# Motion Along a Straight Line

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1 Moion Along a Sraigh Line On Sepember 6, 993, Dave Munday, a diesel mechanic by rade, wen over he Canadian edge of Niagara Falls for he second ime, freely falling 48 m o he waer (and rocks) below. On his aemp, he rode in a seel chamber wih an airhole. Munday, keen on surviving his plunge ha had killed oher sunmen, had done considerable research on he physics and engineering aspecs of he plunge. If he fell sraigh down, how could we predic he speed a which he would hi he waer? The answer is in his chaper. 5

2 6 CHAPTER Moion Along a Sraigh Line - Moion The world, and everyhing in i, moves. Even a seemingly saionary hing, such as a roadway, moves because he Earh is moving. No only is he Earh roaing and orbiing he Sun, bu he Sun is also moving hrough space. The moion of objecs can ake many differen forms. For example, a moving objec s pah migh be a sraigh line, a curve, a circle, or somehing more complicaed. The eniy in moion migh be somehing simple, like a ball, or somehing complex, like a human being or galaxy. In physics, when we wan o undersand a phenomenon such as moion, we begin by exploring relaively simple moions. For his reason, in he sudy of moion we sar wih kinemaics, which focuses on describing moion, raher han on dynamics, which deals wih he causes of moion. Furher, we begin our sudy of kinemaics by developing he conceps required o measure moion and mahemaical ools needed o describe hem in one dimension (or in D). Only hen do we exend our sudy o include a consideraion of he causes of moion and moions in wo and hree dimensions. Furher simplificaions are helpful. Thus, in his chaper, our descripion of he moion of objecs is resriced in wo ways. The moion of he objec is along a sraigh line. The moion may be purely verical (ha of a falling sone), purely horizonal (ha of a car on a level highway), or slaned (ha of an airplane rising a an angle from a runway), bu i mus be a sraigh line.. The objec is effecively a paricle because is size and shape are no imporan o is moion. By paricle we mean eiher (a) a poin-like objec wih dimensions ha are small compared o he disance over which i moves (such as he size of he Earh relaive o is orbi around he Sun), (b) an exended objec in which all is pars move ogeher (such as a falling baskeball ha is no spinning), or (c) ha we are only ineresed in he pah of a special poin associaed wih he objec (such as he bel buckle on a walking person). We will sar by inroducing very precise definiions of words commonly used o describe moion like speed, velociy, and acceleraion. These definiions may conflic wih he way hese erms are used in everyday speech. However, by using precise definiions raher han our casual definiions, we will be able o describe and predic he characerisics of common moions in graphical and mahemaical erms. These mahemaical descripions of phenomena form he basic vocabulary of physics and engineering. Alhough our reamen may seem ridiculously formal, we need o provide a foundaion for he analysis of more complex and ineresing moions. READING EXERCISE - Which of he following moions are along a sraigh line (a) a sring of cars raveling up and down along a roller coaser, (b) a cannonball sho sraigh up, (c) a car raveling along a sraigh ciy sree, (d) a ball rolling along a sraigh ramp iled a a 45 angle. READING EXERCISE - In realiy here are no poin paricles. Rank he following everyday iems from mos paricle-like o leas paricle-like (a) a -m-all long jumper relaive o a 5 m disance covered in a jump, (b) a piece of lead sho from a shogun shell relaive o is range of 5 m, (c) he Earh of diameer 3 6 m relaive o he approximae diameer of is orbi abou he Sun of 3 m.

3 Posiion and Displacemen Along a Line 7 - Posiion and Displacemen Along a Line Defining a Coordinae Sysem In order o sudy moion along a sraigh line, we mus be able o specify he locaion of an objec and how i changes over ime. A convenien way o locae a poin of ineres on an objec is o define a coordinae sysem. Houses in Cosa Rican owns are commonly locaed wih addresses such as meers eas of he Pos Office. In order o locae a house, a disance scale mus be agreed upon (meers are used in he example), a reference poin or origin mus be specified (in his case he Pos Office), and a direcion (in his case eas). Thus, in locaing an objec ha can move along a sraigh line, i is convenien o specify is posiion by choosing a one-dimensional coordinae sysem. The sysem consiss of a poin of reference known as he origin (or zero poin), a line ha passes hrough he chosen origin called a coordinae axis, one direcion along he coordinae axis, chosen as posiive and he oher direcion as negaive, and he unis we use o measure a quaniy. We have labeled he coordinae axis as he x axis, in Fig. -, and placed an origin on i. The direcion of increasing numbers (coordinaes) is called he posiive direcion, which is oward he righ in Fig. -. The opposie direcion is he negaive direcion. Figure - is drawn in he radiional fashion, wih negaive coordinaes o he lef of he origin and posiive coordinaes o he righ. I is also radiional in physics o use meers as he sandard scale for disance. However, we have freedom o choose oher unis and o decide which side of he origin is labeled wih negaive coordinaes and which is labeled wih posiive coordinaes. Furhermore, we can choose o define an x axis ha is verical raher han horizonal, or inclined a some angle. In shor, we are free o make choices abou how we define our coordinae sysem. Good choices make describing a siuaion much easier. For example, in our consideraion of moion along a sraigh line, we would wan o align he axis of our onedimensional coordinae sysem along he line of moion. In Chapers 5 and 6, when we consider moions in wo dimensions, we will be using more complex coordinae sysems wih a se of muually perpendicular coordinae axes. Choosing a coordinae sysem ha is appropriae o he physical siuaion being described can simplify your mahemaical descripion of he siuaion. To describe a paricle moving in a circle, you would probably choose a wo-dimensional coordinae sysem in he plane of he circle wih he origin placed a is cener. Posiive direcion Negaive direcion x(m) Negaive coordinae Posiive coordinae values values Origin FIGURE - Posiion is deermined on an axis ha is marked in unis of meers and ha exends indefiniely in opposie direcions. Defining Posiion as a Vecor Quaniy The reason for choosing our sandard one-dimensional coordinae axis and oriening i along he direcion of moion is o be able o define he posiion of an objec relaive o our chosen origin, and hen be able o keep rack of how is posiion changes as he objec moves. I urns ou ha he posiion of an objec relaive o a coordinae sysem can be described by a mahemaical eniy known as a vecor. This is because, in order o find he posiion of an objec, we mus specify boh how far and in which direcion he objec is from he origin of a coordinae sysem. A VECTOR is a mahemaical eniy ha has boh a magniude and a direcion. Vecors can be added, subraced, muliplied, and ransformed according o well-defined mahemaical rules. There are oher physical quaniies ha also behave like vecors such as velociy, acceleraion, force, momenum, and elecric and magneic fields. However, no all physical quaniies ha have signs associaed wih hem are vecors. For example, emperaures do no need o be described in erms of a coordinae

4 8 CHAPTER Moion Along a Sraigh Line Posiion vecor of magniude.5 m poining in a negaive direcion m m m m m FIGURE - A posiion vecor can be represened by an arrow poining from he origin of a chosen coordinae sysem o he locaion of he objec. x sysem, and single numbers, such as T 5 C or T C, are sufficien o describe hem. The minus sign, in his case, does no signify a direcion. Mass, disance, lengh, area, and volume also have no direcions associaed wih hem and, alhough heir values depend on he unis used o measure hem, heir values do no depend on he orienaion of a coordinae sysem. Such quaniies are called scalars. A SCALAR is defined as a mahemaical quaniy whose value does no depend on he orienaion of a coordinae sysem and has no direcion associaed wih i. In general, a one-dimensional vecor can be represened by an arrow. The lengh of he arrow, which is inherenly posiive, represens he magniude of he vecor and he direcion in which he arrow poins represens he direcion associaed wih he vecor. We begin his sudy of moion by inroducing you o he properies of onedimensional posiion and displacemen vecors and some of he formal mehods for represening and manipulaing hem. These formal mehods for working wih vecors will prove o be very useful laer when working wih wo- and hree-dimensional vecors. A one-dimensional posiion vecor is defined by he locaion of he origin of a chosen one-dimensional coordinae sysem and of he objec of ineres. The magniude of he posiion vecor is a scalar ha denoes he disance beween he objec and he origin. For example, an objec ha has a posiion vecor of magniude 5 m could be locaed a he poin 5 m or 5 m from he origin. On a convenional x axis, he direcion of he posiion vecor is posiive when he objec is locaed o he righ of he origin and negaive when he objec is locaed o he lef of he origin. For example, in he sysem shown in Fig. -, if a paricle is locaed a a disance of 3 m o he lef of he origin, is posiion vecor has a magniude of 3 m and a direcion ha is negaive. One of many ways o represen a posiion vecor is o draw an arrow from he origin o he objec s locaion, as shown in Fig. -, for an objec ha is.5 m o he lef of he origin. Since he lengh of a vecor arrow represens he magniude of he vecor, is lengh should be proporional o he disance from he origin o he objec of ineres. In addiion, he direcion of he arrow should be from he origin o he objec. Insead of using an arrow, a posiion vecor can be represened mahemaically. In order o develop a useful mahemaical represenaion we need o define a uni vecor associaed wih our x axis. A UNIT VECTOR FOR A COORDINATE AXIS is a dimensionless vecor ha poins in he direcion along a coordinae axis ha is chosen o be posiive. I is cusomary o represen a uni vecor ha poins along he posiive x axis wih he symbol î (i-ha), alhough some exs use he symbol xˆ (x-ha) insead. When considering hree-dimensional vecors, he uni vecors poining along he designaed posiive y axis and z axis are denoed by ĵ and ˆk, respecively. These vecors are called uni vecors because hey have a dimensionless value of one. However, you should no confuse he use of word uni wih a physical uni. Uni vecors should be shown on coordinae axes as small poiners wih no physical unis, such as meers, associaed wih hem. This is shown in Fig. -3 for he x axis uni vecor. Since he scale used in he coordinae sysem has unis, i is essenial ha he unis always be associaed wih he number describing he locaion of an objec along an axis. Figure -3 also shows how he uni vecor is used o creae a posiion vecor corresponding o an objec locaed a posiion.5 meers on our x axis. To do his we srech or muliply he uni vecor by he magniude of he posiion vecor, which

5 Posiion and Displacemen Along a Line 9 m m m m m x = (.5 m) î î Dimensionless uni vecor poining in he posiive direcion (.5 m)î x FIGURE -3 Arrows represening () a dimensionless uni vecor, î, poining in he posiive x direcion; () a vecor represening he uni vecor muliplied by.5 meers; and (3) a vecor muliplied by.5 meers and invered by muliplicaion by o creae he posiion vecor x (.5 m)î. This posiion vecor has a magniude of.5 meers and poins in a negaive direcion. is.5 m. Noe ha we are using he coordinae axis o describe a posiion in meers relaive o an origin, so i is essenial o include he unis wih he number. This muliplicaion of he dimensionless uni vecor by.5 m creaes a.5-m-long vecor ha poins in he same direcion as he uni vecor. I is denoed by (.5 m)î. However, he vecor we wan o creae poins in he negaive direcion, so he vecor poining in he posiive direcion mus be invered using a minus sign. The posiion vecor we have creaed is denoed as x. I can be divided ino wo pars a vecor componen and a uni vecor, (.5 m)î. In his example, he x-componen of he posiion vecor, denoed as x, is.5 m. Here he quaniy.5 m wih no minus sign in fron of i is known as he magniude of his posiion vecor. In general, he magniude is denoed as x. Thus, he one-dimensional posiion vecor for he siuaion shown in Fig. -3 is denoed mahemaically using he following symbols x x xî (.5 m)î. The x-componen of a posiion vecor, denoed x, can be posiive or negaive depending on which side of he origin he paricle is. Thus, in one dimension in erms of absolue values, he vecor componen x is eiher x or x, depending on he objec s locaion. In general, a componen of a vecor along an axis, such as x in his case, is no a scalar since our x-componen will change sign if we choose o reverse he orienaion of our chosen coordinae sysem. In conras, he magniude of a posiion vecor is always posiive, and i only ells us how far away he objec is from he origin, so he magniude of a vecor is always a scalar quaniy. The sign of he componen ( or ) ells us in which direcion he vecor is poining. The sign will be negaive if he objec is o he lef of he origin and posiive if i is o he righ of he origin. Defining Displacemen as a Vecor Quaniy The sudy of moion is primarily abou how an objec s locaion changes over ime under he influence of forces. In physics he concep of change has an exac mahemaical definiion. CHANGE is defined as he difference beween he sae of a physical sysem (ypically called he final sae) and is sae a an earlier ime (ypically called he iniial sae). This definiion of change is used o define displacemen. DISPLACEMENT is defined as he change of an objec s posiion ha occurs during a period of ime.

6 3 CHAPTER Moion Along a Sraigh Line r = +7 m i x = +5 m î x = + m i ˆ (a) (b) ˆ ( x ) = m î ( x ) = 5 m i r = 7 m î x = +5 m î ( x ) = 5 m î r = m (c) FIGURE -4 The wide arrow shows he displacemen vecor r for hree siuaions leading o (a) a posiive displacemen, (b) a negaive displacemen, and (c) zero displacemen. ˆ Since posiion can be represened as a vecor quaniy, displacemen is he difference beween wo vecors, and hus, is also a vecor. So, in he case of moion along a line, an objec moving from an iniial posiion x o anoher final posiion x a a laer ime is said o undergo a displacemen r, given by he difference of wo posiion vecors r x x xî (displacemen vecor), (-) where he symbol is used o represen a change in a quaniy, and he symbol signifies ha he displacemen r is given by x x because we have chosen o define i ha way. As you will see when we begin o work wih vecors in wo and hree dimensions, i is convenien o consider subracion as he addiion of one vecor o anoher ha has been invered by muliplying he vecor componen by. We can use his idea of defining subracion as he addiion of an invered vecor o find displacemens. Le s consider hree siuaions (a) A paricle moves along a line from x (5 m)î o x ( m)î. Since r x x x ( x ), r ( m)î (5 m)î ( m)î ( 5 m)î (7 m)î. The posiive resul indicaes ha he moion is in he posiive direcion (oward he righ in Fig. -4a). (b) A paricle moves from x ( m)î o (5 m)î. Since x r x x x ( x ), r (5 m)î ( m)î (5 m)î ( m)î ( 7 m)î. (c) The negaive resul indicaes ha he displacemen of he paricle is in he negaive direcion (oward he lef in Fig. -4b). A paricle sars a 5 m, moves o m, and hen reurns o 5 m. The displacemen for he full rip is given by r x ( x x x ), where x (5 m)î and x (5 m)î r (5 m)î ( 5 m)î ( m)î and he paricle s posiion hasn changed, as in Fig. -4c. Since displacemen involves only he original and final posiions, he acual number of meers raced ou by he paricle while moving back and forh is immaerial. If we ignore he sign of a paricle s displacemen (and hus is direcion), we are lef wih he magniude of he displacemen. This is he disance beween he original and final posiions and is always posiive. I is imporan o remember ha displacemen (or any oher vecor) has no been compleely described unil we sae is direcion. We use he noaion r for displacemen because when we have moion in more han one dimension, he noaion for he posiion vecor is r. For a one-dimensional moion along a sraigh line, we can also represen he displacemen as x. The magniude of displacemen is represened by surrounding he displacemen vecor symbol wih absolue value signs {magniude of displacemen} r or x READING EXERCISE -3 Can a paricle ha moves from one posiion wih a negaive value, o anoher posiion wih a negaive value, undergo a posiive displacemen?

7 Velociy and Speed 3 TOUCHSTONE EXAMPLE - Displacemens Three pairs of iniial and final posiions along an x axis represen he locaion of objecs a wo successive imes (pair ) 3 m, 5 m; (pair ) 3 m, 7 m; (pair 3) 7 m, 3 m. (a) Which pairs give a negaive displacemen? SOLUTION The Key Idea here is ha he displacemen is negaive when he final posiion lies o he lef of he iniial posiion. As shown in Fig. -5, his happens when he final posiion is more negaive han he iniial posiion. Looking a pair, we see ha he final posiion, 5 m, is posiive while he iniial posiion, 3 m, is negaive. This means ha he displacemen is from lef (more negaive) o righ (more posiive) and so he displacemen is posiive for pair. x ( 5 m)î ( 3 m)î ( 5 m)î (3 m)î ( 8 m)î. For pair he same argumen yields x ( 7 m)î ( 3 m)î ( 7 m)î (3 m)î ( 4 m)î. Finally, he displacemen for pair 3 is x ( 3 m)î ( 7 m)î ( 3 m)î ( 7 m)î ( m)î. (c) Wha is he magniude of each posiion vecor? 8m 6 m 4 m m m m 4 m 6 m 8 m Pair Pair For pair he siuaion is differen. The final posiion, 7 m, lies o he lef of he iniial posiion, 3 m, so he displacemen is negaive. For pair 3 he final posiion, 3 m, is o he lef of he origin while he iniial posiion, 7 m, is o he righ of he origin. So he displacemen is from he righ of he origin o is lef, a negaive displacemen. (b) Calculae he value of he displacemen in each case using vecor noaion. SOLUTION The Key Idea here is o use Eq. - o calculae he displacemen for each pair of posiions. I ells us he difference beween he final posiion and he iniial posiion, in ha order, Pair 3 FIGURE -5 Displacemen associaed wih hree pairs of iniial and final posiions along an x axis. x x x (displacemen). (-) For pair he final posiion is x ( 5 m)î and he iniial posiion is x ( 3 m)î, so he displacemen beween hese wo posiions is jus x SOLUTION Of he six posiion vecors given, one of hem namely x ( 3 m)î appears in all hree pairs. The remaining hree posiions are x ( 5 m)î, x 3 ( 7 m)î, and x 4 ( 7 m)î. The Key Idea here is ha he magniude of a posiion vecor jus ells us how far he poin lies from he origin wihou regard o wheher i lies o he lef or o he righ of he origin. Thus he magniude of our firs posiion vecor is 3 m since he posiion specified by x ( 3 m)î is 3 m o he lef of he origin. I s no 3 m, because magniudes only specify disance from he origin, no direcion. For he same reason, he magniude of he second posiion vecor is jus 5 m while he magniude of he hird and he fourh are each 7 m. The fac ha he hird poin lies 7 m o he lef of he origin while he fourh lies 7 m o he righ doesn maer here. (d) Wha is he value of he x-componen of each of hese posiion vecors? SOLUTION To answer his quesion you need o remember wha is mean by he componen of a vecor. The key equaion relaing a vecor in one dimension o is componen along is direcion is x x î, where x (wih he arrow over i) is he vecor iself and x (wih no arrow over i) is he componen of he vecor in he direcion specified by he uni vecor î. So he componen of x ( 3 m)î is 3 m, while ha of x ( 5 m)î is jus 5 m, and x ( 7 m)î has as is componen along he î direcion ( 7 m) while for x ( 7 m)î i s jus ( 7 m). In oher words, he componen of a vecor in he direcion of î is jus he signed number (wih is unis) ha muliplies î. -3 Velociy and Speed Suppose a suden sands sill or speeds up and slows down along a sraigh line. How can we describe accuraely and efficienly where she is and how fas she is moving? We will explore several ways o do his.

8 3 CHAPTER Moion Along a Sraigh Line Represening Moion in Diagrams and Graphs Moion Diagrams Now ha you have learned abou posiion and displacemen, i is quie easy o describe he moion of an objec using picures or skeches o char how posiion changes over ime. Such a represenaion is called a moion diagram. For example, Fig. -6 shows a suden whom we rea as if she were concenraed ino a paricle locaed a he back of her bel. She is sanding sill a a posiion x (. m)î from a poin on a sidewalk ha we choose as our origin. Figure -7 shows a more complex diagram describing he suden in moion. Suppose we see ha jus as we sar iming her progress wih a sopwach (so. s), he back of her bel is.47 m o he lef of our origin. The x-componen of her posiion is hen x.47 m. The suden hen moves oward he origin, almos reaches he origin a.5 s, and hen coninues moving o he righ so ha her x-componen of posiion has increasingly posiive values. I is imporan o recognize ha jus as we chose an origin and direcion for our coordinae axis, we also chose an origin in ime. If we had chosen o sar our iming seconds earlier, hen he new moion diagram would show he back of her bel as being a x.47 m a s. FIGURE -6 A moion diagram of a suden sanding sill wih he back of her bel a a horizonal disance of. m o he lef of a spo of he sidewalk designaed as he origin. m m Origin m m x =. s =. s =. s FIGURE -7 A moion diagram of a suden saring o walk slowly. The horizonal posiion of he back of her bel sars a a horizonal disance of.47 m o he lef of a spo designaed as he origin. She is speeding up for a few seconds and hen slowing down. 3 m m m Origin m m x x(m) (s) 3 3 FIGURE -8 The graph of he x-componen of posiion for a suden who is sanding sill a x. m for a leas 3 seconds. Graphs Anoher way o describe how he posiion of an objec changes as ime passes is wih a graph. In such a graph, he x-componen of he objec s posiion, x, can be ploed as a funcion of ime,. This posiion ime graph has alernae names such as a graph of x as a funcion of, x(), or x vs.. For example, Fig. -8 shows a graph of he suden sanding sill wih he back of her bel locaed a a horizonal posiion of. m from a spo on he sidewalk ha is chosen as he origin. The graph of seady moion shown in Fig. -8 is no more informaive han he picure or a commen ha he suden is sanding sill for 3 seconds a a cerain locaion. Bu i s anoher sory when we consider he graph of a moion. Figure -9 is a graph of a suden s x-componen of posiion as a funcion of ime. I represens he same informaion depiced in he moion diagram in Fig. -7. Daa on he suden s moion are firs recorded a. s when he x-componen of her posiion is x.47 m. The suden hen moves oward x. m, passes hrough ha poin a abou.5 s, and hen moves on o increasingly larger posiive values of x while slowing down.

9 Velociy and Speed 33 x(m) (s) 3 3 =. s =. s =. s 3 m m m m m x FIGURE -9 A graph ha represens how he posiion componen, x, of he walking suden shown in Fig. -7 changes over ime. The moion diagram, shown below he graph, is associaed wih he graph a hree poins in ime as indicaed by he arrows. Alhough he graph of he suden s moion in Fig. -9 seems absrac and quie unlike a moion diagram, i is richer in informaion. For example, he graph allows us o esimae he moion of he suden a imes beween hose for which posiion measuremens were made. Equally imporan, we can use he graph o ell us how fas he suden moves a various imes, and we deal wih his aspec of moion graphs nex. Wha can moion diagrams and x vs. graphs ell us abou how fas and in wha direcion somehing moves along a line? I is clear from an examinaion of he moion diagram a he boom of Fig. -9 ha he suden covers he mos disance and so appears o be moving mos rapidly beween he wo imes. s and.5 s. Bu his ime inerval is also where he slope (or seepness) of he graph has he greaes magniude. Recall from mahemaics ha he average slope of a curve beween wo poins is defined as he raio of he change in he variable ploed on he verical axis (in his case he x-componen of her posiion) o he change in he variable ploed on he horizonal axis (in his case he ime). Hence, on posiion vs. ime graphs (such as hose shown in Fig. -8 and Fig. -9), average slope x x x (definiion of average slope). (-3) Since ime moves forward,, so always has a posiive value. Thus, a slope will be posiive whenever x x, so x is posiive. In his case a sraigh line connecing he wo poins on he graph slans upward oward he righ when he suden is moving along he posiive x-direcion. On he oher hand, if he suden were o move backwards in he direcion along he x axis we chose o call negaive, hen x x. In his case, he slope beween he wo imes would be negaive and he line connecing he poins would slan downward o he righ. Average Velociy For moion along a sraigh line, he seepness of he slope in an x vs. graph over a ime inerval from o ells us how fas a paricle moves. The direcion of moion is indicaed by he sign of he slope (posiive or negaive). Thus, his slope or raio x/ is a special quaniy ha ells us how fas and in wha direcion somehing moves. We haven given he raio x/ a name ye. We do his o emphasize he fac

10 34 CHAPTER Moion Along a Sraigh Line ha he ideas associaed wih figuring ou how fas and in wha direcion somehing moves are more imporan han he names we assign o hem. However, i is inconvenien no o have a name. The common name for his raio is average velociy, which is defined as he raio of displacemen vecor x for he moion of ineres o he ime inerval in which i occurs. This vecor can be expressed in equaion form as x x î x x î (definiion of D average velociy), (-4) where x and x are componens of he posiion vecors a he final and iniial imes. Here we use angle brackes o denoe he average of a quaniy. Also, we use he special symbol for equaliy o emphasize ha he erm on he lef is equal o he erm on he righ by definiion. The ime change is a posiive scalar quaniy because we never need o specify is direcion explicily. In defining v we are basically muliplying he displacemen vecor, x, by he scalar (/ ). This acion gives us a new vecor ha poins in he same direcion as he displacemen vecor. FIGURE - Calculaion of he slope of he line ha connecs he poins on he curve a. s and.5 s. The x- componen of he average velociy is given by his slope. x(m) 3 <v> Slope of he line = x =.4 m/s (s) 3 x =. m (.4 m) =. m =.5 s. s =.5 s (s) x(m) Figure - shows how o find he average velociy for he suden moion represened by he graph shown in Fig. -9 beween he imes. s and.5 s. The average velociy during ha ime inerval is x î x x î. m (.4 m) (.5 s. s) î (.4 m s)î. The x-componen of he average velociy along he line of moion, v x.4 m/s, is simply he slope of he sraigh line ha connecs he poin on he curve a he beginning of our chosen inerval and he poin on he curve a he end of he inerval. Since our suden is speeding up and slowing down, he values of v and v x will in general be differen when calculaed using oher ime inervals. Average Speed Someimes we don care abou he direcion of an objec s moion bu simply wan o keep rack of he disance covered. For insance, we migh wan o know he oal disance a suden walks (number of seps imes disance covered in each sep). Our suden could be pacing back and forh wearing ou her shoes wihou having a vecor displacemen. Similarly, average speed, s, is a differen way of describing how fas an objec moves. Whereas he average velociy involves he paricle s displacemen x, which is a vecor quaniy, he average speed involves he oal disance covered (for example, he produc of he lengh of a sep and he number of seps he suden ook), which is independen of direcion. So average speed is defined as

11 Velociy and Speed 35 s oal disance (definiion of average speed). (-5) Since neiher he oal disance raveled nor he ime inerval over which he ravel occurred has an associaed direcion, average speed does no include direcion informaion. Boh he oal disance and he ime period are always posiive, so average speed is always posiive oo. Thus, an objec ha moves back and forh along a line can have no vecor displacemen, so i has zero velociy bu a raher high average speed. A oher imes, while he objec is moving in only one direcion, he average speed s is he same as he magniude of he average velociy v. However, as you can demonsrae in Reading Exercise -4, when an objec doubles back on is pah, he average speed is no simply he magniude of he average velociy v. Insananeous Velociy and Speed You have now seen wo ways o describe how fas somehing moves average velociy and average speed, boh of which are measured over a ime inerval. Clearly, however, somehing migh speed up and slow down during ha ime inerval. For example, in Fig. -9 we see ha he suden is moving more slowly a. s han she is a.5 s, so her velociy seems o be changing during he ime inerval beween. s and.5 s. The average slope of he line seems o be increasing during his ime inerval. Can we refine our definiion of velociy in such a way ha we can deermine he suden s rue velociy a any one insan in ime? We envision somehing like he almos insananeous speedomeer readings we ge as a car speeds up and slows down. Defining an insan and insananeous velociy is no a rivial ask. As we noed in Chaper, he ime inerval of second is defined by couning oscillaions of radiaion absorbed by a cesium aom. In general, even our everyday clocks work by couning oscillaions in an elecronic crysal, pendulum, and so on. We associae insans in ime wih posiions on he hands of a clock, and ime inervals wih changes in he posiion of he hands. For he purpose of finding a velociy a an insan, we can aemp o make he ime inerval we use in our calculaion so small ha i has almos zero duraion. Of course he displacemen we calculae also becomes very small. So insananeous velociy along a line like average velociy is sill defined in erms of he raio of x /. Bu we have his raio passing o a limi where ges closer and closer o zero. Using sandard calculus noaion for his limi gives us he following definiion x v lim dx d (definiion of D insananeous velociy). (-6) In he language of calculus, he INSTANTANEOUS VELOCITY is he rae a which a paricle s posiion vecor, x, is changing wih ime a a given insan. In passing o he limi he raio x/ is no necessarily small, since boh he numeraor and denominaor are geing small ogeher. The firs par of his expression, x v v x î lim, ells us ha we can find he (insananeous) velociy of an objec by aking he slope of a graph of he posiion componen vs. ime a he poin associaed wih ha

13 Describing Velociy Change 37 SOLUTION We already know he ime inerval wlk (.5 h) for he walk, bu we lack he ime inerval dr for he drive. However, we know ha for he drive he displacemen x dr is 8.4 km and he average velociy v dr x is 7 km/h. A Key Idea o use here comes from Eq. -4 This average velociy is he raio of he displacemen for he drive o he ime inerval for he drive, v dr x x dr dr. Posiion (km) x Driving Truck sops Walking Saion x (=.4 km) Rearranging and subsiuing daa hen give us Therefore, dr x dr v dr x dr wlk. h.5 h.6 h. (c) Wha is your average velociy v x from he beginning of your drive o your arrival a he saion? Find i boh numerically and graphically. SOLUTION The Key Idea here again comes from Eq. -4 v x for he enire rip is he raio of he displacemen of.4 km for he enire rip o he ime inerval of.6 h for he enire rip. Wih Eq. -4, we find i is v x x 8.4 km 7 km h.4 km.6 h. h. 6.8 km/h 7 km/h. To find v x graphically, firs we graph x() as shown in Fig. -, where he beginning and arrival poins on he graph are he origin and he poin labeled Saion. The Key Idea here is ha your average velociy in he x direcion is he slope of he sraigh line (=.6 h)..4 Time (h).6 FIGURE - The lines marked Driving and Walking are he posiion ime plos for he driving and walking sages. (The plo for he walking sage assumes a consan rae of walking.) The slope of he sraigh line joining he origin and he poin labeled Saion is he average velociy for he rip, from beginning o saion. connecing hose poins; ha is, i is he raio of he rise ( x.4 km) o he run (.6 h), which gives us v x 6.8 km/h. (d) Suppose ha o pump he gasoline, pay for i, and walk back o he ruck akes you anoher 45 min. Wha is your average speed from he beginning of your drive o your reurn o he ruck wih he gasoline? SOLUTION The Key Idea here is ha your average speed is he raio of he oal disance you move o he oal ime inerval you ake o make ha move. The oal disance is 8.4 km. km. km.4 km. The oal ime inerval is. h.5 h h. Thus, Eq. -5 gives us s.4 km.37 h 9. km/h. -4 Describing Velociy Change The suden shown in Fig. -9 is clearly speeding up and slowing down as she walks. We know ha he slope of her posiion vs. ime graph over small ime inervals keeps changing. Now ha we have defined velociy, i is meaningful o develop a mahemaical descripion of how fas velociy changes. We see wo approaches o describing velociy change. We could deermine velociy change over an inerval of displacemen magniude, x, and use v/ x as our measure. Alernaively, we could use he raio of velociy change o he inerval of ime,, over which he change occurs or ( v/ ). This is analogous o our definiion of velociy. Boh of our proposals are possible ways of describing velociy change neiher is righ or wrong. In he fourh cenury B.C.E., Arisole believed ha he raio of velociy change o disance change was probably consan for any falling objecs. Almos years laer, he Ialian scienis Galileo did experimens wih ramps o slow down he moion of rolling objecs. Insead he found ha i was he second raio, v /, ha was consan.

14 38 CHAPTER Moion Along a Sraigh Line Our modern definiion of acceleraion is based on Galileo s idea ha v / is he mos useful concep in he descripion of velociy changes in falling objecs. Whenever a paricle s velociy changes, we define i as having an acceleraion. The average acceleraion, a, over an inerval is defined as a v v v (definiion of D average acceleraion). (-7) When he paricle moves along a line (ha is, an x axis in one-dimensional moion), a (v x v x ) ( ) î. I is imporan o noe ha an objec is acceleraed even if all ha changes is only he direcion of is velociy and no is speed. Direcional changes are imporan as well. Insananeous Acceleraion If we wan o deermine how velociy changes during an insan of ime, we need o define insananeous acceleraion (or simply acceleraion) in a way ha is similar o he way we defined insananeous velociy a lim v dv d (definiion of D insananeous acceleraion). (-8) In he language of calculus, he ACCELERATION of a paricle a any insan is he rae a which is velociy is changing a ha insan. Using his definiion, we can deermine he acceleraion by aking a ime derivaive of he velociy, v. Furhermore, since velociy of an objec moving along a line is he derivaive of he posiion, v, wih respec o ime, we can wrie a dv d d d dx d d x d (D insananeous acceleraion). (-9) Equaion -9 ells us ha he insananeous acceleraion of a paricle a any insan is equal o he second derivaive of is posiion, x, wih respec o ime. Noe ha if he objec is moving along an x axis, hen is acceleraion can be expressed in erms of he x-componen of is acceleraion and he uni vecor î along he x axis as a a x î dv x d î so a x dv x d. Figure -c shows a plo of he x-componen of acceleraion of an elevaor cab. Compare he graph of he x-componen of acceleraion as a funcion of ime (a x vs. ) wih he graph of he x-componen of velociy as a funcion of ime (v x vs. ) in par b. Each poin on he a x vs. graph is he derivaive (slope or angen) of he corresponding poin on he v x vs. graph. When v x is consan (a eiher or 4 m/s), is ime derivaive is zero and hence so is he acceleraion. When he cab firs begins o move, he v x vs. graph has a posiive derivaive (he slope is posiive), which means ha a x is posiive. When he cab slows o a sop, he derivaive or slope of he v x vs. graph is negaive; ha is, a x is negaive. Nex compare he slopes of he v x vs. graphs during he

15 Describing Velociy Change 39 x x axis Posiion (m) Velociy (m/s) Acceleraion (m/s ) 5 5 x() x x = 4. m a = 3. s 5 a b Time (s) (a) Slope v of x() 4 3 a a a b 3 Slope of v() b 3 x = 4 m a = 8. s a() 4 5 (c) 6 v() Time (s) 7 (b) Posiive acceleraion 7 c 8 c 8 Negaive Acceleraion c 9 d d d 9 FIGURE - (a) The x vs. graph for an elevaor cab ha moves upward along an x axis. (b) The v x vs. graph for he cab. Noe ha i is he derivaive of he x vs. graph (v x dx/d). (c) The a x vs. graph for he cab. I is he derivaive of he v x vs. graph (a x dv x /d). The sick figures along he boom sugges imes ha a passenger migh feel ligh and long as he elevaor acceleraes downward or heavy and squashed as he elevaor acceleraes upward. wo acceleraion periods. The slope associaed wih he cab s sopping is seeper, because he cab sops in half he ime i ook o ge up o speed. The seeper slope means ha he magniude of he sopping acceleraion is larger han ha of he acceleraion as he car is speeding up, as indicaed in Fig. -c. Acceleraion has boh a magniude and a direcion and so i is a vecor quaniy. The algebraic sign of is componen a x represens he direcion of velociy change along he chosen v x axis. When acceleraion and velociy are in he same direcion (have he same sign) he objec will speed up. If acceleraion and velociy are in opposie direcions (and have opposie signs) he objec will slow down. I is imporan o realize ha speeding up is no always associaed wih an acceleraion ha is posiive. Likewise, slowing down is no always associaed wih an acceleraion ha is negaive. The relaive direcions of an objec s velociy and acceleraion deermine wheher he objec will speed up or slow down. Since acceleraion is defined as any change in velociy over ime, whenever an objec moving in a sraigh line has an acceleraion i is eiher speeding up, slowing down, or urning around. Beware! In lisening o common everyday language, you will probably hear he word acceleraion used only o describe speeding up and he word deceleraion o mean slowing down. I s bes in sudying physics o use he more formal definiion of acceleraion as a vecor quaniy ha describes boh he magniude

16 4 CHAPTER Moion Along a Sraigh Line and direcion of any ype of velociy change. In shor, an objec is acceleraing when i is slowing down as well as when i is speeding up. We sugges avoiding he use of he erm deceleraion while rying o learn he formal language of physics. The fundamenal uni of acceleraion mus be a velociy (displacemen/ime) divided by a ime, which urns ou o be displacemen divided by ime squared. Displacemen is measured in meers and ime in seconds in he SI sysem described in Chaper. Thus, he official uni of acceleraion is m/s. You may encouner oher unis. For example, large acceleraions are ofen expressed in erms of g unis where g is direcly relaed o he magniude of he acceleraion of a falling objec near he Earh s surface. A g uni is given by g 9.8 m s. (-) On a roller coaser, you have brief acceleraions up o 3g, which, in sandard SI unis, is (3)(9.8 m/s ) or abou 9 m/s. A more exreme example is shown in he phoographs of Fig. -3, which were aken while a rocke sled was rapidly acceleraed along a rack and hen rapidly braked o a sop. FIGURE -3 Colonel J.P. Sapp in a rocke sled as i is brough up o high speed (acceleraion ou of he page) and hen very rapidly braked (acceleraion ino he page). READING EXERCISE -8 A ca moves along an x axis. Wha is he sign of is acceleraion if i is moving (a) in he posiive direcion wih increasing speed, (b) in he posiive direcion wih decreasing speed, (c) in he negaive direcion wih increasing speed, and (d) in he negaive direcion wih decreasing speed? TOUCHSTONE EXAMPLE -3 Posiion and Moion A paricle s posiion on he x axis of Fig. - is given by wih x in meers and in seconds. x 4 m (7 m s) ( m s 3 ) 3, (a) Find he paricle s velociy funcion v x () and acceleraion funcion a x (). SOLUTION One Key Idea is ha o ge he velociy funcion v x (), we differeniae he posiion funcion x() wih respec o ime. Here we find v x (7 m s) 3 ( m s 3 ) (7 m s) (3 m s 3 ) wih v x in meers per second. Anoher Key Idea is ha o ge he acceleraion funcion a x (), we differeniae he velociy funcion v x () wih respec o ime. This gives us

17 Consan Acceleraion A Special Case 4 wih a x in meers per second squared. (b) Is here ever a ime when v x? SOLUTION Seing v x () yields which has he soluion a x 3 ( m s 3 ) (6 m s 3 ), (7 m s) (3 m s 3 ), 3 s. Thus, he velociy is zero boh 3 s before and 3 s afer he clock reads. (c) Describe he paricle s moion for. SOLUTION The Key Idea is o examine he expressions for x(), v x (), and a x (). A, he paricle is a x() 4 m and is moving wih a velociy of v x () 7 m/s ha is, in he negaive direcion of he x axis. Is acceleraion is a x (), because jus hen he paricle s velociy is no changing. For 3 s, he paricle sill has a negaive velociy, so i coninues o move in he negaive direcion. However, is acceleraion is no longer bu is increasing and posiive. Because he signs of he velociy and he acceleraion are opposie, he paricle mus be slowing. Indeed, we already know ha i urns around a 3 s. Jus hen he paricle is as far o he lef of he origin in Fig. - as i will ever ge. Subsiuing 3 s ino he expression for x(), we find ha he paricle s posiion jus hen is x 5 m. Is acceleraion is sill posiive. For 3 s, he paricle moves o he righ on he axis. Is acceleraion remains posiive and grows progressively larger in magniude. The velociy is now posiive, and i oo grows progressively larger in magniude. -5 Consan Acceleraion A Special Case If you wach a small seel ball bobbing up and down a he end of a spring, you will see he velociy changing coninuously. Bu insead of eiher increasing or decreasing a a seady rae, we have a very nonuniform paern of moion. Firs he ball speeds up and slows down moving in one direcion, hen i urns around and speeds up and hen slows down in he oher direcion, and so on. This is an example of a nonconsan acceleraion ha keeps changing in ime. Alhough here are many examples of nonconsan acceleraions, we also observe a surprising number of examples of consan or nearly consan acceleraion. As we already discussed, Galileo discovered ha if we choose o define acceleraion in erms of he raio v/, hen a falling ball or a ball ossed ino he air ha slows down, urns around, and speeds up again is always increasing is velociy in a downward direcion a he same rae provided he ball is moving slowly enough ha air drag is negligible. There are many oher common moions ha involve consan acceleraions. Suppose you measure he imes and corresponding posiions for an objec ha you suspec has a consan acceleraion. If you hen calculae he velociies and acceleraions of he objec and make graphs of hem, he graphs will resemble hose in Fig. -4. Some examples of moions ha yield similar graphs o hose shown in Fig. -4 include a car ha you accelerae as soon as a raffic ligh urns green; he same car when you apply is brakes seadily o bring i o a smooh sop; an airplane when firs aking off or when compleing a smooh landing; or a dolphin ha speeds up suddenly afer being sarled. Derivaion of he Kinemaic Equaions Because consan acceleraions are common, i is useful o derive a special se of kinemaic equaions o describe he moion of any objec ha is moving along a line wih a consan acceleraion. We can use he definiions of acceleraion and velociy and an assumpion abou average velociy o derive he kinemaic equaions. These equaions allow us o use known values of he vecor componens describing posiions, velociies, and acceleraions, along wih ime inervals o predic he moions of consanly acceleraed objecs. v x () x() x x Posiion componen vs. ime x x() Slope varies Velociy componen vs. ime v (v v(), ) v a x () v (x, ) (v, ) (a) Slope = a x (b) Acceleraion componen vs. ime a a x = consan Slope = (c) (x, ) FIGURE -4 (a) The posiion componen x() of a paricle moving wih consan acceleraion. (b) Is velociy componen v x (), given a each poin by he slope of he curve in (a). (c) Is (consan) componen of acceleraion, a x, equal o he (consan) slope of v x ().

18 4 CHAPTER Moion Along a Sraigh Line Le s sar he derivaion by noing ha when he acceleraion is consan, he average and insananeous acceleraions are equal. As usual we place our x axis along he line of he moion. We can now use vecor noaion o wrie so ha a a x î a, a (v x v x ) î, (-) where a x is he componen of acceleraion along he line of moion of he objec. We can use he definiion of average acceleraion (Eq. -7) o express he acceleraion componen a x in erms of he objec s velociy componens along he line of moion, where v x and v x are he objec s velociy componens along he line of moion, a x (v x v x ). (-) This expression allows us o derive he kinemaic equaions in erms of he vecor componens needed o consruc he acual one-dimensional velociy and acceleraion vecors. The subscrips and in mos of he equaions in his chaper, including Eq. -, refer o iniial and final imes, posiions, and velociies. If we solve Eq. - for v x, hen he x-componen of velociy a ime is v x v x a x ( ) v x a x (primary kinemaic [a x consan] equaion), or v x a. (-3) This equaion is he firs of wo primary equaions ha we will derive for use in analyzing moions involving consan acceleraion. Before we move on, we should hink carefully abou wha he expression represens in his equaion I represens he ime inerval in which we are racking he moion. In a manner similar o wha we have done above, we can rewrie Eq. -4, he expression for he average velociy along he x axis, v v x î x î (x x ) î. Hence, he x-componen of he average velociy is given by v x <v x > v <v x > = v x + v x Solving for x gives v x (x x ) ( ). x x v x ( ). (-4) v x < > FIGURE -5 When he acceleraion is consan, hen we assume (wihou rigorous proof) ha he average velociy componen in a ime inerval is he average of he velociy componens a he beginning and end of he inerval. In his equaion x is he x-componen of he posiion of he paricle a and v x is he componen along he x axis of average velociy beween and a laer ime. Noe ha unless he velociy is consan, he average velociy componen along he x axis, v x, is no equal o he insananeous velociy componen, v x. However, we do have a plausible alernaive for expressing he average velociy componen in he special case when he acceleraion is consan. Figure -5 depics he fac ha velociy increases in a linear fashion over ime for a consan acceleraion. I seems reasonable o assume ha he componen along he x axis of he average velociy over any ime inerval is he average of he componens for he in-

19 Consan Acceleraion A Special Case 43 sananeous velociy a he beginning of he inerval, v x, and he insananeous velociy componen a he end of he inerval, v x. So we expec ha when a velociy increases linearly, he average velociy componen over a given ime inerval will be v x v x v x. (-5) Using Eq. -3, we can subsiue v x a x ( ) for v x o ge v x v x v x a x ( ) v x a x( ). (-6) Finally, subsiuing his equaion ino Eq. -4 yields x x v x ( ) a x( ) (primary kinemaic [a x consan] equaion),(-7) or x v x a x This is our second primary equaion describing moion wih consan acceleraion. Figures -4a and -6 show plos of Eq. -7. x(m) x =.8m (, x ) =.4s 3 (s) FIGURE -6 A fan on a low-fricion car is urned on a abou. s bu isn hrusing fully unil =.4 s. Daa for he graph were colleced wih a compuer daa acquisiion sysem oufied wih an ulrasonic moion deecor. Beween.4 s and abou 5.4 s he car appears o be undergoing a consan acceleraion as i slows down, urns around, and speeds up again. Thus, he consan acceleraion kinemaic equaions can be used o describe is moion bu only during moion wihin ha ime inerval. Thus, we can se o.4 s and x o.8 m. These wo equaions are very useful in he calculaion of unknown quaniies ha can be used o characerize consanly acceleraed moion. There are five or six quaniies conained in our primary equaions (Eqs. -3 and -7). The simples kinemaic calculaions involve siuaions in which all bu one of he quaniies is known in one of he primary equaions. In more complex siuaions, boh equaions are needed. Typically for a complex siuaion, we need o calculae more han one unknown. To do his, we find he firs unknown using one of he primary equaions and use he resul in he oher equaions o find he second unknown. This mehod is illusraed in he nex secion and in Touchsone Examples -4 and -6. The primary equaions above, v x v x a x ( ) v x a x (Eq. -3), and x x v x ( ) a x ( ) (Eq. -7), are derived direcly from he definiions of velociy and acceleraion, wih he condiion ha he acceleraion is consan. These wo equaions can be combined in hree ways o yield hree addiional equaions. For example, solving for v x in v x v x a x ( ) and subsiuing he resul ino x x v x ( ) a x ( ) gives us v x v x a x (x x ). We recommend ha you learn he wo primary equaions and use hem o derive oher equaions as needed. Then you will no need o remember so much. Table - liss our wo primary equaions. Noe ha a really nice alernaive o using he wo

20 44 CHAPTER Moion Along a Sraigh Line T ABLE - Equaions of Moion wih Consan Acceleraion Equaion Number -3-7 Primary Vecor Componen Equaion* v x v x a x ( ) x x v x ( ) a x ( ) *A reminder In cases where he iniial ime is chosen o be zero i is imporan o remember ha whenever he erm ( ) is replaced by jus, hen acually represens a ime inerval of over which he moion of ineres akes place. equaions in Table - is o use he firs of he equaions (Eq. -3) along wih he expression for he average velociy componen in Eq. -5, v x x v x v x (an alernaive primary equaion), o derive all he oher needed equaions. The derivaions of he kinemaic equaions ha we presen here are no rigorous mahemaical proofs bu raher wha we call plausibiliy argumens. However, we know from he applicaion of he kinemaic equaions o consanly acceleraed moions ha hey do adequaely describe hese moions. x axis x = 48 m Analyzing he Niagara Falls Plunge A he beginning of his chaper we asked quesions abou he moion of he seel chamber holding Dave Munday as he plunged ino he waer afer falling 48 m from he op of Niagara Falls. How long did he fall ake? Tha is,? How fas was he chamber moving when i hi he waer? (Wha is v?) As you will learn in Chaper 3, if no significan air drag is presen, objecs near he surface of he Earh fall a a consan acceleraion of magniude a x 9.8 m/s. Thus, he kinemaic equaions can be used o calculae he ime of fall and he impac speed. Le s sar by defining our coordinae sysem. We will ake he x axis o be a verical or up down axis ha is aligned wih he downward pah of he seel chamber. We place he origin a he boom of he falls and define up o be posiive as shown in Fig. -7. (Laer when considering moions in wo and hree dimensions, we will ofen denoe verical axes as y axes and horizonal axes as x axes, bu hese changes in symbols will no affec he resuls of calculaions.) We know ha he value of he verical displacemen is given by x x ( m) ( 48 m) 48 m x = m Origin FIGURE -7 A coordinae sysem chosen o analyze he fall of a seel chamber holding a man who falls 48 m from he op o he boom of Niagara Falls. and ha he velociy is geing larger in magniude in he downward (negaive direcion). Since he velociy is downward and he objec is speeding up, he verical acceleraion is also downward (in he negaive direcion). Is componen along he axis of moion is given by a x 9.8 m/s. Finally, we assume ha Dave Munday s capsule dropped from res, so v x m/s. Thus we can find he ime of fall ( ) using Eq. -7. Solving his equaion for he ime elapsed during he fall ( ) when he iniial velociy v x is zero gives (x x ) ( 48 m) a x 3.3 s 3. s. 9.8 m/s This is a fas rip indeed!

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