Linear Motion Notes

8/12/2019 Linear Motion Notes

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LINEAR MOTION

More than 2000 years ago, the ancientGreek scientists were familiar withsome of the ideas of physics thatwe study today. They had a good under-standing of some of the properties oflight, but they were confused aboutmotion. Great progress in understand-ing motion occurred with Galileo andhis study of balls rolling on inclinedplanes, as discussed in the previouschapter. In this chapter, we look atmotion in more detail.

You can describe the motion ofan object by its position, speed,direction, and acceleration.

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Do Objects Fall Faster the LongerThey Fall?1. Attach washers to a 3.5 m long string at the

following distances from one end: 0.0 m,0.11 m, 0.44 m, 0.99 m, 1.76 m and 2.76 m.

2. Stand on a chair or desk with the string andwashers suspended over a piece of metalsuch as a pie tin. The washer tied to the endof the string should be just touching themetal surface.

3. Release the string and listen to the rate atwhich the washers hit the metal. You maywish to perform a second trial to confirm yourobservations.

Analyze and Conclude1. Observing Describe the time intervals

between sounds.2. Predicting What would you hear if the wash-

ers were evenly spaced on the string?3. Making Generalizations What can you say

about the distance traveled by a falling objectduring each second of fall?

discover!

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LINEARMOTION

Objectives Explain how you can tell an

object is moving. (4.1)

Describe how you can calculatespeed. (4.2) Distinguish between speed and

velocity. (4.3) Describe how you can calculate

acceleration. (4.4) Describe the acceleration of an

object in free fall. (4.5) Describe how the distance

fallen per second changes foran object in free fall. (4.6)

Describe what the slope ofa speed-versus-time graphrepresents. (4.7)

Describe how air resistanceaffects the motion of fallingobjects. (4.8)

Explain the relationshipbetween velocity andacceleration. (4.9)

discover!

MATERIALS washers, string, pietin, meter stick

EXPECTED OUTCOME The timebetween the sounds will bethe same.

ANALYZE AND CONCLUDEThe time between thesounds was the same.The sounds would be heardat progressively shortertime intervals.The distance traveledduring each second of fallincreases.

TEACHING TIP The distancebetween masses wascalculated using d 5 1/2 gt 2 with time intervals of 0.15 s.

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CHAPTER 4 LINEAR MOTION 47

4.1 Motion Is Relative

Everything moves. Even things that appear to be at rest move. Theymove with respect to the sun and stars. When we describe the motionof one object with respect to another, we say that the object is movingrelative to the other object. A book that is at rest, relative to the tableit lies on, is moving at about 30 kilometers per second relative to thesun. The book moves even faster relative to the center of our galaxy.When we discuss the motion of something, we describe its motionrelative to something else. An object is moving if its positionrelative to a fixed point is changing. When we say that a spaceshuttle moves at 8 kilometers per second, we mean its movementrelative to Earth below. When we say a racing car in the Indy 500reaches a speed of 300 kilometers per hour, of course we mean rela-tive to the track. Unless stated otherwise, when we discuss the speeds

of things in our environment, we mean speed with respect to thesurface of Earth even though Earth moves around the sun, as shownin Figure 4.1. Motion is relative.CONCEPT

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How can you tell if an object is moving?

FIGURE 4.1Although you may be atrest relative to Earthssurface, youre movingabout 100,000 km/h rela-tive to the sun.

A hungry mosquito seesyou resting in a hammockin a 3-meter per secondbreeze. How fast and inwhat direction shouldthe mosquito fly in orderto hover above you forlunch?Answer: 4.1

think!

FIGURE 4.2The racing cars in the Indy 500move relative to the track.

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4.1 Motion isRelative

Key Termrelative

Teaching Tip Begin witha description of motion thatwill be more quantitative thanlater material. This idea thatthe course does not becomeprogressively more difficultshould lessen anxiety for studentswho have read Chapter 4 andfound it intimidating. Spend less

time on Chapter 4 to have timefor more interesting physics later!

An object is movingif its position relative

to a fixed point is changing.

Te a c h i n g R e s o u r c e s

Reading and StudyWorkbook

Presentation EXPRESS

Interactive Textbook Conceptual Physics Alive!

DVDs Linear Motion

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Wisdom is knowing what tooverlook; good teaching isknowing what to omit. Donot spend too much time onthis chapter. Time spent onkinematics is time not spent onwhy satellites dont crash toEarth, why high temperaturesand high voltages can be safe totouch, why rainbows are round,and how nuclear reactions keepEarths interior molten.


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Any combination of units for distance and time that are usefuland convenient are legitimate for describing speed. Miles per hour(mi/h), kilometers per hour (km/h), centimeters per day (the speedof a sick snail?), or light-years per century are all legitimate units forspeed. The slash symbol (/) is read as per. Throughout this book,well primarily use the units meters per second (m/s) for speed.Table 4.1 shows some comparative speeds in different units.

4.2 SpeedBefore the time of Galileo, people described moving things as simplyslow or fast. Such descriptions were vague. Galileo is credited asbeing the first to measure speed by considering the distance coveredand the time it takes. Speed is how fast an object is moving. Youcan calculate the speed of an object by dividing the distancecovered by time.

Speed distancetime

For example, if a cheetah, such as the one shown in Figure 4.3,covers 50 meters in a time of 2 seconds, its speed is 25 m/s.

If a cheetah can maintain aconstant speed of 25 m/s,it will cover 25 meters

every second. At this rate,how far will it travel in10 seconds? In 1 minute?Answer: 4.2.1

think!

FIGURE 4.3A cheetah is the fastest

land animal over dis-tances less than 500meters and can achievepeak speeds of 100 km/h.

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4.2 SpeedKey Termsspeed, instantaneous speed,average speed

Teaching Tip Define speedby writing speed 5 distance/timeon the board while givingexamplesautomobilespeedometers, etc. Use the deltasymbol ( D ) from the beginning,even though the text waits untillater. So speed 5 D distance/ D time.Stress that the slash in yournotation is a division sign thatmeans peras in so manykilometers per so many hours.Then walk across the floor with1-m strides once per second and

explain that you cover a distanceof 1 meter per second; henceyour speed is 1 m/s.

Ask If you ride a bike adistance of 5 m in 1 s, what isyour speed? What is your speedfor 10 m in 2 s? For 100 m in 20 s?Each answer is 5 m/s.

Teaching Tip Youllappreciably improve yourinstruction if you allow somethinking time (say 3 s) after youask a question. If no one answersin 5 s, your class may need aneasier question first.

Teaching Tidbit Faster thanthe cheetah is the cosmopolitansailfish, the fastest fish over shortdistances (>100 km/h).


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Instantaneous Speed A car does not always move at thesame speed. A car may travel down a street at 50 km/h, slowto 0 km/h at a red light, and speed up to only 30 km/hbecause of traffic. You can tell the speed of the car at anyinstant by looking at the cars speedometer, such as theone in Figure 4.4. The speed at any instant is calledthe instantaneous speed. A car traveling at 50 km/hmay go at that speed for only one minute. If the carcontinued at that speed for a full hour, it would cover50 km. If it continued at that speed for only half anhour, it would cover only half that distance, or 25 km.In one minute, the car would cover less than 1 km.

Average Speed In planning a trip by car, the driver often wantsto know how long it will take to cover a certain distance. The carwill certainly not travel at the same speed all during the trip. Thedriver cares only about the average speed for the trip as a whole.The average speed is the total distance covered divided by the time.

average speed total distance coveredtime interval

Average speed can be calculated rather easily. For example, if wedrive a distance of 60 kilometers during a time of 1 hour, we say ouraverage speed is 60 kilometers per hour (60 km/h). Or, if we travel240 kilometers in 4 hours,

average speed total distance coveredtime interval 240 km4 h

60 km/h

Note that when a distance in kilometers (km) is divided by a time inhours (h), the answer is in kilometers per hour (km/h).

Since average speed is the distance covered divided by the time oftravel, it does not indicate variations in the speed that may take placeduring the trip. In practice, we experience a variety of speeds on mosttrips, so the average speed is often quite different from the instanta-neous speed. Whether we talk about average speed or instantaneous

speed, we are talking about the rates at which distance is traveled.If we know average speed and travel time, the distance traveled iseasy to find. A simple rearrangement of the definition above gives

total distance covered average speed travel time

For example, if your average speed is 80 kilometers per hour on a4-hour trip, then you cover a total distance of 320 kilometers.

N EPT

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How can you calculate speed?

The speedometer in everycar also has an odometerthat records the distancetraveled. If the odometerreads zero at the begin-ning of a trip and 35 kma half hour later, what isthe average speed?Answer: 4.2.2

think!

FIGURE 4.4The speedometer for aNorth American car givesreadings of instantaneousspeed in both mi/h andkm/h. Odometers for theU.S. market give readingsin miles; those for theCanadian market givereadings in kilometers.

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You can calculatethe speed of an

object by dividing the distancecovered by time.



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High school students easilyconfuse the concepts of speed,velocity, and accelerationbecause the concepts aredifficult to distinguish andbecause they usually appearearly in the course. Beware ofspending too much time on

Chapter 4. Make the distinctionsamong speed, velocity, andacceleration and move on. Dontget bogged down in Chapter 4.


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Change in Speed When straight-line motion is considered, it iscommon to use speed and velocity interchangeably. When the direc-tion is not changing, acceleration may be expressed as the rate atwhich speed changes.

acceleration (along a straight line) change in speed

time interval

Speed and velocity are measured in units of distance per time.Since acceleration is the change in velocity or speed per time interval,its units are those of speed per time. If we speed up, without chang-ing direction, from zero to 10 km/h in 1 second, our change in speedis 10 km/h in a time interval of 1 s. Our acceleration along a straightline is

acceleration 10 km/h schange in speed

time interval

10 km/h1 s

The acceleration is 10 km/h .s, which is read as 10 kilometersper hour-second. 4.4.2. Note that a unit for time appears twice: oncefor the unit of speed and again for the interval of time in which thespeed is changing.CONCEPT

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How do you calculate acceleration?

Change in Direction Acceleration also applies to changes indirection. If you ride around a curve at a constant speed of 50 km/h, you feel the effects of acceleration as your body tends to move towardthe outside of the curve. You may round the curve at constant speed,but your velocity is not constant, because your direction is changingevery instant. Your state of motion is changing: you are accelerating.It is important to distinguish between speed and velocity.Acceleration is defined as the rate of change in velocity, rather thanspeed.Acceleration, like velocity, is a vector quantity because it isdirectional. The acceleration vector points in the direction the veloc-ity is changing, as shown in Figure 4.7. If we change speed, direction,or both, we change velocity and we accelerate.

In 5 seconds a car movingin a straight line increasesits speed from 50 km/hto 65 km/h, while a truckgoes from rest to 15 km/hin a straight line. Whichundergoes greateracceleration? What isthe acceleration ofeach vehicle?Answer: 4.4.2

think!

FIGURE 4.7When you accelerate in thedirection of your velocity,you speed up; against yourvelocity, you slow down; atan angle to your velocity,your direction changes.

Mathematics is thelanguage of nature.

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Teaching Tip Refer studentsto Appendix B, Working withUnits in Physics, which explainsin detail how to add, subtract,multiply, and divide quantitiesthat contain both numbers and

units of measurement. Ask What is the acceleration

of a car that goes from rest to100 km/h in 10 s? 10 km/h ? sWhat is the acceleration of amechanical part that movesfrom 0 to 10 m/s in a time of 1 s? 10 m/s 2

You can calculate theacceleration of an

object by dividing the change inits velocity by time.



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Although the accelerationof free fall ( g ) at Earthssurface is about 9.8 m/s 2 (32 ft/s 2), I round this offto 10 m/s 2 to more easilyestablish the velocity anddistance relationships. Whenmaking measurements in the

lab the more precise 9.8 m/s2 should be used.


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4.5 Free Fall: How Fast

An apple falls from a tree. Does it accelerate while falling? We knowit starts from a rest position and gains speed as it falls. We know thisbecause it would be safe to catch if it fell a meter or two, but not if itfell from a high-flying balloon. Thus, the apple must gain more speed

during the time it drops from a great height than during the shortertime it takes to drop a meter. This gain in speed indicates that theapple does accelerate as it falls.

Falling Objects Gravity causes the apple to accelerate downwardonce it begins falling. In real life, air resistance affects the accelerationof a falling object. Lets imagine there is no air resistance and thatgravity is the only thing affecting a falling object. An object mov-ing under the influence of the gravitational force only is said to bein free fall. Freely falling objects are affected only by gravity. Table4.2 shows the instantaneous speed at the end of each second of fallof a freely falling object dropped from rest. The elapsed time is thetime that has elapsed, or passed, since the beginning of any motion,in this case the fall.

Note in Table 4.2 the way the speed changes. During each secondof fall, the instantaneous speed of the object increases by an addi-tional 10 meters per second. This gain in speed per second is theacceleration.

acceleration 10 m/s 2change in speed

time interval

10 m/s

1 s

Note that when the change in speed is in m/s and the time interval isin s, the acceleration is in m/s 2, which is read as meters per secondsquared. The unit of time, the second, occurs twiceonce for theunit of speed and again for the time interval during which the speedchanges.

During the span of the

second time interval inTable 4.2, the objectbegins at 10 m/s andends at 20 m/s. What isthe average speed ofthe object during this1-second interval? Whatis its acceleration?Answer: 4.5.1

think!

Free fall to a sky divermeans fall before theparachute is opened,usually with lots of

air resistance. Physicsterms and everydayterms often mean dif-ferent things.

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4.5 Free Fall:How Fast

Key Termsfree fall, elapsed time

Teaching Tip Drop an objectfrom your outstretched hand and

ask if it accelerated. Ask what itmeans to say it accelerated at arate of 10 m/s 2. Explain that inevery second of fall the objects

speed increases by 10 m/s. Thisis made clearer if studentssuppose the falling object isequipped with a speedometer.Even without a knowledge ofphysics, most people instinctivelyknow that the speed of a fallingobject increases with time. (Thatswhy one wouldnt hesitate tocatch a baseball dropped froma height of 1 m, but would bequite reluctant to catch the samebaseball dropped from a tallbuilding.) Your students shouldnow understand that if a freelyfalling object were equippedwith a speedometer, the speedreading would increase by 10 m/sfor each successive second of fall.

Ask If an object at rest isdropped from the top of a cliff,how fast will it be going at theend of 1 s? 10 m/s

Teaching Tip Explain theanswer to the above Ask andthen repeat the process. Askfor the speed at the end of 2 s,and then for 10 s. This leadsyou into stating the relationD v 5 g D t , which you can expressin shorthand notation. Theinformation in Table 4.2 is showngraphically in Figure 4.11.


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The acceleration of an object in free fall is about 10 metersper second squared (10 m/s 2). For free fall, it is customary to use theletter g to represent the acceleration because the acceleration is dueto gravity. Although g varies slightly in different parts of the world,its average value is nearly 10 m/s 2. More accurately, g is 9.8 m/s 2, butit is easier to see the ideas involved when it is rounded off to 10 m/s 2.

Where accuracy is important, the value of 9.8 m/s 2 should be used forthe acceleration during free fall. Note in Table 4.2 that the instanta-neous speed of an object falling from rest is equal to the accelerationmultiplied by the amount of time it falls, the elapsed time.

instantaneous speed acceleration elapsed time

The instantaneous speed v of an object falling from rest after anelapsed time t can be expressed in equation form 4.5

v gt

The letter v symbolizes both speed and velocity. Take a momentto check this equation with Table 4.2. You will see that whenever theacceleration g = 10 m/s 2 is multiplied by the elapsed time in seconds,the result is the instantaneous speed in meters per second.

The average speed of any object moving in a straight line withconstant acceleration is calculated the way we find the average of anytwo numbers: add them and divide by 2. For example, the averagespeed of a freely-falling object in its first second of fall is the sum ofits initial and final speed, divided by 2. So, adding the initial speed

of zero and the final speed of 10 m/s, and then dividing by 2, we get5 m/s. Average speed and instantaneous speed are usually verydifferent.

What would the speed-ometer reading on thefalling rock shown inFigure 4.8 be 4.5 secondsafter it drops from rest?How about 8 secondsafter it is dropped?Answer: 4.5.2

think!

In Figure 4.8 we imagine a freely-falling boulder with a speed-ometer attached. As the boulder falls, the speedometer shows that theboulder acquires 10 m/s of speed each second. This 10 m/s gain eachsecond is the boulders acceleration.

FIGURE 4.8

If a falling rock were somehowequipped with a speedometer,in each succeeding second of fallits reading would increase by the

same amount, 10 m/s. Table 4.2shows the speed we would readat various seconds of fall.

Since acceleration isa vector quantity, itsbest to say the acceler-ation due to gravity is10 m/s 2 down .

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Teaching Tip Distinguishbetween the acceleration offree fall and the deceleration aball undergoes when it strikesthe floor. Deceleration uponimpact is considerably greaterthan 10 m/s 2 because the velocitychange occurs in a much briefertime. Explain that deceleration issimply acceleration opposite thedirection of velocity.

Teaching Tidbit Theacceleration due to gravity is10 m/s each second all the waydown.

Teaching Tidbit Wisdom?Knowing not to throw a brickstraight up.


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Can You Catch a Falling Bill?1. Have a friend hold a dollar bill so

the midpoint hangs between yourfingers.

2. Have your friend release the billwithout warning. Try to catch it!

3. Think How much reaction time do you have whenyour friend drops the bill?

discover!

Rising Objects So far, we have been look-ing at objects moving straight downward due to

gravity. Now consider an object thrown straightup. It continues to move upward for a while,then it comes back down. At the highest point,when the object is changing its direction ofmotion from upward to downward, its instanta-neous speed is zero. It then starts downward justas if it had been dropped from rest at that height.

During the upward part of this motion, theobject slows from its initial upward velocity tozero velocity. We know the object is accelerat-ing because its velocity is changing. How muchdoes its speed decrease each second? It shouldcome as no surprise that the speed decreases atthe same rate it increases when moving down-wardat 10 meters per second each second.So as Figure 4.9 shows, the instantaneous speed at points of equal elevation in the path is thesame whether the object is moving upward ordownward. The velocities are different of course,

because they are in opposite directions. Duringeach second, the speed or the velocity changesby 10 m/s. The acceleration is 10 m/s 2 downwardthe entire time, whether the object is movingupward or downward.

CONCEPT

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. . What is the acceleration of an

object in free fall?

FIGURE 4.9The change in speedeach second is thesame whether theball is going upwardor downward.

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discover!

MATERIALS dollar billEXPECTED OUTCOME It will beimpossible for the friend tocatch the falling dollar billbecause the reaction time isgreater than the time it takes

the bill to fall 8 cm.THINK Less than 1/8 of asecond to catch the bill

Teaching Tip Distinguishbetween velocity andacceleration by asking, Whatis the velocity at the top, inFigure 4.9? (zero, for an instant)and What is the acceleration

at the top? (10 m/s 2; constantacceleration remains the samegoing up, at the top, and goingdown).

The acceleration ofan object in free fall

is about 10 meters per second

squared (10 m/s2).



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This should be enoughinformation for a class period.After any questions, discussion,and examples, state that youare going to pose a differentquestion in the next classperiodasking not how fast,but how far.


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4.6 Free Fall: How Far

How fast something moves is entirely different from how far itmovesspeed and distance are not the same thing. To understand thedifference, return to Table 4.2. At the end of the first second, the fallingobject has an instantaneous speed of 10 m/s. Does this mean it falls a

distance of 10 meters during this first second? No. Heres where thedifference between instantaneous speed and average speed comes in.The initial speed of fall is zero and takes a full second to get to 10 m/s.So the average speed is half way between zero and 10 m/sthats 5 m/s,as discussed earlier. So during the first second, the object has an averagespeed of 5 m/s and falls a distance of 5 m.

Table 4.3 shows the total distance moved by a freely falling objectdropped from rest. At the end of one second, it has fallen 5 meters.At the end of 2 seconds, it has dropped a total distance of 20 meters.At the end of 3 seconds, it has dropped 45 meters altogether. Foreach second of free fall, an object falls a greater distance than it didin the previous second. These distances form a mathematical pattern 4.6.1:at the end of time t , the object has fallen a distance d of gt 2.12

An apple drops from atree and hits the groundin one second. What is itsspeed upon striking theground? What is its aver-age speed during the onesecond? How high aboveground was the applewhen it first dropped?Answer: 4.6

think!We used freely falling objects to describe the relationship between

distance traveled, acceleration, and velocity acquired. In our examples, we used the acceleration of gravity, g 10 m/s 2. But acceleratingobjects need not be freely falling objects. A car accelerates when westep on the gas or the brake pedal. Whenever an objects initial speedis zero and the acceleration a is constant, that is, steady and non- jerky, the equations 4.6.2 for the velocity and distance traveled are

12v at and d at

2

CONCEPT

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second change?

FIGURE 4.10Pretend that a falling rockis somehow equipped withan odometer . The readingsof distance fallen increase

with time and are shown inTable 4.3.

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4.6 Free Fall:How Far

Teaching Tip Ask how fast an object is falling 1 s after itis released from rest; after 2 s;after 10 s. Now pose the verydifferent question How far does an object fall in 1 s? Howfar it falls is entirely differentfrom the free-fall case. Ask fora written response and then askif the students could explain totheir neighbors why the distanceis only 5 m rather than 10 m. Ifnecessary, explain that in freefall, the object does not maintaina speed of 10 m/s throughout thesecond of fall. Its average speedduring this time is only 5 m/s,

where v avg 5 (v f 1 v i)/2. Point outthat d 5 vt only holds when v isthe average speed or velocity.

Teaching Tip CompareTable 4.3 with Table 4.2 to stressthe difference between speedand distance.

Ask How far will a freelyfalling object that is releasedfrom rest fall in 2 s? 20 mIn 10 s? 500 m In 0.5 s? 1.25 m

Ask Consider a rock throwndownward with an initialvelocity of 15 m/s. What is theacceleration of the rock after onesecond? 9.8 m/s 2 , or g

I usually derive d 5 1/2 gt 2 on the board, as is done inNote 4.6.1, and say that thederivation is a sidelight to thecoursesomething that wouldbe the crux of a follow-upphysics course. The derivationis not something that I expectof them, but is to show thatd 5 1/2 gt 2 is a reasonedstatement that doesnt justpop up from nowhere.


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4.7 Graphs of MotionEquations and tables are not the only way to describe relationshipssuch as velocity and acceleration. Another way is to use graphs thatvisually describe relationships. Since youll develop basic graphingskills in the laboratory, we wont make a big deal of graphs. Here well

simply show the graphs for Tables 4.2 and 4.3.Speed-Versus-Time Figure 4.11 is a graph of the speed-versus-time data in Table 4.2. Note that speed v is plotted on the vertical axisand time t is plotted on the horizontal axis. In this case, the curvethat best fits the points forms a straight line. The straightness ofthe curve indicates a linear relationship between speed and time.For every increase of 1 s, there is the same 10 m/s increase in speed.Mathematicians call this linearity, and the graph shows whythecurve is a straight line. Since the object is dropped from rest, the line

starts at the origin, where both v and t are zero. If we double t, wedouble v; if we triple t, we triple v; and so on. This particular linearityis called a direct proportion, and we say that time and speed aredirectly proportional to each other.

The slope of a line on agraph is RISE/RUN.

The curve is a straight line, so its slope is constantlike aninclined plane. Slope is the vertical change divided by the horizontalchange for any part of the line. On a speed-versus-time graphthe slope represents speed per time, or acceleration. Note that foreach 10 m/s of vertical change there is a corresponding horizontalchange of 1 s. We see the slope is 10 m/s divided by 1 s, or 10 m/s 2.The straight line shows the acceleration is constant. If the accelera-tion were greater, the slope of the graph would be steeper. For moreinformation about slope, see Appendix C.

FIGURE 4.11A speed-versus-timegraph of the datafrom Table 4.2 islinear.

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For each second offree fall, an object

falls a greater distance than it didin the previous second.


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4.7 Graphs ofMotion

Teaching Tip Point out thatthe nonzero value of g (even atthe top of a trajectory) is shownin Figure 4.11. The slope (i.e.,acceleration) is the same at allpoints on the graph.

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What I do not do is ask for thetime of fall for a freely fallingobject, given the distance.Why? Unless the distance givenis the familiar 5 m, algebraicmanipulation is called for. Idrather put my energy and theirsinto straight physics! If mostof your class have made thedistinction between velocityand acceleration, move on toNewtons second and third lawsof motion and the tasty physicsthat follows.


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Distance-Versus-Time Figure 4.12 is a graph of the distance-versus-time data in Table 4.3. Distance d is plotted on the verticalaxis, and time t is on the horizontal axis. The result is a curved line.The curve shows that the relationship between distance traveled andtime is nonlinear . The relationship shown here is quadratic and thecurve is parabolic when we double t, we do not double d; we qua-

druple it. Distance depends on time squared!

A curved line also has a slope different at different points. If

you look at the graph in Figure 4.12 you can see that the curve hasa certain slant or steepness at every point. This slope changes fromone point to the next. The slope of the curve on a distance-versus-time graph is very significant. It is speed, the rate at which distance iscovered per unit of time. In this graph the slope steepens (becomesgreater) as time passes. This shows that speed increases as timepasses. In fact, if the slope could be measured accurately, you wouldfind it increases by 10 meters per second each second.

ONCEPT

CHECK . . . . .

. What does a slope of a speed-versus-time graph

represent?

How fast something

falls is entirely differentthan how far it falls.From rest, how fast isgiven by v gt ; howfar by d 12 gt

2.

For:

Visit:

Web Code:

Links on graphing speed,velocity, acceleration www.SciLinks.org csn 0407

FIGURE 4.12A distance-versus-time graph of thedata from Table 4.3 is parabolic.

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On a speed-versus-time graph the slope

represents speed per time, oracceleration.



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Shouldnt learning time-consuming graph plotting occurin a math class? Since its moremath than physics I spend no class time on graphs, which allowsme to cover rainbows and beyondin my course. If you must teachgraphical analysis, consider doingso at the end of your course after youve covered physicsrather than math material. Itsinteresting to note that thereare no laws of physics in thischapter on motion, nor in thechapter on projectiles thatfollows. I recommend movingquickly through this and thefollowing chapter, making thedistinction between velocity andacceleration, and then focus onthe physics of Newtons secondlaw in Chapter 6!


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4.8 Air Resistance and Falling ObjectsDrop a feather and a coin and we notice the coin reaches the floor farahead of the feather. Air resistance is responsible for these differentaccelerations. This fact can be shown quite nicely with a closed glasstube connected to a vacuum pump. The feather and coin are placed

inside. When the tube is inverted with air inside, the coin falls muchmore rapidly than the feather. The feather flutters through the air.But if the air is removed with a vacuum pump and the tube is quicklyinverted, the feather and coin fall side by side with the same accelera-tion, g , as shown in Figure 4.13.

Air resistance noticeably slows the motion of things withlarge surface areas like falling feathers or pieces of paper. But airresistance less noticeably affects the motion of more compactobjects like stones and baseballs. In many cases the effect of air resis-tance is small enough to be neglected. With negligible air resistance,falling objects can be considered to be falling freely. Air resistance willbe covered in more detail in Chapter 6.

N EPT

CHECK . . . .

. .

How does air resistance affect falling objects?

4.9 How Fast, How Far, How QuicklyHow Fast Changes

Some of the confusion that occurs in analyzing the motion of fallingobjects comes about from mixing up how fast with how far. Whenwe wish to specify how fast something freely falls from rest aftera certain elapsed time, we are talking about speed or velocity. Theappropriate equation is v gt . When we wish to specify how farthat object has fallen, we are talking about distance. The appropriateequation is d gt 212 . Velocity or speed (how fast) and distance (howfar) are entirely different from each other.

One of the most confusing concepts encountered in this book isacceleration, or how quickly does speed or velocity change. Whatmakes acceleration so complex is that it is a rate of a rate . It is oftenconfused with velocity, which is itself a rate (the rate at which distanceis covered). Acceleration is not velocity, nor is it even a change invelocity. Acceleration is the rate at which velocity itself changes.

Please be patient with yourself if you find that you require severalhours to achieve a clear understanding of motion. It took people nearly2000 years from the time of Aristotle to Galileo to achieve as much!CONCEPT

CHECK . . . . .

. What is the relationship between velocityand acceleration?

FIGURE 4.13A feather and a coinaccelerate equallywhen there is no airaround them.

59

4.8 Air Resistanceand Falling Objects Air resistance

noticeably shows themotion of things with largesurface areas like falling feathersor pieces of paper. But air

resistance less noticeably affectsthe motion of more compactobjects like stones or baseballs.



Laboratory Manual 14, 15



4.9 How Fast, HowFar, How Quickly HowFast Changes Acceleration is the

rate at which velocityitself changes.



Concept-DevelopmentPractice Book 4-2



CONCEPT

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CHECK . . .

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CONCEPT

CHECK . . .

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60

Hang time can beseveral seconds in thesport of sailsurfingquite a bitdifferent when the air

plays a major role!

Hang TimeSome people are gifted with great jumping ability. Leaping straightup, they seem to hang in theair. Ask your friends to estimatethe hang time of the great jumpersthe amount of time a jumper is airborne (feet off theground). One or two seconds?Several? Nope. Surprisingly,the hang time of the greatest jumpers is almost always less than1 second! Our perception of alonger hang time is one of many

illusions we have about nature.Jumping ability is best measuredby a standing vertical jump. Standfacing a wall, and with feet flaton the floor and arms extendedupward, make a mark on thewall at the top of your reach.Then make your jump, and at thepeak, make another mark. Thedistance between these two marksmeasures your vertical leap.Heres the physics. When you leapupward, jumping force is appliedonly as long as your feet are stillin contact with the ground. Thegreater the force, the greater yourlaunch speed and the higher the jump is. It is important to notethat as soon as your feet leave theground, whatever upward speedyou attain immediately decreasesat the steady rate of g, 10 m/s2.Maximum height is attained whenyour upward speed decreasesto zero. You then begin falling,gaining speed at exactly the samerate, g. If you land as you took off,upright with legs extended, thentime rising equals time falling.Hang time is the sum of rising andfalling times.

The relationship between rising orfalling time and vertical height isgiven by

d gt 212If we know the vertical height, wecan rearrange this expression toread

t 2d gNo basketball player is known tohave exceeded a jump of 1.25meters (4 feet). Settingd equal4.9to 1.25 m, and using the moreprecise 9.8 m/s2 for g, we find thatt , half the hang time, is

t 0.50 s 2d g 2(1.25 m)9.8 m/s 2Doubling this, we see such arecord hang time would be1 second!Weve only been talking aboutvertical motion. How aboutrunning jumps? Well learn inChapter 5 that hang time dependsonly on the jumpers vertical speedat launch; it does not depend onhorizontal speed. While airborne,the jumpers horizontal speedremains constant but the verticalspeed undergoes acceleration.Interesting physics!

Physics of Sports

60

Physics of Sports

Ha ng TimeTEACHING TIP This is fascinatinginformation, especially foryour sports-minded students.A widely held misconceptionis that the great jumpers stayairborne for considerably morethan a second. Note that a 2-shangtime means 1 s up andanother 1 s down. How fardoes an object fall in 1 s? Along 5 m, or 16 ft! Thats quitea jump! Have students calculatetheir own hang times frommeasurements of their ownpersonal vertical jumps.


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REVIEW


Concept Summary

An object is moving if its position relative toa fixed point is changing.

You can calculate the speed of an object bydividing the distance covered by time.

Speed is a description of how fast anobject moves; velocity is how fast and

in what direction it moves. You can calculate the acceleration of anobject by dividing the change in its velocityby time.

The acceleration of an object in free fallis about 10 meters per second squared(10 m/s 2).

For each second of free fall, an object falls agreater distance than it did in the previous

second. On a speed-versus-time graph the slope

represents speed per time, or acceleration.

Air resistance noticeably slows the motionof things with large surface areas like fallingfeathers or pieces of paper. But air resistanceless noticeably affects the motion of morecompact objects like stones and baseballs.

Acceleration is the rate at which velocityitself changes.

Key Terms

relative (p. 47)speed (p. 48)instantaneous

speed (p. 49)average speed (p. 49)

velocity (p. 50)acceleration (p. 51)free fall (p. 53)elapsed time (p. 53)

4.1 The mosquito should fly toward you intothe breeze. When above you it should flyat 3 meters per second in order to hover atrest above you. Unless its grip on your skinis strong enough after landing, it must con-tinue flying at 3 meters per second to keepfrom being blown off. Thats why a breezeis an effective deterrent to mosquito bites.

4.2.1 In 10 s the cheetah will cover 250 m, andin 1 minute (or 60 s) it will cover 1500 m.

4.2.2 average speed

70 km/h.

total distance coveredtime interval

35 km0.5 h

At some point, the speedometer would haveto exceed 70 km/h.

4.3 Both cars have the same speed, but theyhave opposite velocities because they aremoving in opposite directions.

4.4.1 We see that the speed increases by 5 km/hduring each 1-s interval. The accelerationis therefore 5 km/h .s during each interval.

4.4.2 The car and truck both increase their speedby 15 km/h during the same time interval,

so their acceleration is the same.4.5.1 The average speed will be 15 m/s. The acceleration will be 10 m/s 2.

4.5.2 The speedometer readings would be45 m/s and 80 m/s, respectively.

4.6 When it hits the ground, the apples speedwill be 10 m/s. Its average speed is 5 m/s,and it starts 5 m above the ground.

think! Answers

For:

Visit:

Web Code:

Self-Assessment

PHSchool.com

csa 04004

61

REVIEWTeaching Tip Consider

building a pair of tracks like Aand C in Think and Rank 24.Equal lengths of channel ironused for shelf brackets caneasily be bent with a vise. The

tracks help make the distinctionbetween the speed and timewhen the ball reaches the end ofthe tracks. Ball speed will be thesame at the ends of the tracks,but time of travel will differ.


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4


ASSESS


17. What does the slope of the curve on avelocity-versus-time graph represent?

Section 4.8

18. Does air resistance increase or decrease theacceleration of a falling object?

Section 4.9

19. What is the appropriate equation for howfast an object freely falls from a position ofrest? For how far that object falls?

Think and Rank

Rank each of the following sets of scenarios inorder of the quantity or property involved. Listthem from left to right. If scenarios have equalrankings, then separate them with an equal sign.

(e.g., A = B)20. Jogging Jake runs along a train flatcar that

moves at the velocities shown. From greatestto least, rank the relative velocities of Jake asseen by an observer on the ground. (Call thedirection to the right positive.)

21. Below we see before and after snapshots ofa cars velocity. The time interval betweenbefore and after for each is the same.

a. Rank the cars in terms of the change in ve-locity, from most positive to most negative.(Negative numbers rank lower than positiveones, and remember, tie scores can be partof your ranking.)

b. Rank them in terms of acceleration, fromgreatest to least.

22. These are drawings of same-size balls ofdifferent masses thrown straight downward.The speeds shown occur immediately afterleaving the throwers hand. Ignore air resis-tance. Rank their accelerations from greatestto least. Or are the accelerations the samefor each?

63

17. Acceleration 18. Decrease 19. v 5 gt ; d 5 1/2 gt 2

Think and Rank

20. D, C, A, B 21. a. A 5 C, D, B

b. A5

C, D, B 22. A 5 B 5 C 5 D


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4 ASSESS

64

23. A track is made of a piece of channel metalbent as shown. A ball is released from rest atthe left end of the track and continues pastthe various points. Rank the ball at points A,B, C, and D, from fastest to slowest. (Again,watch for tie scores.)

24. A ball is released from rest at the left end ofthree different tracks. The tracks are bentfrom pieces of metal of the same length.

a. From fastest to slowest, rank the tracks interms of the speed of the ball at the end.Or, do all balls have the same speed there?

b. From longest to shortest,rank the tracks in termsof the time for the ball toreach the end. Or do allballs reach the end in thesame time?

c. From greatest to least, rank the tracksin terms of the average speed of the ball.Or do the balls all have the same averagespeed on all three tracks?

25. In the speed versus time graphs, all times t are in s and all speeds v are in m/s.

a. From greatest to least, rank the graphs interms of the greatest speed at 10 seconds.

b. From greatest to least, rank the graphs interms of the greatest acceleration.

c. From greatest to least, rank the graphs interms of the greatest distance covered in10 seconds.

Plug and Chug These are to familiarize you with the centralequations of the chapter.

Average speed total distance coveredtime interval 26. Calculate your average walking speed when

you step 1 meter in 0.5 second.

27. Calculate the speed of a bowling ball thatmoves 8 meters in 4 seconds.

28. Calculate your average speed if you run50 meters in 10 seconds.

Distance average speed time

29. Calculate the distance (in km) that Charlieruns if he maintains an average speed of8 km/h for 1 hour.

4 ASSESS (continued)

64

23. C, B 5 D, A 24. a. B, A 5 C

b. C, B, Ac. C, B, A

25. a. A, C, Bb. C, A 5 B (Both show zeroacceleration.)c. A, C, B

Plug and Chug 26. Average speed 5 total distance

covered/time interval 5 1 m/0.5 s 5 2 m/s.

27. Average speed 5 total distancecovered/time interval 5 8 m/4 s 5 2 m/s.

28. Average speed 5 total distancecovered/time interval 5

50 m/10 s5

5 m/s. 29. Distance 5 average speed 3 time 5 8 km/h 3 1 h 5 8 km.

30 Di t 5 d 3


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4


ASSESS


38. On a distant planet a freely-fall-ing object has an acceleration of 20m/s 2. Calculate the speed an objectdropped from rest on this distantplanet acquires in 1.5 seconds.

Distance fallen in free fall, from rest:d gt 212

39. A sky diver steps from a high-flyinghelicopter. If there were no air resis-tance, how fast would she be falling atthe end of a 12-second jump?

40. Calculate the vertical distancean object dropped from restwould cover in 12 seconds if itfell freely without air resistance.

41. An apple drops from a tree andhits the ground in 1.5 seconds. Calculatehow far it falls.

Think and Explain 42. Light travels in a straight line at a con-

stant speed of 300,000 km/s. What is thelights acceleration?

43. a. If a freely falling rock were equipped witha speedometer, by how much would itsspeed readings increase with each secondof fall?

b. Suppose the freely falling rock weredropped near the surface of a planetwhere g 20 m/s 2. By how much wouldits speed readings change each second?

30. Calculate the distance you will travel if youmaintain an average speed of 10 m/s for40 seconds.

31. Calculate the distance (in km) you willtravel if you maintain an average speed of10 km/h for 1/2 hour.

Acceleration change of velocity

time interval

32. Calculate the acceleration of a car (inkm/h/s) that can go from rest to 100 km/hin 10 s.

33. Calculate the acceleration of a bus that goesfrom 10 km/h to a speed of 50 km/h in 10seconds.

34. Calculate the acceleration of a ball thatstarts from rest and rolls down a ramp and

gains a speed of 25 m/s in 5 seconds. From a rest position:

acceleration time

v at

Instantaneousspeed

35. Calculate the instantaneous speed (in m/s)at the 10-second mark for a car that acceler-ates at 2 m/s 2 from a position of rest.

36. Calculate the speed (in m/s) of a skate-boarder who accelerates from rest for3 seconds down a ramp at an accelerationof 5 m/s 2.

Velocity acquired in free fall, from rest:v gt

37. Calculate the instantaneous speed ofan apple 8 seconds after being droppedfrom rest.

65

30. Distance 5 average speed 3 time 5 10 m/s 3 40 s 5 400 m.

31. Distance 5 average speed3 time 5 10 km/h 3 1/2 h 5 5 km.

32. Acceleration 5 D v / D t 5 (100 km/h)/10 s 5 10 km/h/s.

33. Acceleration 5 D v / D t 5

(50 km/h2

10 km/h)/ 10 s5

(40 km/h)/10 s 5 4 km/h/s. 34. Acceleration 5 D v / D t 5

(25 m/s)/5 s 5 5 m/s 2. 35. Instantaneous speed 5

acceleration 3 time 5 2 m/s 2 3 10 s 5 20 m/s.

36. Instantaneous speed 5 acceleration 3 time 5 5 m/s 2 3 3 s 5 15 m/s.

37. v 5 gt 5 10 m/s 2 3 8 s 5 80 m/s 38. v 5 gt 5 20 m/s 2 3 1.5 s 5

30 m/s 39. v 5 gt 5 10 m/s 2 3 12 s 5

120 m/s 40. d 5 1/2 gt 2 5

1/2 (10 m/s 2)(12 s) 2 5 720 m 41. d 5 1/2 gt 2 5

1/2 (10 m/s 2)(1.5 s) 2 5 11.3 m

Think and Explain 42. Zero; its velocity is constant. 43. a. 10 m/s

b. 20 m/s

44 Same; each undergoes the


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4 ASSESS

66

44. Which has more acceleration whenmoving in a straight linea car increasingits speed from 50 to 60 km/h, or a bicyclethat goes from zero to 10 km/h in the sametime? Defend your answer.

45. Correct your friend who says, The dragsterrounded the curve at a constant velocity of100 km/h.

46. What is the acceleration of a car thatmoves at a steady velocity of 100 km/h duenorth for 100 seconds? Explain your answerand state why this question is an exercise incareful reading as well as physics.

47. Tiffany stands at the edge of a cliff andthrows a ball straight up at a certain speedand another ball straight down with the

same initial speed. Neglect air resistance. a. Which ball is in the air for the longesttime?

b. Which ball has the greater speed when itstrikes the ground below?

48. A ball is thrown straight up. What willbe the instantaneous velocity at the top ofits path? What will be its acceleration at thetop? Why are your answers different?

49. Two balls are released simultaneously fromrest at the left ends of the equal-lengthtracks A and B as shown. Which ball reachesthe end of its track first?

50. Refer to the tracks in the previous problem.a. Does the ball on B roll faster in the dip

than the ball rolling along track A?b. On track B, is the speed gained going

down into the dip equal to the speed lostgoing up the dip? If so, do the balls thenhave the same speed at the ends of bothtracks?

c.Is the average speed of the ball on thelower part of track B greater than theaverage speed of the ball on A during thesame time?

d. So overall, which ball has the greateraverage speed? (Do you wish to change your answer to the previous exercise?)

Think and Solve

51. A dragster going at 15 m/s north increasesits velocity to 25 m/s north in 4 seconds.What is its acceleration during this timeinterval?

52. An apple drops from a tree and hits theground in 1.5 s. What is its speed just beforeit hits the ground?

53. On a distant planet a freely falling object

has an acceleration of 20 m/s2

. What verticaldistance will an object dropped from rest onthis planet cover in 1.8 s?

54. If you throw a ball straight upward at aspeed of 10 m/s, how long will it take toreach zero speed? How long will it taketo return to its starting point? How fastwill it be going when it returns to itsstarting point?

4 ASSESS (continued)

66

44. Same; each undergoes thesame change in speed at thesame time.

45. Constant velocity meansconstant direction, so yourfriend should say . . . at aconstant speed of 100 km/h.

46. Zero, for no change in velocityoccurred; misreading thequestion might mean missingthe word steady (nochange).

47. a. Longest time for ballthrown upwardb. Both hit ground with samespeed. (When upward ballreturns to original level it hasthe same speed as ball throwndownward.)

48. Zero; g; concepts of velocityand acceleration differ!

49. Ball on B finishes first, for itgains extra speed in the dip,which makes overall higheraverage speed on track B.

50. a. Yes, ball B is faster in dipthan the ball on track A.b. Yes, speed gained goingdown 5 speed lost going up.So both balls have the samespeed at the end of theirtracks.c. Yes, average speed onlower part of track B isgreater than the speed of theball on Track A.d. Ball on B has greateraverage speed because itmoves faster on the lowerpart of the track. (Manypeople get the wrong answerto 48 due to assumption that

because the balls end up withthe same speed that they rollfor the same time . Not so.)

Think and Solve

51. a 5 D v / D t 5 (25 m/s 2 15 m/s)/4 s 5 2.5 m/s 2

52. v 5 gt 5 10 m/s 2 3 1.5 s 5 15 m/s

53. d 5 1/2 gt 2 5

1/2 (20 m/s 2)(1.8 s) 2 5 32.4 m


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Linear Motion Notes

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