Average velocity is displacement divided by the time ...

[Pages:6]Homework Set #2 VELOCITY

Where an object started and where it stopped does not completely describe the motion of the object. For example, the ground that you're standing on may move 8.0 cm to the left. This motion could take several years and be a sign of the normal slow movement of Earth's tectonic plates. If this motion takes place in just a second, however, you may be experiencing an earthquake or a landslide. Knowing the speed is important when evaluating motion.

Average velocity is displacement divided by the time interval

Consider the car in Figure 4. The car is moving along a highway in a straight line (the x-axis). Suppose that the positions of the car are xi at time ti and xf at time tf. In the time interval t = tf - ti, the displacement of the car is x = xf - xi. The average velocity, vavg, is defined as the displacement divided by the time interval during which the displacement occurred. In SI, the unit of velocity is meters per second, abbreviated as m/s.

AVERAGE VELOCITY

vavg

=

x t

=

xf - xi tf - ti

change in position displacement average velocity = =

change in time time interval

ti

xi tf

xf

Figure 4

The average velocity of this car tells you how fast and in which direction it is moving.

average velocity

the total displacement divided by the time interval during which the displacement occurred

The average velocity of an object can be positive or negative, depending on the sign of the displacement. (The time interval is always positive.) As an example, consider a car trip to a friend's house 370 km to the west (the negative direction) along a straight highway. If you left your house at 10 A.M. and arrived at your friend's house at 3 P.M., your average velocity would be as follows:

x -370 km

vavg =

t

=

5.0 h

= -74

km/h =

74 km/h west

This value is an average. You probably did not travel exactly 74 km/h at every moment. You may have stopped to buy gas or have lunch. At other times, you may have traveled more slowly as a result of heavy traffic. To make up for such delays, when you were traveling slower than 74 km/h, there must also have been other times when you traveled faster than 74 km/h.

The average velocity is equal to the constant velocity needed to cover the given displacement in a given time interval. In the example above, if you left your house and maintained a velocity of 74 km/h to the west at every moment, it would take you 5.0 h to travel 370 km.

Average velocity is not always equal to the average of the initial and final velocities. For instance, if you drive first at 40 km/h west and later at 60 km/h west, your average velocity is not necessarily 50 km/h west.

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For a variety of links related to this

chapter, go to Topic: Motion SciLinks Code: HF60996

Did you know?

The branch of physics concerned with motion and forces is called mechanics. The subset of mechanics that describes motion without regard to its causes is called kinematics.

Motion in One Dimension 43 1

Homework Set #2 SAMPLE PROBLEM A

Average Velocity and Displacement

PROBLEM

During a race on level ground, Andra runs with an average velocity of 6.02 m/s to the east. What is Andra's displacement after 137 s?

SOLUTION

Given:

vavg = 6.02 m/s t = 137 s

Unknown: x = ?

Rearrange the average velocity equation to solve for displacement.

x

vavg

=

t

x = vavg t

CALCULATOR SOLUTION

The calculator answer is 824.74 m, but both the values for velocity and time have three significant figures, so the displacement must be reported as 825 m.

x = vavg t = (6.02 m/s)(137 s) = 825 m to the east

Velocity Check: Practice Problems 1. Heather and Matthew walk with an average velocity of 0.98 m/s eastward. If it takes them 34 min to walk to the PRAstoCreT,IwChEat iAs their displacement?

Average Velocity and Displacement

1. Heather and Matthew walk with an average velocity of 0.98 m/s eastward. If it takes them 34 min to walk to the store, what is their displacement?

2. It takes you 9.25.miInf tJooewraildkewsihthisabnicayvcelreaigneavestloracitgyhotfli1n.2e fmo/rs1t5o mtheinnworitthhfaronmavthereabguesvesltopc-to the museum entrance. What is iytoyuorfd1is2p.5lackemm/ehnsto?uth, how far has he ridden?

3. It takes you 9.5 min to walk with an average velocity of 1.2 m/s to the north from the bus stop to the museum entrance. What is your displacement?

4. Simpson drives his car with an average velocity of 48.0 km/h to the east. How long will it take him to drive 144 km on a straight highway?

5. Look back at item 4. How much time would Simpson save by increasing 3. Simpson drives hishicsaarvweritahgaenvealvoecriatgyetove5l6o.c0itkymof/h48t.o0 tkhme/ehatsot?the east. How long will it take him to drive 144 km

on a straight highway? 6. A bus travels 280 km south along a straight path with an average velocity of 88 km/h to the south. The bus stops for 24 min. Then, it travels 210 km south with an average velocity of 75 km/h to the south. a. How long does the total trip last? b. What is the average velocity for the total trip?

44 Chapter 2

2

to the distance traveled divided by the time interval for the motion.

distance traveled

Homework Set #2

average speed = time of travel

Velocity is not the same as speed

InVeevleorcyidtyaycalanngbueaignet,etrhperteetremdsgsrpaepehdiacnadllyvelocity are used interchangeably. In physics, however, there is an important distinction between these two terms. AsThweevehlaovceityseoefna,nvoeblojeccittycadnebscerdibeetesrmmionteidonif twhiethobbjeoctth's paodsiitrieocntiiosnknaonwdn a nuamt sepreicciaflicvatilmuees(aalmonaggnitistupdaeth).inOdniecawtianygtohodwetefarmstisnoemtheitshiisngtommoavkees.aHgorawpehver,osfpteheedmhoatsionno. Fdiigreucrteio5nr,eoprnelsyenmtsagsuncithuadge.raApnh.oNbojetcict'estahvaetrtaimgeesipsepeldotitsedeqounal totthheehdoirsitzaonnctealtraaxviselaendddpivoisditeiodnbiys tphloetttiemdeoinnttherevvaelrftoicratlhaexims. otion.

The object moves 4.0 m in the time interval between t = 0 s and t = 4.0 s. Likewise, the object mavoevreasgeanspaededdit=ion daislt4a.n0cem trinavtehleedtime interval between t = 4.0 s and t = 8.0 s. From these data, wetismeeetohfattrtahveealverage velocity for each

of these time intervals is +1.0 m/s (because vavg = x/t = 4.0 m/4.0 s). VeBleoccaiutysectahne abveeriangteevreplroecitteyddgoreas pnhoticcahlalnyge, the object is moving with a con-

Thsteavnetlvoecliotycitoyf oafn+o1b.0jemct/cs,aannbdeitdsemteormtioineids riefptrheeseonbtejedctb'sy paosstirtaiiognhtisliknneoownn

attshpeepcoifsiictitoimn-etismaelognrgapiths. path. One way to determine this is to make a graph

of thFeomr oantiyopno. Fsiitgiounr-eti5mreepgreaspehn,tws esuccahn algsoradpehte. rNmoitniceeththeaatvteirmageeisvepllooctiteydboyn

thderhaowriinzgonatsatlraaixgihstalnindepboestwitieoen iasnpyltowttoedpoointtshoenvtehretigcraalpahx.isT.he slope of this

liTnheeinodbicjeactetsmthoeveasve4r.a0gemveinlotchitey tbimetwe eienntetrhvealpboesittwioenesnatn=d t0ims easnrdeptr=ese4n.0t-s.

Liekdewbiyset,htehse poobijnetcst. mToobvetstearnuandddeirtsitoannadl t4h.0is mconincetpht,ectoimepairnetetrhveaelqbueattwioenen

t =fo4r.0thseasnlodpte=of8t.0hes.liFnreowmitthhethse deqautaa,twioensfeoertthhaetatvheeragvervaegloecviteylo. city for each

BofectahuesseethtiemaeveirnatgeervSvaelollsopceiistoy+fd1ao.L0eisnmne/ost c(bheacnagues,ethveaovgbj=ectixAs/vmetroav=gine4gV.0ewloimtch/it4ay.0cosn)-.

sthtaenptsolvoseipltoeioc=nit-rrytuisionmef e=+g1 cr.h0acaphmnahgn./esg,ei nainnhdovreiitrzstoicm natloactloicooonrod riisdniranetapetsreessentedvbavyg

a =

strxa=ig xhft - lixnie t tf - ti

on

For any position-time graph, we can also determine the average velocity by

drawing a straight line between any two points on the graph. The slope of this

line indicates the average velocity between the positions and times represent-

ed by these points. To better understand this concept, compare the equation

for the slope of the line with the equation for the average velocity.

Position (m)

Position (m)

Integrating Technology

Visit go. for the activity "Methods of Transportation."

Keyword HF6MODX

16.0

12.0

8.0

4.0

0 0 2.0 4.0 6.0 8.0

Time (s)

Figur1e6.50

The motion of an object moving

with 1co2n.0stant velocity will provide a

straight-line graph of position ver-

sus tim8e..0The slope of this graph

indicates the velocity.

4.0

0 0 2.0 4.0 6.0 8.0

Time (s) Figure 5

The motion of an object moving with constant veOlobjceictyt 1will provide a straight-line graph of position versus time. The slope of this graph indicates the velocity.

Position

Slope of a Line

Average Velocity

slope

=

rise change in vertical coordinates run1ar.o=Buno cdhoaktnhegoenei dngahe oToraifzboal nettaablAlceotbooop rodkwiniitsahtmesdoimveednsoionvcnaesvg

x xf - xi =2.T=ra ve l Car A Netw Yotfrk-ttoi Miami

travels from at a speed of

1.75 m ? 2.25 m. If the book ends up at its initial 25 m/s. Car B travels from New

Figure 6 repporessiteinonts, wsthraaitgihsti-tlsindeispgrlaacpehmseonft?pIofsiitticoonm-vpelertseuss-ittsimeYfoorrkthtoreCehdicifa-go, also at a speed of

Object 2 Object 3

ferent objects.mOotbiojenctin123hass, wahactonissittasnatveproasgietivelovceiltoyc?iWtyhbatecisause25itms /ps.oAsrietiothne velocities of the cars

increases unifoitrsmalvyerwaigtehstpimeeed.?Thus, the slope of this line is positivee.qOubalj?ecEtx2plahians.

Time

zero velocity because its position does not change (the object is at rest). Hence,

Figure 6

the slope of this line is zero. Object 3 has a constant negative velocity because its

These position-versus-time graphs show that object 1 moves with a

position decreases with time. As a result, the slope of this line is negative.

constant positive velocity. Object 2

1. Book on a Table A book is moved once 2. Travel Car A trisavaetlrsesftr.oOmbject 3 moves with a

around the edge of a tabletop with dimensions New York to Miami atcoansstpaenetdneogfative velocity.

Instantaneo1.7u5s vmel?oc2it.2y5mma.yIfntohtebbeotohke esandmseuapsaatveitrsaigneitivael loc2i5tym/s. Car B travels from New

Now considpmeroosatiitnoiononbi,nwje2hc3tatwsi,shwiothssaetdipissopilstaisctieaomvneervnaegt?resIuvfesitlotcicmoitmye?pgWlertahepashtitiisss noY2to5ramk /ststo.rAaCirgehhitcthaegov,ealolscoitaietsaosfptheeedcaorfs line, but a cuitrsvaev,earsagine sFpiegeudr?e 7. The object moves through largeerqaunald? Elaxrpglaeirn.

3

displacements as each second passes. Thus, its velocity increases with time.

instantaneous velocity

Homework Set #2

Velocity vs. Speed Check: Practice Problem

An athlete swims from the north end to the south end of a 50.0 m pool in 20.0 s and makes the return trip to the starting position in 25.0 s.

a. What is the average velocity for the first half of the swim?

b. What is the average velocity for the second half of the swim?

c. What is the average velocity for the roundtrip?

d. What is the average speed for the round trip?

4

Introduction to Vectors

Distinguish between a scalar

and a vector.

SC

Homework Set #2

Add and subtract vectors by

In t

using the graphical method.

ited

lar

SCALARS AND VECTORS

Multiply and divide vectors by scalars.

the

wor

by

In the chapter "Motion in One Dimension," our discussion of motion was lim-

d.

des

ited to two directions, forward and backward. Mathematically, we described

s

these directions of motion with a positive or negative sign. That method

Vec

works only for motion in a straight line. This chapter explains a method of

describing the motion of objects that do not travel along a straight line.

scalar

Eac

Vectors indicate direction; scalars do not

a physical quantity that has

eith

magnitude but no direction

ma

Each of the physical quantities encountered in this book can be categorized as

and

either a scalar quantity or a vector quantity. A scalar is a quantity that has vector

has

magnitude but no direction. Examples of scalar quantities are speed, volume, a physical quantity that has both and the number of pages in this textbook. A vector is a physical quantity that magnitude and direction

A disp

has both direction and magnitude.

trip

As we look back to the chapter "Motion in One Dimension," we can see that

qua

oth

displacement is an example of a vector quantity. An airline pilot planning a

spe

trip must know exactly how far and which way to fly. Velocity is also a vector

eas

quantity. If we wish to describe the velocity of a bird, we must specify both its

speed (say, 3.5 m/s) and the direction in which the bird is flying (say, north-

east). Another example of a vector quantity is acceleration.

Vectors are represented by boldface symbols

In physics, quantities are often represented by symbols, such as t for

time. To help you keep track of which symbols represent vector quan-

tities and which are used to indicate scalar quantities, this book will

use boldface type to indicate vector quantities. Scalar quantities will

be in italics. For example, the speed of a bird is written as v = 3.5 m/s.

But a velocity, which includes a direction, is written as v = 3.5 m/s to

the northeast. When writing a vector on your paper, you can distin-

guish it from a scalar by drawing an arrow above the abbreviation for

a quantity, such as v = 3.5 m/s to the northeast.

One way to keep track of vectors and their directions is to use

diagrams. In diagrams, vectors are shown as arrows that point in the direction of the vector. The length of a vector arrow in a diagram is proportional to the vector's magnitude. For example, in Figure 1

Figure 1

The lengths of the vector arrows represent the magnitudes of these

the arrows represent the velocities of the two soccer players running two soccer players' velocities.

s se

toward the soccer ball.

82 Chapter 3

5

Homework Set #2 Extra Practice: Below is a position vs. time graph.

1. What is the velocity of the object? Is there any difference between the magnitude of the speed and the magnitude of the velocity for this object?

2. How long will it take the object to travel 1 km? 6

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