period oftime, then we say that the car is travelling with constant speed (oruniform motion). Some vehicles have cruise control to keep themmoving along at a fairly constant speed. Few objects maintain constantspeed over a lengthy period of time, however, often because of friction.However, we often assume constant (uniform) speed for the purposes ofsimplifying our physics calculations.

The average speed of an object is the same as its instantaneous speedif that object is travelling at constant speed.

Sample Problem 1Eiko skates to school, a total distance of 4.5 km(Figure 2). She has to slow down twice to cross busystreets, but overall the journey takes her 0.62 h. Whatis Eiko’s average speed during the trip?

∆d = 4.5 km

∆t = 0.62 h

vav = ?

vav = �∆∆

dt

�

= �40..562

kmh

�

= 7.3 km/h

Eiko’s average skating speed is 7.3 km/h.

The Greeks did not have aconcept of speed until Autolycusof Pitane (about 300 B.C.), definedconstant speed as a speed inwhich equal distances aretraversed in equal times. We usethe same definition today.

Did You Know ?

Figure 2

In-line skating is often a quick way to get to school.

Figure 1

At this moment the instantaneous speed of the car is zero,but this will change when the traffic light becomes green.

356 Chapter 9

Sample Problem 2Imagine that you are riding on the Cariboo Dayliner,in the dome car of course (Figure 3), and you see asign that reads 120 km. You decide, after seeingseveral such signs, that you are going to measure theelapsed time between the next two signs, which are 10 km apart. You read the elapsed time as 390.6 s.Determine the speed of the train in kilometres perhour during the elapsed time.

∆d = 10 km

∆t = 390.6 s � �16m0

isn

� � �60

1

m

h

in� = 0.1085 h

vav = ?

vav = �∆∆

dt

�

= �0.

11008

k5m

h�

= 92 km/h

The average speed of your train is 92 km/h.

Sample Problem 3Kira is trying to predict the time required to ride her bike tothe nearby beach. She knows that the distance is 45 km and,from other trips, that she can usually average about 20 km/h,including slowing down for climbing hills. Predict how longthe trip will take.

Alternatively, you can substitute and then rearrange the equation.

Kira should be able to make the trip in about 2.3 h.

vav = �∆∆d

t�

20 �khm� = ��

45∆kt

m�

∆t =

= 2.3 h

45 km�

20 �khm�

∆d = 45 km

vav = 20 km/h

∆t = ?

Figure 3

The Cariboo Dayliner yields a scenic view of Canada.

There is an easy method of convertingbetween km/h and m/s.

1 m/s = 3.6 km/h

exactly, because all of the conversionfactors used were exact values, bydefinition.

vav = �∆∆d

t�

∆t = �∆va

d

v

�

=

= 2.3 h

45 km�

20 �khm�

∆d = 45 km

vav = 20 km/h

∆t = ?

On April 15, 1999, a Japanesemagnetically levitated trainbroke its own world speedrecord by travelling at 552 km/h.

Did You Know ?

Distance and Speed 357

Note that the units in Sample Problem 3 simplify as follows:

= km � �khm� = h

Sample Problem 4Janna has a summer job helping with bison research (Figure 4). She notesthat they graze (move and eat grass) at an average speed of about 110 m/hfor about 7.0 h/d. What distance, in kilometres, will the herd travel in twoweeks (14 d)?

According to Janna’s observations, after two weeks the bison will havecovered a distance of about 11 km.

km��khm�

vav = 110 m/h

∆t = 7.0 �hd

� × 14 d = 98 h

∆d = ?

vav = �∆∆

dt

� or

∆d = vav∆t

= 110 �mh� × 98 h

= 11 km

Figure 4

A few buffalo still roam.

vav = �∆∆d

t�

110 �mh� = �

9∆8

dh

�

∆d = 110 �mh� × 98 h

= 11 km

358 Chapter 9

Understanding Concepts1. (a) How is average speed different from instantaneous

speed?

(b) When are they the same?

2. A car and a truck travel along the same highway with thecar moving faster than the truck.

(a) How do their distances travelled compare after thesame length of time?

(b) How do their times compare after travelling the samedistance?

3. Holidays might mean a multiday trip to be taken by foot,boat, train, or automobile. The Trans Canada Trail (Figure 5),for example, has become a popular hiking and cyclingvacation route.

(a) If two hikers walk the Trans Canada Trail for 6.0 h, andcovered 31 km, what is their average speed for the day?

(b) If three bike riders on the Trail cycle for 6.0 h one day, andcover 85 km, what is their average speed for the day?

(c) Mary walked for 2.1 h along a portion of the TransCanada Trail at a speed of 3.6 km/h. What distance didMary travel?

(d) What length of time would it take a hiker to travel a totaldistance of 25.0 km at an average speed of 5.2 km/h?

4. The cruise control of a car is set at 90.0 km/h. Whatdistance is travelled by the car during 2.50 h?

5. Show that 1 m/s = 3.6 km/h.

6. Use the conversion factor in question 5 to convert

(a) 92 km/h to m/s;

(b) 21 m/s to km/h.

7. (a) The Breitling Orbiter 3 balloon (Figure 6) set worldrecords in 1999 by travelling 40 814 km in 19 d, 21 h, and47 min. On March 1, 1999, the balloon lifted off from avillage in the Swiss Alps. It eventually landed in Egypt onMarch 21. Calculate the average speed of the balloon.

(b) Using the average speed you calculated in (a), whatlength of time did it take the Breitling Orbiter 3 to crossthe Atlantic Ocean, a distance of 6670 km?

(c) In the final leg of the round-the-world trip, the balloonflew for 18 h at an average speed of 210 km/h. How fardid it travel?

8. In 1997, Thrust SSC, the world’s fastest jet-engine car,travelled 604 m at an average speed of 341 m/s.

(a) What length of time did this take?

(b) Convert 341 m/s to kilometres per hour.

9. In a marathon race, one runner moving at 5.0 m/s passes asecond runner moving at 4.5 m/s. What is the distancebetween the runners 10 min after the one runner passedthe other?

Figure 5

HudsonBay

Labrador Sea

Fort McMurray

Norman Wells

Tuktoyaktuk

Inuvik

Fort Smith

Saskatoon

Thunder Bay

Kenora

ancouver

Edmonton

Quebec

Regina

toria

Fredericton

BritishColumbia

BritishColumbia

Banff

Calgary

Manitoba

AlbertaAlberta

Manitoba

Windsor

OntarioOntario

QuebecQuebec

Sault Ste. Marie

Toronto

North Bay

Ottawa

Montreal Halifax

Moncton

Charlottetown

Sydney

St. Jo

hn’s

P.E.I.P.E.I.

Nova Scotia

Nova Scotia

The Trans Canada TraiThe Trans Canada Trai

N

ATLANTICOCEAN

Medicine Hat

Whitehorse

Saskatchewan

Saskatchewan

Winnipeg

N e w f o u n d l a n d

a n d L a b r a d o r

N e w f o u n d l a n d

a n d L a b r a d o r

Dawson City

0 200 400 600

kilometres

N u n a v u tN u n a v u t(1999)

Lambert Conformal Conic Projection

75˚75˚

30˚

45˚

60˚75˚90˚105˚120˚135˚

60˚

N.B.N.B.

YukonTerritory

YukonTerritory

Territories

Northwest

NorthwestTerritories

Figure 6

Distance and Speed 359

10. The “hand” of the Canadarm (Figure 7) used on the spaceshuttle can move up to 60 cm/s without a load attached.

(a) What is the minimum time for the Canadarm’s hand tomove 1.20 m?

(b) When the Canadarm is moving an object, the speed isslightly less than 60 cm/s. To move the same distanceof 1.20 m, will the time be more or less than youranswer to (a)? Explain your answer.

(c) The Canadarm takes 30 s to move some equipment fromthe cargo bay. During this time, the space shuttle moves232 km through space. What is the speed of the spaceshuttle in kilometres per second? in kilometres per hour?

11. The following lab report describes how recordings weretaken during a road trip. Complete the Analysis andEvaluation section using the observation, recorded in Table 2.

QuestionWhat is the average speed for a multiple-stage car trip tomy grandparents’ home?

Evidence

Analysis and Evaluation(a) During which time periods was the car moving?

Calculate the speed during each of these separatetime periods.

(b) Calculate the average speed for the trip as a whole,from the beginning of Stage 1 to the end of Stage 5.

(c) Explain the differences in your answers for (a) and(b).

(d) Evaluate the experimental design. What would havemade the design more efficient? Would thismodification have changed the answer?

Reflecting12. Sample Problem 4 on page 357 shows the calculation of

the distance moved by the bison herd after 14 d. Is Jannalikely to find the bison 11 km away? What does thecalculated distance not take into account? Whatinformation is missing in the distance the bison move?

Figure 7

8A

Evaluating the EvidenceSKILLS HANDBOOK: 8A

A Trip to the Grandparents’Time Initial Final Initial Final

period or odometer odometer time timestage of reading reading (h:min) (h:min)

trip (km) (km)

1 36 252.1 36 260.7 8:04 8:142 36 260.7 36 260.7 8:14 8:323 36 260.7 36 542.3 8:32 11:304 36 542.3 36 542.3 11:30 11:525 36 542.3 36 709.6 11:52 13:27

Table 2

2 Laws are created, tested, and then used. How would youtest the average speed equation (a law)?

3 Relate what you have just learned to driving. How does acar’s speed affect the distance it travels during the driver’s“reaction time”?

Challenge