chapter Describing Motion chapter outline chapter overview unit one

Transcription

1 Describing Moion chaper 2 chaper oeriew The main purpose of his chaper is o proide clear definiions and illusraions of he erms used in physics o describe moion, such as he moion of he car described in his chaper s opening example. Speed, elociy, and acceleraion are crucial conceps for he analysis of moion in laer chapers. Precise descripion is he firs sep o undersanding. Wihou i, we remain awash in ague ideas ha are no defined well enough o es our explanaions. Each numbered opic in his chaper builds on he preious secion, so i is imporan o obain a clear undersanding of each opic before going on. The disincions beween speed and elociy and elociy and acceleraion are paricularly imporan. chaper ouline 1 Aerage 2 Velociy. 3 Acceleraion. 4 Graphing 5 Uniform and insananeous speed. How do we describe how fas an objec is moing? How does insananeous speed differ from aerage speed? How do we inroduce direcion ino descripions of moion? Wha is he disincion beween speed and elociy? How do we describe changes in moion? Wha is he relaionship beween elociy and acceleraion? moion. How can graphs be used o describe moion? How can he use of graphs help us gain a clearer undersanding of speed, elociy, and acceleraion? acceleraion. Wha happens when an objec acceleraes a a seady rae? How do he elociy and disance raeled ary wih ime when an objec is uniformly acceleraing? uni one 18

2 2.1 Aerage and Insananeous Speed 19 Imagine ha you are in your car sopped a an inersecion. Afer waiing for cross raffic, you pull away from he sop sign, acceleraing eenually o a speed of 56 kilomeers per hour (35 miles per hour). You mainain ha speed unil a dog runs in fron of your car and you hi he brakes, reducing your speed rapidly o 10 km/h (fig. 2.1). Haing missed he dog, you speed up again o 56 km/h. Afer anoher block, you come o anoher sop sign and reduce your speed gradually o zero. We can all relae o his descripion. Measuring speed in miles per hour (MPH) may be more familiar han he use of kilomeers per hour (km/h), bu speedomeers in cars now show boh. The use of he erm acceleraion o describe an increase in speed is also common. In physics, howeer, hese conceps ake on more precise and specialized meanings ha make hem een more useful in describing exacly wha is happening. These meanings are someimes differen from hose in eeryday use. The erm acceleraion, for example, is used by physiciss o describe any siuaion in which elociy is changing, een when he speed may be decreasing or he direcion of he moion may be changing. How would you define he erm speed if you were explaining he idea o a younger broher or siser? Does elociy mean he same hing? Wha abou acceleraion is he noion ague or does i hae a precise meaning? Is i he same hing as elociy? Clear definiions are essenial o deeloping clear explanaions. The language used by physiciss differs from our eeryday language, een hough he ideas are relaed and he same words are used. Wha are he exac meanings ha physiciss aach o hese conceps, and how can hey help us o undersand moion? figure 2.1 As he car brakes for he dog, here is a sudden change in speed. 2.1 Aerage and Insananeous Speed Since driing or riding in cars is a common aciiy in our daily lies, we are familiar wih he concep of speed. Mos of us hae had experience in reading a speedomeer (or perhaps failing o read i carefully enough o aoid he aenion of law enforcemen). If you describe how fas somehing is moing, as we did in our example in he inroducion, you are alking abou speed. How is aerage speed defined? Wha does i mean o say ha we are raeling a a speed of 55 MPH? I means ha we would coer a disance of 55 miles in a ime of 1 hour if we raeled seadily a ha speed. Carefully noe he srucure of his descripion: here is a number, 55, and some unis or dimensions, miles per hour. Numbers and unis are boh essenial pars of a descripion of speed. The erm miles per hour implies ha miles are diided by hours in arriing a he speed. This is exacly how we would compue he aerage speed for a rip: suppose, for example, ha we rael a disance of 260 miles in a ime of 5 hours, as shown on he road map of figure 2.2. The aerage speed is hen 260 miles diided by 5 hours, which is equal o 52 MPH. This ype of compuaion is familiar o mos of us. We can also express he definiion of aerage speed in a word equaion as Aerage speed equals he disance raeled diided by he ime of rael. or Aerage speed We can represen his same definiion wih symbols by wriing s d, disance raeled. ime of rael

3 20 Chaper 2 Describing Moion Kingman 120 mi 2.4 h Flagsaff example box 2.1 Uni Conersions 1 km miles 1 mile km 140 mi 2.6 h Coner 70 kilomeers per hour o miles per hour. a70 km h miles b a b 43.5 MPH km Coner 70 kilomeers per hour o meers per second. a70 km h b a1000 m b m/h km figure 2.2 Phoenix A road map showing a rip of 260 miles, wih driing imes for he wo legs of he rip. Bu 1 h m/h 3600 s/h (60 min) a m/s s b 3600 s min Lines drawn hrough he unis indicae cancellaion. where he leer s represens he speed, d represens disance, and represens he ime. As noed in chaper 1, leers or symbols are a compac way of saying wha could be said wih a lile more effor and space wih words. Judge for yourself which is he more efficien way of expressing his definiion of aerage speed. Mos people find he symbolic expression easier o remember and use. The aerage speed ha we hae jus defined is he rae a which disance is coered oer ime. Raes always represen one quaniy diided by anoher. Gallons per minue, pesos per dollar, and poins per game are all examples of raes. If we are considering ime raes, he quaniy ha we diide by is ime, which is he case wih aerage speed. Many oher quaniies ha we will be considering inole ime raes. Wha are he unis of speed? Unis are an essenial par of he descripion of speed. Suppose you say ha you were doing 70 wihou saing he unis. In he Unied Saes, ha would probably be undersood as 70 MPH, since ha is he uni mos frequenly used. In Europe, on he oher hand, people would probably assume ha you are alking abou he considerably slower speed of 70 km/h. If you do no sae he unis, you will no communicae effeciely. I is easy o coner from one uni o anoher if he conersion facors are known. For example, if we wan o coner kilomeers per hour o miles per hour, we need o know he relaionship beween miles and kilomeers. A kilomeer is roughly 6 10 of a mile (0.6214, o be more precise). As shown in example box 2.1, 70 km/h is equal o 43.5 MPH. The process inoles muliplicaion or diision by he appropriae conersion facor. Unis of speed will always be a disance diided by a ime. In he meric sysem, he fundamenal uni of speed is meers per second (m/s). Example box 2.1 also shows he conersion of kilomeers per hour o meers per second, done as a wo-sep process. As you can see, 70 km/h can also be expressed as 19.4 m/s or roughly 20 m/s. This is a conenien size for discussing he speeds of ordinary objecs. (As shown in example box 2.2, he conenien uni for measuring he growh of grass has a ery differen size.) example box 2.2 Sample Quesion: Waching Grass Grow Quesion: The unis km/h or m/s hae an appropriae size for moing cars or people. Many oher processes moe much more slowly, hough. Wha unis would hae an appropriae size for measuring he aerage speed wih which a blade of grass grows? Answer: When grass is well ferilized and waered, i is no unusual for i o grow 3 o 6 cenimeers in he course of a week. This can be seen by measuring he lengh of he clippings afer mowing. If we measured he speed in m/s, we would obain an exremely small number ha would no proide a good inuiie sense of he rae of growh. The unis of cm/week or mm/day would proide a beer indicaion of his speed.

4 2.1 Aerage and Insananeous Speed 21 able 2.1 Familiar Speeds in Differen Unis 20 MPH 32 km/h 9 m/s 40 MPH 64 km/h 18 m/s 60 MPH 97 km/h 27 m/s 80 MPH 130 km/h 36 m/s 100 MPH 160 km/h 45 m/s Table 2.1 shows some familiar speeds expressed in miles per hour, kilomeers per hour, and meers per second o gie you a sense of heir relaionships. Wha is insananeous speed? If we rael a disance of 260 miles in 5 hours, as in our earlier example, is i likely ha he enire rip akes place a a speed of 52 MPH? Of course no; he speed goes up and down as he road goes up and down, when we oerake slower ehicles, when res breaks occur, or when he highway parol looms on he horizon. If we wan o know how fas we are going a a gien insan in ime, we read he speedomeer, which displays he insananeous speed (fig. 2.3). How does insananeous speed differ from aerage speed? The insananeous speed ells us how fas we are going a a gien insan bu ells us lile abou how long i will ake o rael seeral miles, unless he speed is held consan. The aerage speed, on he oher hand, allows us o compue how long a rip migh ake bu says lile abou he ariaion in speed during he rip. A more complee descripion of how he speed of a car aries during a porion of a rip could be proided by a graph such as ha shown in figure 2.4. Each poin on his graph represens he insananeous speed a he ime indicaed on he horizonal axis. Een hough we all hae some inuiie sense of wha insananeous speed means from our experience in reading figure A speedomeer wih wo scales for measuring insananeous speed, MPH and km/h km/h MPH Speed (km/h) figure 2.4 Passing he ruck Turn ono highway Behind a slow ruck Sopligh Time (minues) Variaions in insananeous speed for a porion of a rip on a local highway. speedomeers, compuing his quaniy presens some problems ha we did no encouner in defining aerage speed. We could say ha insananeous speed is he rae ha disance is being coered a a gien insan in ime, bu how do we compue his rae? Wha ime ineral should we use? Wha is an insan in ime? Our soluion o his problem is simply o choose a ery shor ineral of ime during which a ery shor disance is coered and he speed does no change drasically. If we know, for example, ha in 1 second a disance of 20 meers was coered, diiding 20 meers by 1 second o obain a speed of 20 m/s would gie us a good esimae of he insananeous speed, proided ha he speed did no change much during ha single second. If he speed was changing rapidly, we would hae o choose an een shorer ineral of ime. In principle, we can choose ime inerals as small as we wish, bu in pracice, i can be hard o measure such small quaniies. If we pu hese ideas ino a word definiion of insananeous speed, we could sae i as Insananeous speed is he rae a which disance is being coered a a gien insan in ime. I is found by compuing he aerage speed for a ery shor ime ineral in which he speed does no change appreciably. Insananeous speed is closely relaed o he concep of aerage speed bu inoles ery shor ime inerals. When discussing raffic flow, aerage speed is he criical issue, as shown in eeryday phenomenon box 2.1.

5 22 Chaper 2 Describing Moion eeryday phenomenon Transiions in Traffic Flow The Siuaion. Jennifer commues ino he ciy on a freeway eery day for work. As she approaches he ciy, he same paerns in raffic flow seem o show up in he same places each day. She will be moing wih he flow of raffic a a speed of approximaely 60 MPH when suddenly hings will come o a screeching hal. The raffic will be sop-and-go briefly and hen will sele ino a waelike mode wih speeds arying beween 10 and 30 MPH. Unless here is an acciden, his will coninue for he res of he way ino he ciy. The raffic in he upper lanes is flowing freely wih adequae spacing o allow higher speeds. The higher-densiy raffic in he lower lanes moes much more slowly. box 2.1 Wha causes hese paerns? Why does he raffic sop when here is no apparen reason such as an acciden? Why do ramp raffic lighs seem o help he siuaion? Quesions like hese are he concern of he growing field of raffic engineering. The Analysis. Alhough a full analysis of raffic flow is complex, here are some simple ideas ha can explain many of he paerns ha Jennifer obseres. The densiy of ehicles, measured in ehicles per mile, is a key facor. Adding ehicles a enrance ramps increases he densiy. The spacing beween ehicles aries wih speed so ha speed and densiy are inerrelaed. When Jennifer and oher commuers are raeling a 60 MPH, hey need o keep a spacing of seeral car lenghs beween ehicles. Mos driers do his wihou hinking abou i, alhough here are always some who follow oo closely or ailgae. Tailgaing runs he risk of rear-end collisions when he raffic suddenly slows. When more ehicles are added a an enrance ramp, he densiy mus increase, reducing he disance beween ehicles. As he disance beween ehicles decreases, driers should reduce heir speed o mainain a safe sopping disance. If his occurred uniformly, here would be a gradual decrease in he aerage speed of he raffic o accommodae he greaer densiy. This is no wha usually happens, howeer. (coninued) We find an aerage speed by diiding he disance raeled by he ime required o coer ha disance. Aerage speed is herefore he aerage rae a which disance is being coered. Insananeous speed is he rae ha disance is being coered a a gien insan in ime and is found by considering ery small ime inerals or by reading a speedomeer. Aerage speed is useful for esimaing how long a rip will ake, bu insananeous speed is of more ineres o he highway parol. 2.2 Velociy Do he words speed and elociy mean he same hing? They are ofen used inerchangeably in eeryday language, bu physiciss make an imporan disincion beween he wo erms. The disincion has o do wih direcion: which way is he objec moing? This disincion urns ou o be essenial o undersanding Newon s heory of moion (inroduced in chaper 4), so i is no jus a maer of whim or jargon. Wha is he difference beween speed and elociy? Imagine ha you are driing a car around a cure (as illusraed in figure 2.5) and ha you mainain a consan speed of 60 km/h. Is your elociy also consan in his case? The answer is no, because elociy inoles he direcion of moion as well as how fas he objec is going. The direcion of moion is changing as he car goes around he cure. To simply sae his disincion, speed as we hae defined i ells us how fas an objec is moing bu says nohing abou he direcion of he moion. Velociy includes he idea of direcion. To specify a elociy, we mus gie boh

6 2.2 Velociy 23 A significan proporion of driers will aemp o mainain heir speed of 50 o 60 MPH een when densiies hae increased beyond he poin where his is adisable. This creaes an unsable siuaion. A some poin, usually near an enrance ramp, he densiy becomes oo large o susain hese speeds. A his poin here is a sudden drop in aerage speed and a large increase in he local densiy. As shown in he drawing, cars can be separaed by less han a car lengh when hey are sopped or moing ery slowly. Once he aerage speed of a few ehicles has slowed o less han 10 MPH, ehicles moing a 50 o 60 MPH begin o pile up behind his slower moing jam. Because his does no happen smoohly, some ehicles mus come o a complee sop, furher slowing he flow. A he fron end of he jam, on he oher hand, he densiy is reduced due o he slower flow behind. Cars can hen sar moing a a speed consisen wih he new densiy, perhaps around 30 MPH. If eery ehicle moed wih he appropriae speed, flow would be smooh and he increased densiy could be safely accommodaed. More ofen, howeer, oeranxious driers exceed he appropriae speed, causing flucuaions in he aerage speed as ehicles begin o pile up again. Noice ha we are using aerage speed wih wo differen meanings in his discussion. One is he aerage speed of an indiidual ehicle as is insananeous speed increases and decreases. The oher is he aerage speed of he oerall raffic flow inoling many ehicles. When he raffic is flowing freely, he aerage speed of differen ehicles may differ. When he raffic is in a slowly moing jam, he aerage speeds of differen ehicles are essenially he same, a leas wihin a gien lane. Traffic lighs a enrance ramps ha permi ehicles o ener one-a-a-ime a appropriae inerals can help o smoohly inegrae he added ehicles o he exising flow. This reduces he sudden changes in speed caused by a rapid increase in densiy. Once he densiy increases beyond he cerain leel, howeer, a slowing of raffic is ineiable. The abrup change from low-densiy, high-speed flow o higherdensiy, slow flow is analogous o a phase ransiion from a gas o a liquid. (Phase ransiions are discussed in chaper 10.) Traffic engineers hae used his analogy o beer undersand he process. If we could auomaically conrol and coordinae he speeds of all he ehicles on he highway, he highway migh carry a much greaer olume of raffic a a smooh rae of flow. Speeds could be adjused o accommodae changes in densiy and smaller ehicle separaions could be mainained a higher speeds because he ehicles would all be moing in a synchronized fashion. Beer echnology may someday achiee his dream. figure The direcion of he elociy changes as he car moes around he cure, so ha he elociy 2 is no he same as he elociy 1 een hough he speed has no changed. B 1 N A E is size or magniude (how fas) and is direcion (norh, souh, eas, up, down, or somewhere in beween). If you ell me ha an objec is moing 15 m/s, you hae old me is speed. If you ell me ha i is moing due wes a 15 m/s, you hae old me is elociy. A poin A on he diagram in figure 2.5, he car is raeling due norh a 60 km/h. A poin B, because he road cures, he car is raeling norhwes a 60 km/h. Is elociy a poin B is differen from is elociy a poin A (because he direcions are differen). The speeds a poin A and B are he same. Direcion is irrelean in specifying he speed of he objec. I has no effec on he reading on your speedomeer. Changes in elociy are produced by forces acing upon he car, as we will discuss furher in chaper 4. The mos imporan force inoled in changing he elociy of a car is he fricional force exered on he ires of he car by he road surface. A force is required o change eiher he size or he direcion of he elociy. If no ne force were acing on he car, i would coninue o moe a consan speed in a sraigh line. This happens someimes when here is ice or oil on he road surface, which can reduce he fricional force o almos zero.

7 24 Chaper 2 Describing Moion sudy hin Science has always relied on picures and chars o ge poins across. Throughou he book, a number of conceps will be inroduced and illusraed. In he illusraions, he same color will be used for cerain phenomena. Wha is a ecor? Blue arrows are elociy ecors. Green arrows depic acceleraion ecors. Red arrows depic force ecors. Purple arrows show momenum, a concep we will explore in chaper 7. Velociy is a quaniy for which boh he size and direcion are imporan. We call such quaniies ecors. To describe hese quaniies fully, we need o sae boh he size and he direcion. Velociy is a ecor ha describes how fas an objec is moing and in wha direcion i is moing. Many of he quaniies used in describing moion (and in physics more generally) are ecor quaniies. These include elociy, acceleraion, force, and momenum, o name a few. Think abou wha happens when you hrow a rubber ball agains a wall, as shown in figure 2.6. The speed of he ball may be abou he same afer he collision wih he wall as i was before he ball hi he wall. The elociy has clearly changed in he process, hough, because he ball is moing in a differen direcion afer he collision. Somehing has happened o he moion of he ball. A srong force had o be exered on he ball by he wall o produce his change in elociy. 2 figure 2.7 elociy ecor. 10 m/s 20 m/s The lengh of he arrow shows he size of he The elociy ecors in figures 2.5 and 2.6 are represened by arrows. This is a naural choice for depicing ecors, since he direcion of he arrow clearly shows he direcion of he ecor, and he lengh can be drawn proporional o he size. In oher words, he larger he elociy, he longer he arrow (fig. 2.7). In he ex, we will represen ecors by prining heir symbols in boldface and larger han oher symbols: is hus he symbol for elociy. A fuller descripion of ecors can be found in appendix C. How do we define insananeous elociy? In considering auomobile rips, aerage speed is he mos useful quaniy. We do no really care abou he direcion of moion in his case. Insananeous speed is he quaniy of ineres o he highway parol. Insananeous elociy, howeer, is mos useful in considering physical heories of moion. We can define insananeous elociy by drawing on our earlier definiion of insananeous speed. Insananeous elociy is a ecor quaniy haing a size equal o he insananeous speed a a gien insan in ime and haing a direcion corresponding o ha of he objec s moion a ha insan. Insananeous elociy and insananeous speed are closely relaed, bu elociy includes direcion as well as size. I is changes in insananeous elociy ha require he inerenion of forces. These changes will be emphasized when we explore Newon s heory of mechanics in chaper 4. We can also define he concep of aerage elociy, bu ha is a much less useful quaniy for our purposes han eiher insananeous elociy or aerage speed. figure The direcion of he elociy changes when a ball bounces from a wall. The wall exers a force on he ball in order o produce his change. To specify he elociy of an objec, we need o sae boh how fas and in wha direcion he objec is moing; elociy is a ecor quaniy. Insananeous elociy has a magniude equal o he insananeous speed and poins in he direcion ha he objec is moing. Changes in insananeous elociy are where he acion is, so o speak, and we will consider hese in more deail when we discuss acceleraion in secion 2.3.

8 2.3 Acceleraion Acceleraion Acceleraion is a familiar idea. We use he erm in speaking of he acceleraion of a car away from a sop sign or he acceleraion of a running back in fooball. We feel he effecs of acceleraion on our bodies when a car s elociy changes rapidly and een more srikingly when an eleaor lurches downward, leaing our somachs slighly behind (fig. 2.8). These are all acceleraions. You can hink of your somach as an acceleraion deecor a roller-coaser gies i a real workou! Undersanding acceleraion is crucial o our sudy of moion. Acceleraion is he rae a which elociy changes. (Noe ha we said elociy, no speed.) I plays a cenral role in Newon s heory of moion. How do we go abou finding a alue of an acceleraion, hough? As wih speed, i is conenien o sar wih a definiion of aerage acceleraion and hen exend i o he idea of insananeous acceleraion. How is aerage acceleraion defined? How would we go abou proiding a quaniaie descripion of an acceleraion? Suppose ha your car, poining due eas, sars from a full sop a a sop sign, and is elociy increases from zero o 20 m/s as shown in figure 2.9. The change in elociy is found simply by subracing he iniial elociy from he final elociy (20 m/s 0 m/s 20 m/s). Acceleraion deecor = 0 = 0 figure 2.9 To find is rae of change, howeer, we also need o know he ime needed o produce his change. If i ook jus 5 seconds for he elociy o change, he rae of change would be larger han if i ook 30 seconds. Suppose ha a ime of 5 seconds was required o produce his change in elociy. The rae of change in elociy could hen be found by diiding he size of he change in elociy by he ime required o produce ha change. Thus he size of he aerage acceleraion, a, is found by diiding he change in elociy of 20 m/s by he ime of 5 seconds, a 20 m/s 5 s 4 m/s/s. = 20 m/s = 5 s A car, saring from res, acceleraes o a elociy of 20 m/s due eas in a ime of 5 s. The uni m/s/s is usually wrien m/s 2 and is read as meers per second squared. I is easier o undersand i, howeer, as meers per second per second. The car s elociy (measured in m/s) is changing a a rae of 4 m/s eery second. Oher unis could be used for acceleraion, bu hey will all hae his same form: disance per uni of ime per uni of ime. In discussing he acceleraion of a car on a drag srip, for example, he uni miles per hour per second is someimes used. The quaniy ha we hae jus compued is he size of he aerage acceleraion of he car. The aerage acceleraion is found by diiding he oal change in elociy for some ime ineral by ha ime ineral, ignoring possible differences in he rae of change of elociy ha migh be occurring wihin he ime ineral. Is definiion can be saed in words as Aerage acceleraion is he change in elociy diided by he ime required o produce ha change. figure 2.8 Your acceleraion deecor senses he downward acceleraion of he eleaor. a We can resae i in symbols as or change in elociy Acceleraion elapsed ime a.

9 26 Chaper 2 Describing Moion Because change is so imporan in his definiion, we hae used he special symbol (he Greek leer dela) o mean a change in a quaniy. Thus is a compac way of wriing he change in elociy, which oherwise would be expressed as 2 1, since a change is he difference beween wo quaniies. Because he concep of change is criical, his noaion will appear ofen. The idea of change is all-imporan. Acceleraion is no elociy oer ime. I is he change in elociy diided by ime. I is common for people o associae large acceleraions wih large elociies, when in fac he opposie is ofen rue. The acceleraion of a car may be larges, for example, when i is jus saring up and is elociy is near zero. The rae of change of elociy is greaes hen. On he oher hand, a car can be raeling a 100 MPH bu sill hae a zero acceleraion if is elociy is no changing. Wha is insananeous acceleraion? Insananeous acceleraion is similar o aerage acceleraion wih an imporan excepion. Jus as wih insananeous speed or elociy, we are now concerned wih he rae of change a a gien insan in ime. I is insananeous acceleraion ha our somachs respond o. I can be defined as Insananeous acceleraion is he rae a which elociy is changing a a gien insan in ime. I is compued by finding he aerage acceleraion for a ery shor ime ineral during which he acceleraion does no change appreciably. If he acceleraion is changing wih ime, choosing a ery shor ime ineral guaranees ha he acceleraion compued for ha ime ineral will no differ oo much from he insananeous acceleraion a any ime wihin he ineral. This is he same idea used in finding an insananeous speed or insananeous elociy. 1 = 8 m/s 1 figure 2.10 Δ = 12 m/s 1 Δ + = a The acceleraion ecor is in he same direcion as he elociy ecors when he elociy is increasing. ecors, as shown in figure Because he iniial elociy 1 is larger han he final elociy 2, he change in elociy mus poin in he opposie direcion o produce a shorer 2 arrow. The acceleraion is also in he opposie direcion o he elociy, since i is in he direcion of he change in elociy. In Newon s heory of moion, he force required o produce his acceleraion would also be opposie in direcion o he elociy. I mus push backward on he car o slow i down. The erm acceleraion describes he rae of any change in an objec s elociy. The change could be an increase (as in our iniial example), a decrease, or a change in direcion. The erm applies een o decreases in elociy (deceleraions). To a physicis hese are simply acceleraions wih a direcion opposie ha of he elociy. If a car is braking while raeling in a sraigh line, is elociy is decreasing and is acceleraion is negaie if he elociy is posiie. This siuaion is illusraed in he sample exercise in example box 2.3. The minus sign is an imporan par of he resul in he example in example box 2.3 because i indicaes ha he change in elociy is negaie. The elociy is geing smaller. We can call i a deceleraion if we like, bu i is 2 2 = 20 m/s 2 Wha is he direcion of an acceleraion? Like elociy, acceleraion is a ecor quaniy. Is direcion is imporan. The direcion of he acceleraion ecor is ha of he change in elociy. If, for example, a car is moing in a sraigh line and is elociy is increasing, he change in elociy is in he same direcion as he elociy iself, as shown in figure The change in elociy mus be added o he iniial elociy 1 o obain he final elociy 2. All hree ecors poin forward. The process of adding ecors can be readily seen when we represen he ecors as arrows on a graph. (More informaion on ecor addiion can be found in appendix C.) If he elociy is decreasing, howeer, he change in elociy poins in he opposie direcion o he wo elociy 1 = 20 m/s figure Δ = 12 m/s 1 Δ + = a 2 2 = 8 m/s The elociy and acceleraion ecors for decreasing elociy: and a are now opposie in direcion o he elociy. The acceleraion a is proporional o. 2

10 2.4 Graphing Moion 27 example box 2.3 Sample Exercise: Negaie Acceleraions The drier of a car seps on he brakes, and he elociy drops from 20 m/s due eas o 10 m/s due eas in a ime of 2.0 seconds. Wha is he acceleraion? 1 20 m/s due eas 2 10 m/s due eas 2.0 s a? a a 5.0 m/s 2 due wes Noice ha when we are dealing jus wih he magniude of a ecor quaniy, we do no use he boldface noaion. The sign can indicae direcion, howeer, in a problem inoling sraigh-line moion. he same hing as a negaie acceleraion. One word, acceleraion, coers all siuaions in which he elociy is changing m/s 20 m/s 2.0 s 10 m/s 2.0 s 5 m/s 2 2 figure 2.12 Δ a 1 2 A change in he direcion of he elociy ecor also inoles an acceleraion, een hough he speed may be consan. Acceleraion is he rae of change of elociy and is found by diiding he change in he elociy by he ime required o produce ha change. Any change in elociy inoles an acceleraion, wheher an increase or a decrease in speed, or a change in direcion. Acceleraion is a ecor haing a direcion corresponding o he direcion of he change in elociy, which is no necessarily he same direcion as he insananeous elociy iself. The concep of change is crucial. The graphical represenaions in secion 2.4 will help you isualize changes in elociy as well as in oher quaniies. 1 Can a car be acceleraing when is speed is consan? Wha happens when a car goes around a cure a consan speed? Is i acceleraing? The answer is yes, because he direcion of is elociy is changing. If he direcion of he elociy ecor is changing, he elociy is changing. This means ha here mus be an acceleraion. This siuaion is illusraed in figure The arrows in his drawing show he direcion of he elociy ecor a differen poins in he moion. The change in elociy is he ecor ha mus be added o he iniial elociy 1 o obain he final elociy 2. The ecor represening he change in elociy poins oward he cener of he cure, and herefore, he acceleraion ecor also poins in ha direcion. The size of he change is represened by he lengh of he arrow. From his we can find he acceleraion. Acceleraion is inoled wheneer here is a change in elociy, regardless of he naure of ha change. Cases like figure 2.12 will be considered more fully in chaper 5 where circular moion is discussed. 2.4 Graphing Moion I is ofen said ha a picure is worh a housand words, and he same can be said of graphs. Imagine rying o describe he moion depiced in figure 2.4 precisely in words and numbers. The graph proides a quick oeriew of wha ook place. A descripion in words would be much less efficien. In his secion, we will show how graphs can also help us o undersand elociy and acceleraion. Wha can a graph ell us? How can we produce and use graphs o help us describe moion? Imagine ha you are waching a baery-powered oy car moing along a meer sick (fig. 2.13). If he car is moing slowly enough, you could record he car s posiion while also recording he elapsed ime using a digial wach. A regular ime inerals (say, eery 5 seconds), you would noe he alue of he posiion of he fron of he car on he meer sick and wrie hese alues down. The resuls migh be somehing like hose shown in able 2.2.

11 28 Chaper 2 Describing Moion figure 2.13 A oy car moing along a meer sick. Is posiion can be recorded a differen imes. How do we graph hese daa? Firs, we creae eenly spaced inerals on each of wo perpendicular axes, one for disance raeled (or posiion) and he oher for ime. To show how disance aries wih ime, we usually pu ime on he horizonal axis and disance on he erical axis. Such a graph is shown in figure 2.14, where each daa poin from able 2.2 is ploed and a line is drawn hrough he poins. To make sure ha you undersand his process, choose differen poins from able 2.2 and find where hey are locaed on he graph. Where would he poin go if he car was a 21 cenimeers a 25 seconds? The graph summarizes he informaion presened in he able in a isual forma ha makes i easier o grasp a a able 2.2 Posiion of he Toy Car along he Meer Sick a Differen Times Time Posiion 0 s 0 cm 5 s 4.1 cm 10 s 7.9 cm 15 s 12.1 cm 20 s 16.0 cm 25 s 16.0 cm 30 s 16.0 cm 35 s 18.0 cm 40 s 20.1 cm 45 s 21.9 cm 50 s 24.0 cm 55 s 22.1 cm 60 s 20.0 cm glance. The graph also conains informaion on he elociy and acceleraion of he car, alhough ha is less obious. For example, wha can we say abou he aerage elociy of he car beween 20 and 30 seconds? Is he car moing during his ime? A glance a he graph shows us ha he disance is no changing during ha ime ineral, so he car is no moing. The elociy is zero during ha ime, which is represened by a horizonal line on our graph of disance ersus ime. Wha abou he elociy a oher poins in he moion? The car is moing more rapidly beween 0 and 20 seconds han i is beween 30 and 50 seconds. The disance cure is rising more rapidly beween 0 and 20 seconds han beween 30 and 50 seconds. Since more disance is coered in he same ime, he car mus be moing faser here. A seeper slope o he cure is associaed wih a larger speed. In fac, he slope of he disance-ersus-ime cure a any poin on he graph is equal o he insananeous elociy of he car.* The slope indicaes how rapidly he disance is changing wih ime a any insan in ime. The rae of change of disance wih ime is he insananeous speed according o he definiion gien in secion 2.1. Since he moion akes place along a sraigh line, we can hen represen he direcion of he elociy wih plus or minus signs. There are only wo possibiliies, forward or backward. We hen hae he insananeous elociy, which includes boh he size (speed) and direcion of he moion. When he car raels backward, is disance from he saring poin decreases. The cure goes down, as i does beween 50 and 60 seconds. We refer o his downwardsloping porion of he cure as haing a negaie slope and also say ha he elociy is negaie during his porion of he moion. A large upward slope represens a large insananeous elociy, a zero slope (horizonal line) a zero elociy, and a downward slope a negaie (backward) elociy. Looking a he slope of he graph ells us all we need o know abou he elociy of he car. Velociy and acceleraion graphs These ideas abou elociy can be bes summarized by ploing a graph of elociy agains ime for he car (fig. 2.15). The elociy is consan whereer he slope of he disanceersus-ime graph of figure 2.14 is consan. Any sraighline segmen of a graph has a consan slope, so he elociy changes only where he slope of he graph in figure 2.14 changes. If you compare he graph in figure 2.15 o he graph in figure 2.14 carefully, hese ideas should become clear. *Since he mahemaical definiion of slope is he change in he erical coordinae d diided by he change in he horizonal coordinae, he slope, d/, is equal o he insananeous elociy, proided ha is sufficienly small. I is possible o grasp he concep of slope, howeer, wihou appealing o he mahemaical definiion.

12 2.4 Graphing Moion Time (s) figure 2.14 Disance ploed agains ime for he moion of he oy car. The daa poins are hose lised in able 2.2. Velociy (cm/s) Disance (cm) Time (s) figure 2.15 Insananeous elociy ploed agains ime for he moion of he oy car. The elociy is greaes when disance raeled is changing mos rapidly. Acceleraion (cm/s 2 ) figure Time (s) An approximae skech of acceleraion ploed agains ime for he oy-car daa. The acceleraion is non-zero only when he elociy is changing. 60 Wha can we say abou he acceleraion from hese graphs? Since acceleraion is he rae of change of elociy wih ime, he elociy graph (fig. 2.15) also proides informaion abou he acceleraion. In fac, he insananeous acceleraion is equal o he slope of he elociy-ersusime graph. A seep slope represens a rapid change in elociy and hus a large acceleraion. A horizonal line has zero slope and represens zero acceleraion. The acceleraion urns ou o be zero for mos of he moion described by our daa. The elociy changes a only a few poins in he moion. The acceleraion would be large a hese poins and zero eerywhere else. Since our daa do no indicae how rapidly he changes in elociy acually occur, we do no hae enough informaion o say jus how large he acceleraion is a hose few poins where i is no zero. We would need measuremens of disance or elociy eery enh of a second or so o ge a clear idea of how rapid hese changes are. As we will see in chaper 4, we know ha hese changes in elociy canno occur insanly. Some ime is required. So we can skech an approximae graph of acceleraion ersus ime, as shown in figure The spikes in figure 2.16 occur when he elociy is changing. A 20 seconds, here is a rapid decrease in he elociy represened by a downward spike or negaie acceleraion. A 30 seconds, he elociy increases rapidly from zero o a consan alue, and his is represened by an upward spike or posiie acceleraion. A 50 seconds, here is anoher negaie acceleraion as he elociy changes from a posiie o a negaie alue. If you could pu yourself inside he oy car, you would definiely feel hese acceleraions. (Eeryday phenomenon box 2.2 proides anoher example of how a graph is useful for analyzing moion.) Can we find he disance raeled from he elociy graph? Wha oher informaion can be gleaned from he elociyersus-ime graph of figure 2.15? Think for a momen abou how you would go abou finding he disance raeled if you knew he elociy. For a consan elociy, you can ge he disance simply by muliplying he elociy by he ime, d. In he firs 20 seconds of he moion, for example, he elociy is 0.8 cm/s and he disance raeled is 0.8 cm/s imes 20 seconds, which is 16 cm. This is jus he reerse of wha we used in deermining he elociy in he firs place. We found he elociy by diiding he disance raeled by he ime. How would his disance be represened on he elociy graph? If you recall formulas for compuing areas, you may recognize ha he disance d is he area of he shaded recangle on figure The area of a recangle is found by muliplying he heigh imes he widh, jus wha we hae done here. The elociy, 0.8 cm/s, is he heigh and he

13 30 Chaper 2 Describing Moion eeryday phenomenon The 100-m Dash The Siuaion. A world-class spriner can run 100 m in a ime of a lile under 10 s. The race begins wih he runners in a crouched posiion in he saring blocks, waiing for he sound of he sarer s pisol. The race ends wih he runners lunging across he finish line, where heir imes are recorded by sopwaches or auomaic imers. box 2.2 are easies o isualize by skeching a graph of speed ploed agains ime, as shown. Since he runner raels in a sraigh line, he magniude of he insananeous elociy is equal o he insananeous speed. The runner reaches op speed a approximaely 2 o 3 s ino he race. s (m/s) Zero acceleraion Decreasing acceleraion Consan acceleraion (s) Runners in he saring blocks, waiing for he sarer s pisol o fire. Wha happens beween he sar and finish of he race? How do he elociy and acceleraion of he runners ary during he race? Can we make reasonable assumpions abou wha he elociy-ersus-ime graph looks like for a ypical runner? Can we esimae he maximum elociy of a good spriner? Mos imporanly for improing performance, wha facors affec he success of a runner in he dash? The Analysis. Le s assume ha he runner coers he 100-m disance in a ime of exacly 10 s. We can compue he aerage speed of he runner from he definiion s d/: s 100 m 10 s 10 m/s. Clearly, his is no he runner s insananeous speed hroughou he course of he race, since he runner s speed a he beginning of he race is zero and i akes some ime o accelerae o he maximum speed. The objecie in he race is o reach a maximum speed as quickly as possible and o susain ha speed for he res of he race. Success is deermined by wo hings: how quickly he runner can accelerae o his maximum speed and he alue of his maximum speed. A smaller runner ofen has beer acceleraion bu a smaller maximum speed, while a larger runner someimes akes longer o reach op speed bu has a larger maximum speed. The ypical runner does no reach op speed before raeling a leas 10 o 20 m. If he aerage speed is 10 m/s, he runner s maximum speed mus be somewha larger han his alue, since we know ha he insananeous speed will be less han 10 m/s while he runner is acceleraing. These ideas A graph of speed ersus ime for a hypoheical runner in he 100-m dash. The aerage speed (or elociy) during he ime ha he runner is acceleraing is approximaely half of is maximum alue if he runner s acceleraion is more or less consan during he firs 2 s. If we assume ha he runner s aerage speed during his ime is abou 5.5 m/s (half of 11 m/s), hen he speed hrough he remainder of he race would hae o be abou 11.1 m/s o gie an aerage speed of 10 m/s for he enire race. This can be seen by compuing he disance from hese alues: d (5.5 m/s)(2 s) (11.1 m/s)(8 s) 11 m 89 m 100 m. Wha we hae done here is o make some reasonable guesses for hese alues ha will make he aerage speed come ou o 10 m/s; we hen checked hese guesses by compuing he oal disance. This suggess ha he maximum speed of a good spriner mus be abou 11 m/s (25 MPH). For sake of comparison, a disance runner who can run a 4-min mile has an aerage speed of abou 15 MPH, or 6.7 m/s. The runner s sraegy should be o ge a good jump ou of he blocks, keeping he body low iniially and leaning forward o minimize air resisance and maximize leg drie. To mainain op speed during he remainder of he race, he runner needs good endurance. A runner who fades near he end needs more condiioning drills. For a gien runner wih a fixed maximum speed, he aerage speed depends on how quickly he runner can reach op speed. This abiliy o accelerae rapidly depends upon leg srengh (which can be improed by working wih weighs and oher raining exercises) and naural quickness.

14 2.5 Uniform Acceleraion 31 ime, 20 seconds, is he widh of his recangle on he graph. I urns ou ha we can find he disance his way een when he areas inoled on he graph are no recangles, alhough he process is more difficul when he cures are more complicaed. The general rule is ha he disance raeled is equal o he area under he elociy-ersus-ime cure. When he elociy is negaie (below he ime axis on he graph), he objec is raeling backward and is disance from he saring poin is decreasing. Een wihou compuing he area precisely, i is possible o ge a rough idea of he disance raeled by sudying he elociy graph. A large area represens a large disance. Quick isual comparisons gie a good picure of wha is happening wihou he need for lenghy calculaions. This is he beauy of a graph. A good graph can presen a picure of moion ha is rich in insigh. Disance raeled ploed agains ime ells us no only where he objec is a any ime, bu is slope also indicaes how fas i was moing. The graph of elociy ploed agains ime also conains informaion on acceleraion and on he disance raeled. Producing and sudying such graphs can gie us a more general picure of he moion and he relaionships beween disance, elociy, and acceleraion. change as he moion proceeds. I has he same alue a any ime, which produces a horizonal-line graph. The graph of elociy ploed agains ime for his same siuaion ells a more ineresing sory. From our discussion in secion 2.4, we know ha he slope of a elociy-ersusime graph is equal o he acceleraion. For a uniform posiie acceleraion, he elociy graph should hae a consan upward slope; he elociy increases a a seady rae. A consan slope produces a sraigh line, which slopes upward if he acceleraion is posiie as shown in figure In ploing his graph, we assumed ha he iniial elociy is zero. This graph can also be represened by a formula. The elociy a any ime is equal o he original elociy plus he elociy ha has been gained because he car is acceleraing. The change in elociy is equal o he acceleraion imes he ime, a since acceleraion is defined as /. These ideas resul in he relaionship 0 a. The firs erm on he righ, 0, is he original elociy (assumed o be zero in figure 2.18), and he second erm, a 2.5 Uniform Acceleraion If you drop a rock, i falls oward he ground wih a consan acceleraion, as we will see in he nex chaper. An unchanging or uniform acceleraion is he simples form of acceleraed moion. I occurs wheneer here is a consan force acing on an objec, which is he case for a falling rock as well as for many oher siuaions. How do we describe he resuling moion? The imporance of his quesion was firs recognized by Galileo, who sudied he moion of balls rolling down inclined planes as well as objecs in free fall. In his famous work, Dialogues Concerning Two New Sciences, published in 1638 near he end of his life, Galileo deeloped he graphs and formulas ha are inroduced in his secion and ha hae been sudied by sudens of physics eer since. His work proided he foundaion for much of Newon s hinking a few decades laer. How does elociy ary in uniform acceleraion? Suppose a car is moing along a sraigh road and acceleraing a a consan rae. We hae ploed he acceleraion agains ime for his siuaion in figure The graph is ery simple, bu i illusraes wha we mean by uniform acceleraion. A uniform acceleraion is one ha does no figure 2.17 The acceleraion graph for uniform acceleraion is a horizonal line. The acceleraion does no change wih ime. = 1 2 figure 2.18 Velociy ploed agains ime for uniform acceleraion, saring from res. For his special case, he aerage elociy is equal o one-half he final elociy.

15 32 Chaper 2 Describing Moion a, represens he change in elociy due o he acceleraion. Adding hese wo erms ogeher yields he elociy a any laer ime. A numerical example applying hese ideas o an acceleraing car is found in par a of example box 2.4. The car could no keep on acceleraing indefiniely a a consan rae because he elociy would soon reach incredible alues. No only is his dangerous, bu physical limis imposed by air resisance and oher facors preen his from happening. Wha happens if he acceleraion is negaie? Velociy would decrease raher han increase, and he slope of he elociy graph would slope downward raher han upward. Because he acceleraion is hen negaie, he second erm in he formula for would subrac from he firs erm, causing he elociy o decrease from is iniial alue. The elociy hen decreases a a seady rae. d figure 2.19 As he car acceleraes uniformly, he disance coered grows more and more rapidly wih ime because he elociy is increasing. How does disance raeled ary wih ime? If he elociy is increasing a a seady rae, wha effec does his hae on he disance raeled? As he car moes faser and faser, he disance coered grows more and more rapidly. Galileo showed how o find he disance for his siuaion. We find disance by muliplying elociy by ime, bu in his case we mus use an aerage elociy since he elociy is changing. By appealing o he graph in figure 2.18, we can see ha he aerage elociy should be jus half he final elociy,. If he iniial elociy is zero, he final example box 2.4 Sample Exercise: Uniform Acceleraion A car raeling due eas wih an iniial elociy of 10 m/s acceleraes for 6 seconds a a consan rae of 4 m/s 2. a. Wha is is elociy a he end of his ime? b. How far does i rael during his ime? a m/s 0 a a 4 m/s 2 10 m/s (4 m/s 2 )(6 s) 6 s 10 m/s 24 m/s? 34 m/s 34 m/s due eas b. d a2 (10 m/s)(6 s) 1 2 (4 m/s2 )(6 s) 2 60 m (2 m/s 2 )(36 s 2 ) 60 m 72 m 132 m elociy is a, so muliplying he aerage elociy by he ime yields d 1 2 a2. The ime eners wice, once in finding he aerage elociy and hen again when we muliply he elociy by ime o find he disance.* The graph in figure 2.19 illusraes his relaionship; he disance cure slopes upward a an eer-increasing rae as he elociy increases. This formula and graph are only alid if he objec sars from res as shown in figure Since disance raeled is equal o he area under he elociy-ersus-ime cure (as discussed in secion 2.4), his expression for disance can also be hough of as he area under he riangle in figure The area of a riangle is equal o one-half is base imes is heigh, which produces he same resul. If he car is already moing before i begins o accelerae, he elociy graph can be redrawn as picured in figure The oal area under he elociy cure can hen be spli in wo pieces, a riangle and a recangle, as shown. The oal disance raeled is he sum of hese wo areas, 1 d 0 2 a 2. The firs erm in his formula represens he disance he objec would rael if i moed wih consan elociy 0, and he second erm is he addiional disance raeled because he objec is acceleraing (he area of he riangle in figure 2.20). If he acceleraion is negaie, meaning ha he objec is slowing down, his second erm will subrac from he firs. *Expressing his argumen in symbolic form, i becomes The aerage elociy a d a2 1 2 a2.

16 Summary 33 0 figure 2.20 The elociy-ersus-ime graph redrawn for an iniial elociy differen from zero. The area under he cure is diided ino wo porions, a recangle and a riangle. This more general expression for disance may seem complex, bu he rick o undersanding i is o break i down ino is pars, as jus suggesed. We are merely adding wo erms represening differen conribuions o he oal disance. Each one can be compued in a sraighforward 1 2 d = a 2 d = 0 0 = Δ 0 manner, and i is no difficul o add hem ogeher. The wo porions of he graph in figure 2.20 represen hese wo conribuions. The sample exercise in example box 2.4 proides a numerical example of hese ideas. The car in his example acceleraes uniformly from an iniial elociy of 10 m/s due eas o a final elociy of 34 m/s due eas and coers a disance of 132 meers while his acceleraion is aking place. Had i no been acceleraing, i would hae gone only 60 meers in he same ime. The addiional 72 meers comes from he acceleraion of he car. Acceleraion inoles change, and uniform acceleraion inoles a seady rae of change. I herefore represens he simples kind of acceleraed moion ha we can imagine. Uniform acceleraion is essenial o an undersanding of free fall, discussed in chaper 3, as well as o many oher phenomena. Such moion can be represened by eiher he graphs or he formulas inroduced in his secion. Looking a boh and seeing how hey are relaed will reinforce hese ideas. summary The main purpose of his chaper is o inroduce conceps ha are crucial o a precise descripion of moion. To undersand acceleraion, you mus firs grasp he concep of elociy, which in urn builds on he idea of speed. The disincions beween speed and elociy, and beween elociy and acceleraion, are paricularly imporan. 1 Aerage and insananeous speed. Aerage speed is defined as he disance raeled diided by he ime. I is he aerage rae a which disance is coered. Insananeous speed is he rae a which disance is being coered a a gien insan in ime and requires ha we use ery shor ime inerals for compuaion. 2 Velociy. The insananeous elociy of an objec is a ecor quaniy ha includes boh direcion and size. The size of he elociy ecor is equal o he insananeous speed, and he direcion is ha of he objec s moion. = speed and direcion s d = 3 Acceleraion. Acceleraion is defined as he ime rae of change of elociy and is found by diiding he change in elociy by he ime. Acceleraion is also a ecor quaniy. I can be compued as eiher an aerage or an insananeous alue. A change in he direcion of he elociy can be as imporan as a change in magniude. Boh inole acceleraion. 1 Δ + = a Δ = _ 2