Motion Along a Straight Line

Transcription

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

12 36 CHAPTER Moion Along a Sraigh Line momen in ime. If he graph is a curve raher han a sraigh line, he slope a a poin is acually he angen o he line a ha poin. Alernaively, he second par of he expression, shown in Eq. -6, v dx d, indicaes ha, if we can approximae he relaionship beween and as a coninuous mahemaical funcion such as x (3. m/s ), we can also find he objec s insananeous velociy by aking a derivaive wih respec o ime of he objec s posiion x. When x varies coninuously as ime marches on, we ofen denoe x as a posiion funcion x () o remind us ha i varies wih ime. Insananeous speed, which is ypically called simply speed, is jus he magniude of he insananeous velociy vecor, v. Speed is a scalar quaniy consising of he velociy value ha has been sripped of any indicaion of he direcion he objec is moving, eiher in words or via an algebraic sign. A velociy of ( 5 m/s)î and one of ( 5 m/s)î boh have an associaed speed of 5 m/s. x READING EXERCISE -4 Suppose ha you drive mi due eas o a sore. You suddenly realize ha you forgo your money. You urn around and drive he mi due wes back o your home and hen reurn o he sore. The oal rip ook 3 min. (a) Wha is your average velociy for he enire rip? (Se up a coordinae sysem and express your resul in vecor noaion.) (b) Wha was your average speed for he enire rip? (c) Discuss why you obained differen values for average velociy and average speed. READING EXERCISE -5 Suppose ha you are driving and look down a your speedomeer. Wha does he speedomeer ell you average speed, insananeous speed, average velociy, insananeous velociy or somehing else? Explain. READING EXERCISE -6 The following equaions give he posiion componen, x(), along he x axis of a paricle s moion in four siuaions (in each equaion, x is in meers, is in seconds, and ) () x (3 m/s) ( m) ; () x ( 4 m/s ) ( m); (3) x ( 4 m/s ) ; and (4) x m. (a) In which siuaions is he velociy v of he paricle consan? (b) In which is he vecor v poining in he negaive x direcion? READING EXERCISE -7 In Touchsone Example -, suppose ha righ afer refueling he ruck you drive back o x a 35 km/h. Wha is he average velociy for your enire rip? TOUCHSTONE EXAMPLE - Ou of Gas You drive a bea-up pickup ruck along a sraigh road for 8.4 km a 7 km/h, a which poin he ruck runs ou of gasoline and sops. Over he nex 3 min, you walk anoher. km farher along he road o a gasoline saion. (a) Wha is your overall displacemen from he beginning of your drive o your arrival a he saion? SOLUTION Assume, for convenience, ha you move in he posiive direcion along an x axis, from a firs posiion of x o a second posiion of x a he saion. Tha second posiion mus be a x 8.4 km. km.4 km. Then he Key Idea here is ha your displacemen x along he x axis is he second posiion minus he firs posiion. From Eq. -, we have x x x.4 km.4 km Thus, your overall displacemen is.4 km in he posiive direcion of he x axis. (b) Wha is he ime inerval from he beginning of your drive o your arrival a he saion?

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!

21 Consan Acceleraion A Special Case 45 Nex we can use he ime inerval of he fall in he oher primary kinemaic equaion, Eq. -3, o find he velociy a impac. This gives a componen of impac velociy a he end of he fall of v x v x a x ( ) m/s ( 9.8 m/s )(3.3 s) 3 m/s. The minus sign indicaes ha he impac velociy componen is negaive and is, herefore, in he downward direcion. In vecor noaion, he velociy v v x î is hus v ( 3 m/s)î. Noe ha his is a speed of abou 69 mi/hr. Since he ime inerval was pu ino he calculaion of velociy of impac as an inermediae value, we reained an exra significan figure o use in he nex calculaion. READING EXERCISE -9 The following equaions give he x-componen of posiion x() of a paricle in meers (denoed m) as a funcion of ime in seconds for four siuaions () x (3 m/s) 4 m; () x ( 5 m/s 3 ) 3 (4 m/s) 6 m; (3) x ( m/s ) (4 m/s); (4) x (5 m/s ) 3 m. To which of hese siuaions do he equaions of Table - apply? Explain. TOUCHSTONE EXAMPLE -4 Slowing Down Spoing a police car, you brake your Porsche from a speed of km/h o a speed of 8. km/h during a displacemen of 88. m, a a consan acceleraion. (a) Wha is ha acceleraion? SOLUTION Assume ha he moion is along he posiive direcion of an x axis. For simpliciy, le us ake he beginning of he braking o be a ime, a posiion x. The Key Idea here is ha, wih he acceleraion consan, we can relae he car s acceleraion o is velociy and displacemen via he basic consan acceleraion equaions (Eqs. -3 and -7). The iniial velociy is v x km/h 7.78 m/s, he displacemen is x x 88., and he velociy a he end of ha displacemen is v x 8. km/h. m/s. However, we do no know he acceleraion a x and ime, which appear in boh basic equaions, so we mus solve hose equaions simulaneously. To eliminae he unknown, we use Eq. -3 o wrie (b) How much ime is required for he given decrease in speed? SOLUTION Now ha we know a x, we can use Eq. -8 o solve for v x v x a x. m/s 7.78 m/s.58 m/s 3.5 s. If you are iniially speeding and rying o slow o he speed limi, here is pleny of ime for he police officer o measure your excess speed. You can use one of he alernae equaions for moion wih a consan acceleraion, Eq. -5, o check his resul. The Key Idea here is ha he disance raveled is jus he produc of he average velociy and he elapsed ime, when he acceleraion is consan. The Porsche raveled 88. m while i slowed from km/h down o 8 km/h. Thus is average velociy while i covered he 88. m was v x v x a x, and hen we subsiue his expression ino Eq. -7 o wrie x x v x v x v x a x v x v x a x. (-8) v x ( km/h 8 km/h) 9 km h so he ime i ook o slow down was jus m km h 5. m/s, 36 s Solving for a x and subsiuing known daa hen yields a x v x v x (x x ) (. m/s) (7.78 m/s) (88. m).58 m/s. x x v x which sill isn enough ime o avoid ha speeding icke! 88. m 5. m/s 3.5 s,

22 46 CHAPTER Moion Along a Sraigh Line TOUCHSTONE EXAMPLE -5 Moion Daa Suppose ha you gave a box siing on a carpeed floor a push and hen recorded is posiion hree imes per second as i slid o a sop. The able gives he resuls of such a measuremen. Le s analyze he posiion vs. ime daa for he box sliding on he carpe and use curve fiing and calculus o obain he velociy measuremens. We will use Excel spreadshee sofware o perform our analysis, bu oher compuer- or calculaor-based fiing or modeling sofware can be used. Box Sliding on Carpe [s] x[m] (a) Draw a graph of he x vs. daa and discuss wheher he relaionship appears o be linear or no. SOLUTION The Key Idea here is ha he relaionship beween wo variables is linear if he graph of he daa poins lie more or less along a sraigh line. There are many ways o graph he daa for examinaion by hand, wih a graphing calculaor, wih a spreadshee graphing rouine, or wih oher graphing sofware such as Daa Sudio (available from PASCO scienific) or Graphical Analysis (available from Vernier Sofware and Technology). The graph in Fig. -8 ha shows a curve and so he relaionship beween posiion, x, as a funcion of ime is no linear. x(m) (s) FIGURE -8 Soluion o Touchsone Example -5(a). A graph of posiion versus ime for a box sliding across a carpe. (b) Draw a moion diagram of he box as i comes o res on he carpe. SOLUTION The Key Idea here is o use he daa o skech he posiion along a line a equal ime inervals. In Fig. -9, he black circles represen he locaion of he rear of he box a inervals of /3 of a second. =. s.5.55 =.66 s =.33 s and so on x FIGURE -9 Soluion o Touchsone Example -5(b). A moion diagram for a box sliding across a carpe. (c) Is he acceleraion consan? If so, wha is is componen along he x axis? SOLUTION The Key Idea here is o explore wheher or no he relaionship beween posiion and ime of he box as i slides o a sop can be described wih a quadraic (parabolic) funcion of ime as described in Eq. -7. This can be done by enering he daa ha are given ino a spreadshee or graphing calculaor and eiher doing a quadraic model or a fi o he daa. The oucome of a quadraic model is shown in Fig. -. The x-model column conains he resuls of calculaing x using he equaion x x v x ( ) a x ( ) for each of he imes in he firs column using he iniial posiion, velociy and acceleraion daa shown in he boxes. The line shows he model daa. If he kinemaic equaion fis he daa, hen we can conclude ha he acceleraion componen is a consan given by a x 6.6 m/s. Thus he acceleraion is in he negaive y direcion. a 6.7 (m/s ) v.6 (m/s) x.537 (m) Box Sliding on Carpe (s) x-daa (m) x-model (m) Box sliding on carpe x-daa (m) x-model (m).. (s) FIGURE - Soluion o Touchsone Example (c). Daa and a graph of posiion as funcion of ime for a box sliding over carpe. Acual daa is compared o a model of wha is expeced from Eq. - 7 (assumed consan acceleraion). The value of acceleraion which produced he bes mach beween he model and acual daa is 6.6 m/s.

23 Problems 47 TOUCHSTONE EXAMPLE -6 Disance Covered Figure -b shows a graph of a person riding on a low-fricion car being pulled along wih a bungee cord as shown in Fig. -a. Use informaion from he wo graphs and he kinemaic equaions o deermine approximaely how far he suden moved in he ime inerval beween. s and. s. SOLUTION The Key Idea is ha he iniial velociy can be deermined from he velociy vs. ime graph on he lef and he acceleraion during he ime inerval can be deermined from he acceleraion vs. ime graph on he righ (or by finding he slope of he velociy vs. ime graph on he lef during he ime inerval). Noe ha he velociy a. s is given by v x.4 m/s. The acceleraion during he ime inerval of ineres is given by a x.4 m/s. Since he acceleraion is consan over he ime inerval of ineres, we can use he daa in Eq. -7 o ge x x v x ( ) a x( ) (.4 m/s)(. s. s) (.4 m/s )(. s. s).5 m. Half a meer is no very far! Moion deecor Bungee cord Velociy (m/s) A no pulling B cord is sreching C cord is consan lengh Time (seconds) (a) Acceleraion (m/s ) (b)..8 C cord is.6 consan lengh A no.4 pulling B cord is. sreching Time (seconds) FIGURE - (a) A person riding on a low-fricion car is pulled by anoher person who exers a consan force along a sraigh line by keeping he lengh of a bungee cord consan. (b) These graphs show velociy and acceleraion componens vs. ime for a rider on a car. For he firs.5 s (region A) he car is a res. Beween.5 s and. s (region B) he cord is beginning o srech. Beween. s and. s (region C) a consan force is acing and he acceleraion is also consan. Problems In several of he problems ha follow you are asked o graph posiion, velociy, and acceleraion versus ime. Usually a skech will suffice, appropriaely labeled and wih sraigh and curved porions apparen. If you have a compuer or graphing calculaor, you migh use i o produce he graph. SEC. -3 VELOCITY AND SPEED. Fasball If a baseball picher hrows a fasball a a horizonal speed of 6 km/h, how long does he ball ake o reach home plae 8.4 m away?. Fases Bicycle A world speed record for bicycles was se in 99 by Chris Huber riding Cheeah, a high-ech bicycle buil by hree mechanical engineering graduaes. The record (average) speed was.6 km/h hrough a measured lengh of. m on a deser road. A he end of he run, Huber commened, Cogio ergo zoom! (I hink, herefore I go fas!) Wha was Huber s elapsed ime hrough he. m? 3. Auo Trip An auomobile ravels on a sraigh road for 4 km a 3 km/h. I hen coninues in he same direcion for anoher 4 km a 6 km/h. (a) Wha is he average velociy of he car during his 8 km rip? (Assume ha i moves in he posiive x direcion.) (b) Wha is he average speed? (c) Graph x vs. and indicae how he average velociy is found on he graph. 4. Radar Avoidance A op-gun pilo, pracicing radar avoidance maneuvers, is manually flying horizonally a 3 km/h, jus 35 m above he level ground. Suddenly, he plane encouners errain ha slopes genly upward a 4.3, an amoun difficul o deec visually (Fig. -). How much ime does he pilo have o make a correcion o avoid flying ino he ground? 4.3 FIGURE - Problem m

24 48 CHAPTER Moion Along a Sraigh Line 5. On Inersae You drive on Inersae from San Anonio o Houson, half he ime a 55 km/h and he oher half a 9 km/h. On he way back you ravel half he disance a 55 km/h and he oher half a 9 km/h. Wha is your average speed (a) from San Anonio o Houson, (b) from Houson back o San Anonio, and (c) for he enire rip? (d) Wha is your average velociy for he enire rip? (e) Skech x vs. for (a), assuming he moion is all in he posiive x direcion. Indicae how he average velociy can be found on he skech. 6. Walk Then Run Compue your average velociy in he following wo cases (a) You walk 73. m a a speed of. m/s and hen run 73. m a a speed of 3.5 m/s along a sraigh rack. (b) You walk for. min a a speed of. m/s and hen run for. min a 3.5 m/s along a sraigh rack. (c) Graph x vs. for boh cases and indicae how he average velociy is found on he graph. 7. Posiion and Time The posiion of an objec moving along an x axis is given by x (3 m/s) (4 m/s ) ( m/s 3 ) 3, where x is in meers and in seconds. (a) Wha is he posiion of he objec a,, 3, and 4 s? (b) Wha is he objec s displacemen beween and 4 4 s? (c) Wha is is average velociy for he ime inerval from s o 4 4 s? (d) Graph x vs. for 4 4 s and indicae how he answer for (c) can be found on he graph. 8. Two Trains and a Bird Two rains, each having a speed of 3 km/h, are headed a each oher on he same sraigh rack. A bird ha can fly 6 km/h flies off he fron of one rain when hey are 6 km apar and heads direcly for he oher rain. On reaching he oher rain i flies direcly back o he firs rain, and so forh. (We have no idea why a bird would behave in his way.) Wha is he oal disance he bird ravels? 9. Two Winners On wo differen racks, he winners of he kilomeer race ran heir races in min, 7.95 s and min, 8.5 s. In order o conclude ha he runner wih he shorer ime was indeed faser, how much longer can he oher rack be in acual lengh?. Scampering Armadillo The x graph in Fig. -3 is for an armadillo ha scampers lef (negaive direcion of x) and righ along an x axis. (a) When, if ever, is he animal o (s) he lef of he origin on he axis? When, if ever, is is velociy (b) negaive, (c) posiive, or (d) zero?. Posiion and Time (a) If a paricle s posiion is given by FIGURE -3 Problem. x 4m ( m/s) (3 m/s ) (where is in seconds and x is in meers), wha is is velociy a s? (b) Is i moving in he posiive or negaive direcion of x jus hen? (c) Wha is is speed jus hen? (d) Is he speed larger or smaller a laer imes? (Try answering he nex wo quesions wihou furher calculaion.) (e) Is here ever an insan when he velociy is zero? (f) Is here a ime afer 3 3 s when he paricle is moving in he negaive direcion of x?. Paricle Posiion and Time The posiion of a paricle moving along he x axis is given in cenimeers by x 9.75m (.5 m/s 3 ) 3 where is in seconds. Calculae (a) he average velociy during he ime inerval. s o 3. s; (b) he insananeous velociy a. s; (c) he insananeous velociy a 3. s; (d) he insananeous velociy a.5 s; and (e) he insananeous velociy when he paricle is midway beween is posiions a. s and 3. s (f) Graph x vs. and indicae your answers graphically. 3. Velociy Time Graph How far does he runner whose velociy ime graph is shown in Fig. -4 ravel in 6 s? FIGURE -4 Problem 3 Velociy (m/s) Time (s) SEC. -4 DESCRIBING VELOCITY CHANGE Various Moions Skech a graph ha is a possible descripion of posiion as a funcion of ime for a paricle ha moves along he x axis and, a s, has (a) zero velociy and posiive acceleraion; (b) zero velociy and negaive acceleraion; (c) negaive velociy and posiive acceleraion; (d) negaive velociy and negaive acceleraion. (e) For which of hese siuaions is he speed of he paricle increasing a s? 5. Two Similar Expressions Wha do he quaniies (a) (dx/d) and (b) d x/d represen? (c) Wha are heir SI unis? 6. Frighened Osrich A frighened osrich moves in a sraigh line v wih velociy described by he velociy ime graph of Fig. -5. Skech acceleraion vs. ime. 7. Speed Then and Now A paricle had a speed of 8 m/s a a 4 6 cerain ime, and.4 s laer is speed was 3 m/s in he opposie Time (s) direcion. Wha were he magniude and direcion of he average acceleraion of he paricle during his FIGURE -5 Problem 6..4 s inerval? 8. Sand Then Walk From o 5 5. min, a man sands sill, and from 5 5. min o. min, he walks briskly in a sraigh line a a consan speed of. m/s. Wha are (a) his average velociy v and (b) his average acceleraion a in he ime inerval. min o 8. min? Wha are (c) v and a in he ime inerval 3. min o 9. min? (e) Skech x vs. and v vs., and indicae how he answers o (a) hrough (d) can be obained from he graphs. 9. Paricle Posiion and Time The posiion of a paricle moving along he x axis depends on he ime according o he equaion x c b 3, where x is in meers and in seconds. (a) Wha unis mus c and b have? Le heir numerical values be 3. and.. respecively. (b) A wha ime does he paricle reach is maximum posiive x posiion? From. s o 4 4. s, (c) wha disance does he paricle move and (d) wha is is displacemen? A.,., 3., and 4. s, wha are (e) is velociies and (f) is acceleraions? SEC. -5 CONSTANT ACCELERATION A SPECIAL CASE v. Driver and Rider An auomobile driver on a sraigh road increases he speed a a consan rae from 5 km/h o 55 km/h in.5 min. A bicycle rider on a sraigh road speeds up a a consan rae from res o 3 km/h in.5 min. Calculae heir acceleraions.. Sopping a Muon A muon (an elemenary paricle) moving in a sraigh line eners a region wih a speed of 5. 6 m/s and Velociy (m/s)

25 Problems 49 hen is slowed a he rae of.5 4 m/s. (a) How far does he muon ake o sop? (b) Graph x vs. and v vs. for he muon.. Ralesnake Sriking The head of a ralesnake can accelerae a 5 m/s in sriking a vicim. If a car could do as well, how long would i ake o reach a speed of km/h from res? 3. Acceleraing an Elecron An elecron has a consan acceleraion of 3. m/s î. A a cerain insan is velociy is 9.6 m/sî. Wha is is velociy (a).5 s earlier and (b).5 s laer? 4. Speeding Bulle The speed of a bulle is measured o be 64 m/s as he bulle emerges from a barrel of lengh. m. Assuming consan acceleraion, find he ime ha he bulle spends in he barrel afer i is fired. 5. Comforable Acceleraion Suppose a rocke ship in deep space moves wih consan acceleraion equal o 9.8 m/s, which gives he illusion of normal graviy during he fligh. (a) If i sars from res, how long will i ake o acquire a speed one-enh ha of ligh, which ravels a 3. 8 m/s? (b) How far will i ravel in so doing? 6. Taking Off A jumbo je mus reach a speed of 36 km/h on he runway for akeoff. Wha is he leas consan acceleraion needed for akeoff from a.8 km runway? 7. Even Faser Elecrons An elecron wih iniial velociy v.5 5 m/s eners a region. cm long where i is elecrically acceleraed (Fig. -6). I emerges wih velociy v m/s. Wha is is acceleraion, assumed consan? (Such a process occurs in convenional elevision ses.) 8. Sopping Col. Sapp A world s land speed record was se by Colonel John P. Sapp when in March 954 he rode a rocke-propelled sled ha moved along a rack a km/h. He and he sled were brough o a sop in.4 s. (See Fig. -3) In g unis, wha acceleraion did he experience while sopping? 9. Speed Trap The brakes on your auomobile are capable of slowing down your car a a rae of 5. m/s. (a) If you are going 37 km/h and suddenly see a sae rooper, wha is he minimum ime in which you can ge your car under he 9 km/h speed limi? The answer reveals he fuiliy of braking o keep your high speed from being deeced wih a radar or laser gun.) (b) Graph x vs. and v vs. for such a deceleraion. 3. Judging Acceleraion Figure -7 x (m) depics he moion of a paricle moving along an x axis wih a consan acceleraion. Wha are he magniude and di- 6 recion of he paricle s acceleraion? 4 3. Hiing a Wall A car raveling 56. km/h is 4. m from a barrier when he driver slams on he brakes. The car his he barrier. s laer. (a) Wha is he car s consan acceleraion before impac? (b) How fas is he car raveling a impac? Nonacceleraing region Pah of elecron Source of high volage Acceleraing region. cm FIGURE -6 Problem 7. FIGURE -7 Problem 3. (s) 3. Red and Green Trains A red rain raveling a 7 km/h and a green rain raveling a 44 km/h are headed oward one anoher along a sraigh, level rack. When hey are 95 m apar, each engineer sees he oher s rain and applies he brakes. The brakes slow each rain a he rae of. m/s. Is here a collision? If so, wha is he speed of each rain a impac? If no, wha is he separaion beween he rains when hey sop? 33. Beween Two Poins A car moving wih consan acceleraion covered he disance beween wo poins 6. m apar in 6. s. Is speed as i passes he second poin was 5. m/s. (a) Wha was he speed a he firs poin? (b) Wha was he acceleraion? (c) A wha prior disance from he firs poin was he car a res? (d) Graph x vs. and v vs. for he car from res ( ). 34. Chasing a Truck A he insan he raffic ligh urns green, an auomobile sars wih a consan acceleraion a of. m/s. A he same insan a ruck, raveling wih a consan speed of 9.5 m/s, overakes and passes he auomobile. (a) How far beyond he raffic signal will he auomobile overake he ruck? (b) How fas will he car be raveling a ha insan? 35. Reacion Time To sop a car, firs you require a cerain reacion ime o begin braking; hen he car slows under he consan braking. Suppose ha he oal disance moved by your car during hese wo phases is 56.7 m when is iniial speed is 8.5 km/h, and 4.4 in when is iniial speed is 48.3 km/h. Wha are (a) your reacion ime and (b) he magniude of he deceleraion? 36. Avoiding a Collision When a high-speed passenger rain raveling a 6 km/h rounds a bend, he engineer is shocked o see ha a locomoive has improperly enered ono he rack from a siding and is a disance D 676 m ahead (Fig. -8). The locomoive is moving a 9. km/h. The engineer of he high-speed rain immediaely applies he brakes. (a) Wha mus be he magniude of he resuling consan acceleraion if a collision is o be jus avoided? (b) Assume ha he engineer is a x when, a, he firs spos he locomoive. Skech he x() curves represening he locomoive and, high-speed rain for he siuaions in which a collision is jus avoided and is no quie avoided. High-speed rain FIGURE -8 Problem Going Up An elevaor cab in he New York Marquis Marrio has a oal run of 9 m. Is maximum speed is 35 m/min. Is acceleraion (boh speeding up and slowing) has a magniude of. m/s. (a) How far does he cab move while acceleraing o full speed from res? (b) How long does i ake o make he nonsop 9 m run, saring and ending a res? 38. Shuffleboard Disk A shuffleboard disk is acceleraed a a consan rae from res o a speed of 6. m/s over a.8 m disance by a player using a cue. A his poin he disk loses conac wih he cue and slows a a consan rae of.5 m/s unil i sops. (a) How much D Locomoive

26 5 CHAPTER Moion Along a Sraigh Line ime elapses from when he disk begins o accelerae unil i sops? (b) Wha oal disance does he disk ravel? 39. Elecric Vehicle An elecric vehicle sars from res and acceleraes a a rae of. m/s in a sraigh line unil i reaches a speed of m/s. The vehicle hen slows a a consan rae of. m/s unil i sops. (a) How much ime elapses from sar o sop? (b) How far does he vehicle ravel from sar o sop? 4. Red Car Green Car In Fig. -9 a red car and a green car, idenical excep for he color, move oward each oher in adjacen lanes and parallel o an x axis. A ime, he red car is a x and he green car is a x m. If he red car has a consan velociy of km/h, he cars pass each oher a x 44.5 m, and if i has a consan velociy of 4 km/h, hey pass each oher a x 76.6 m. Wha are (a) he iniial velociy and (b) he acceleraion of he green car? Red car m x P Green car FIGURE -9 Problem Posiion Funcion The posiion of a paricle moving along an x axis is given by x ( m/s ) ( m/s 3 ) 3, where x is in meers and is in seconds. (a) Deermine he posiion, velociy, and acceleraion of he paricle a 3 3. s. (b) Wha is he maximum posiive coordinae reached by he paricle and a wha ime is i reached? (c) Wha is he maximum posiive velociy reached by he paricle and a wha ime is i reached? (d) Wha is he acceleraion of he paricle a he insan he paricle is no moving (oher han a )? (e) Deermine he average velociy of he paricle beween and 3 3 s. x Addiional Problems 4. Kids in he Back! An unresrained child is playing on he fron sea of a car ha is raveling in a residenial neighborhood a 35 km/h. (How many mi/h is his? Is his car going oo fas?) A small dog runs across he road and he driver applies he brakes, sopping he car quickly and missing he dog. Esimae he speed wih which he child srikes he dashboard, presuming ha he car sops before he child does so. Compare his speed wih ha of he world-record m dash, which is run in abou s. 43. The Passa GLX Tes resuls (Car & Driver, February 993, p. 48) on a Volkswagen Passa GLX show ha when he brakes are fully applied i has an average braking acceleraion of magniude 8.9 m/s. If a preoccupied driver who is moving a a speed of 4 mph looks up suddenly and sees a sop ligh 3 m in fron of him, will he have sufficien ime o sop? The weigh of he Volkswagen is 3 5 lb. 44. Velociy and Pace When we drive a car we usually describe our moion in erms of speed or velociy. A speed limi, such as 6 mi/h, is a speed. When runners or joggers describe heir moion, hey ofen do so in erms of a pace how long i akes o go a given disance. A 4-min mile (or beer, 4 minues/mile ) is an example of a pace. (a) Express he speed 6 mi/h as a pace in min/mi. (b) I walk on my readmill a a pace of 7 min/mi. Wha is my speed in mi/h? (c) If I ravel a a speed, v, given in mi/h, wha is my pace, p, given in min/mi? (Wrie an equaion ha would permi easy conversion.) 45. Spiri of America The 9 lb Spiri of America (designed o be he world s fases car) acceleraed from res o a final velociy of 756 mph in a ime of 45 s. Wha would he acceleraion have been in meers per second? Wha disance would he driver, Craig Breedlove, have covered? 46. Driving o New York You and a friend decide o drive o New York from College Park, Maryland (near Washingon, D.C.) on Saurday over he Thanksgiving break o go o a concer wih some friends who live here. You figure you have o reach he viciniy of he ciy a 5 P.M. in order o mee your friends in ime for dinner before he concer. I s abou mi from he enrance o Roue 95 o he viciniy of New York Ciy. You would like o ge on he highway abou noon and sop for a bie o ea along he way. Wha does your average velociy have o be? If you keep an approximaely consan speed (no a realisic assumpion!), wha should your speedomeer read while you are driving? 47. NASA Inernship You are working as a suden inern for he Naional Aeronauics and Space Adminisraion (NASA) and your supervisor wans you o perform an indirec calculaion of he upward velociy of he space shule relaive o he Earh s surface jus 5.5 s afer i is launched when i has an aliude of m. In order o obain daa, one of he engineers has wired a sreamlined flare o he side of he shule ha is genly released by remoe conrol afer 5.5 s. If he flare his he ground 8.5 s afer i is released, wha is he upward velociy of he flare (and hence of he shule) a he ime of is release? (Neglec any effecs of air resisance on he flare.) Noe Alhough he flare idea is ficional, he daa on a ypical shule aliude and velociy a 5.5 s are sraigh from NASA! 48. Cell Phone Figh You are arguing over a cell phone while railing an unmarked police car by 5 m; boh your car and he police car are raveling a km/h. Your argumen divers your aenion from he police car for. s (long enough for you o look a he phone and yell, I won do ha! ). A he beginning of ha. s, he police officer begins emergency braking a 5. m/s. (a) Wha is he separaion beween he wo cars when your aenion finally reurns? Suppose ha you ake anoher.4 s o realize your danger and begin braking. (b) If you oo brake a 5. m/s, wha is your speed when you hi he police car? 49. Reacion Disance When a driver brings a car o a sop by braking as hard as possible, he sopping disance can be regarded as he sum of a reacion disance, which is iniial speed muliplied by he driver s reacion ime, and a braking disance, which is he disance raveled during braking. The following able gives ypical values. (a) Wha reacion ime is he driver assumed o have? (b) Wha is he car s sopping disance if he iniial speed is 5 m/s? Iniial Reacion Braking Sopping Speed (m/s) Disance (m) Disance (m) Disance (m)

27 Addiional Problems 5 5. Tailgaing In his problem we analyze he phenomenon of ailgaing in a car on a highway a high speeds. This means raveling oo close behind he car ahead of you. Tailgaing leads o muliple car crashes when one of he cars in a line suddenly slows down. The quesion we wan o answer is How close is oo close? To answer his quesion, le s suppose you are driving on he highway a a speed of km/h (a bi more han 6 mi/h). The driver ahead of you suddenly pus on his brakes. We need o calculae a number of hings how long i akes you o respond; how far you ravel in ha ime, and how far he oher car ravels in ha ime. (a) Firs le s esimae how long i akes you o respond. Two imes are involved how long i akes from he ime you noice somehing happening ill you sar o move o he brake, and how long i akes o move your foo o he brake. You will need a ruler o do his. Take he ruler and have a friend hold i from he one end hanging sraigh down. Place your humb and forefinger opposie he boom of he ruler. As your friend releases he ruler suddenly, ry o cach i wih your humb and forefinger. Measure how far i falls before you cach i. Do his hree imes and ake he average disance. Assuming he ruler is falling freely wihou air resisance (no a bad assumpion), calculae how much ime i akes you o cach i,. Now esimae he ime,, i akes you o move your foo from he gas pedal o he brake pedal. Your reacion ime is. (b) If you brake hard and fas, you can bring a ypical car o res from km/h (abou 6 mi/h) in 5 seconds.. Calculae your acceleraion, a, assuming ha i is consan.. Suppose he driver ahead of you begins o brake wih an acceleraion a. How far will he ravel before he comes o a sop? (Hin How much ime will i ake him o sop? Wha will be his average velociy over his ime inerval?) (c) Now we can pu hese resuls ogeher ino a fairly realisic siuaion. You are driving on he highway a km/hr and here is a driver in fron of you going a he same speed.. You see him sar o slow immediaely (an unreasonable bu simplifying assumpion). If you are also raveling km/h, how far (in meers) do you ravel before you begin o brake? If you can also produce he acceleraion a when you brake, wha will be he oal disance you ravel before you come o a sop?. If you don noice he driver ahead of you beginning o brake for s, how much addiional disance will you ravel? 3. Discuss, on he basis of hese calculaions, wha you hink is a safe disance o say behind a car a 6 mi/h. Express your disance in car lenghs (abou 5 f). Would you include a safey facor beyond wha you have calculaed here? How much? 5. Tesing he Moion Deecor A moion deecor ha may be used in physics laboraories is shown in Fig. -3. I measures he disance o he neares objec by using a speaker and a microphone. The speaker clicks 3 imes a second. The microphone deecs he sound bouncing back from he neares objec in fron of i. The compuer calculaes he ime delay beween making he sound and receiving he echo. I knows he speed of sound (abou 343 m/s a room emperaure), and from ha i can calculae he disance o he objec from he ime delay. FIGURE -3 Problem 5. (a) If he neares objec in fron of he deecor is oo far away, he echo will no ge back before a second click is emied. Once ha happens, he compuer has no way of knowing ha he echo isn an echo from he second click and ha he deecor isn giving correc resuls any more. How far away does he objec have o be before ha happens? (b) The speed of sound changes a lile bi wih emperaure. Le s ry o ge an idea of how imporan his is. A room emperaure (7 F) he speed of sound is abou 343 m/s. A 6 F i is abou % smaller. Suppose we are measuring an objec ha is really.5 meers away a 7 F. Wha is he ime delay ha he compuer deecs before he echo reurns? Now suppose he emperaure is 6 F. If he compuer deecs a ime delay of bu (because i doesn know he emperaure) calculaes he disance using he speed of sound appropriae for 7 F, how far away does he compuer repor he objec o be? 5. Hiing a Bowling Ball A bowling ball sis on a hard floor a a poin ha we ake o be he origin. The ball is hi some number of imes by a hammer. The ball moves along a line back and forh across he floor as a resul of he his. (See Fig. -3.) The region o he righ of he origin is aken o be posiive, bu during is moion he ball is a imes on boh sides of he origin. Afer he ball has been moving for a while, a moion deecor like he one discussed in Problem 5 is sared and akes he following graph of he ball s velociy. Velociy + A D Time FIGURE -3 Problem 5. Answer he following quesions wih he symbols L (lef), R (righ), N (neiher), or C (can say which). Each quesion refers only o he ime inerval displayed by he compuer. (a) A which side of he origin is he ball for he ime marked A? (b) A he ime marked B, in which direcion is he ball moving? (c) Beween he imes A and C, wha is he direcion of he ball s displacemen? (d) The ball receives a hi a he ime marked D. In wha direcion is he ball moving afer ha hi? 53. Waking he Balrog In The Fellowship of he Ring, he hobbi Peregrine Took (Pippin for shor) drops a rock ino a well while he ravelers are in he caves of Moria. This wakes a balrog (a bad hing) and causes all kinds of rouble. Pippin hears he rock hi he waer 7.5 s afer he drops i. (a) Ignoring he ime i akes he sound o ge back up, how deep is he well? (b) I is quie cool in he caves of Moria, and he speed of sound in air changes wih emperaure. Take he speed of sound o be 34 m/s (i is prey cool in ha par of Moria). Was i OK o ignore he ime i akes sound o ge back up? Discuss and suppor your answer wih a calculaion. 54. Two Balls, Passing in he Nigh* Figure -3 represens he posiion vs. clock reading of he moion of wo balls, A and B, *From A. Arons, A Guide o Inroducory Physics Teaching (New York John Wiley, 99). B C

28 5 CHAPTER Moion Along a Sraigh Line moving on parallel racks. Carefully x skech he figure on your homework paper and answer he following A quesions B (a) Along he axis, mark wih he symbol A any insan or insans a which one ball is passing he oher. (b) Which ball is moving faser a B clock reading B? FIGURE -3 Problem 54. (c) Mark wih he symbol C any insan or insans a which he balls have he same velociy. (d) Over he period of ime shown in he diagram, which of he following is rue of ball B? Explain your answer.. I is speeding up all he ime.. I is slowing down all he ime. 3. I is speeding up par of he ime and slowing down par of he ime. 55. Graph for a Car on a Tiled Airrack wih Spring The graph in Fig. -33 below shows he velociy graph of a car moving on an air rack. The rack has a spring a one end and has is oher end raised. The car is sared sliding up he rack by pressing i agains he spring and releasing i. The clock is sared jus as he car leaves he spring. Take he direcion he car is moving in iniially o be he posiive x direcion and ake he boom of he spring o be he origin. Spring For each of he following quaniies, selec he leer of he graph ha could provide a correc graph of he quaniy for he ball in he siuaion shown (if he verical axis were assigned he proper unis). Use he x and y coordinaes shown in he picure. If none of he graphs could work, wrie N. (a) The x-componen of he ball s posiion (b) The y-componen of he ball s velociy (c) The x-componen of he ball s acceleraion (d) The y-componen of he normal force he ramp exers on he ball (e) The x-componen of he ball s velociy (f) The x-componen of he force of graviy acing on he ball () (4) y x () (5) θ v (3) (6) Glider Air rack (7) (8) (9) Velociy + A B C D FIGURE -33 Problem 55 Time Leers poin o six poins on he velociy curve. For he physical siuaions described below, idenify which of he leers corresponds o he siuaion described. You may use each leer more han once, more han one leer may be used for each answer, or none may be appropriae. If none is appropriae, use he leer N. (a) This poin occurs when he car is a is highes poin on he rack. (b) A his poin, he car is insananeously no moving. (c) This is a poin when he car is in conac wih he spring. (d) A his poin, he car is moving down he rack oward he origin. (e) A his poin, he car has acceleraion of zero. 56. Rolling Up and Down A ball is launched up a ramp by a spring as shown in Fig A he ime when he clock sars, he ball is near he boom of he ramp and is rolling up he ramp as shown. I goes o he op and hen rolls back down. For he graphs shown in Fig. -34, he horizonal axis represens he ime. The verical axis is unspecified. E F FIGURE -34 Problem Model Rocke A model rocke, propelled by burning fuel, akes off verically. Plo qualiaively (numbers no required) graphs of y, v, and a versus for he rocke s fligh. Indicae when he fuel is exhaused, when he rocke reaches maximum heigh, and when i reurns o he ground. 58. Rock Climber A ime, a rock climber accidenally allows a pion o fall freely from a high poin on he rock wall o he valley below him. Then, afer a shor delay, his climbing parner, who is m higher on he wall, hrows a pion downward. The posiions y of he pions versus during he fall are given in Fig Wih wha speed was he second pion hrown? 59. Two Trains As wo rains move along a rack, heir conducors suddenly noice ha hey are headed oward each oher. Figure -36 gives heir velociies v as funcions of ime as he conducors slow he rains. v (m/s) y 3 (s) FIGURE -35 Problem (s) 4 6 FIGURE -36 Problem 59.

29 Addiional Problems 53 The slowing processes begin when he rains are m apar. Wha is heir separaion when boh rains have sopped? 6. Runaway Balloon As a runaway v scienific balloon ascends a 9.6 m/s, one of is insrumen packages breaks free of a harness and free-falls. Figure (s) gives he verical velociy of he package versus ime, from before i breaks free o when i reaches he ground. (a) Wha maximum heigh above he break-free poin does i FIGURE -37 rise? (b) How high was he break-free Problem 6. poin above he ground? 6. Posiion Funcion Two A paricle moves along he x axis wih posi- x() x ion funcion x() as shown in Fig Make rough skeches of he paricle s velociy versus ime and is acceleraion versus ime for his FIGURE moion. -38 Problem Velociy Curve Figure -39 gives v he velociy v(m/s) versus ime (s) for a paricle moving along an x axis. The unis area beween he ime axis and he ploed curve is given for he wo porions of he graph. A A (a one of unis A B he crossing poins in he ploed figure), he paricle s posiion is x 4 m. Wha is is posiion a FIGURE -39 (a) and (b) B? Problem The Moion Deecor Rag This assignmen is based on he Physics Pholk Song CD disribued by Pasco scienific. These songs are also available hrough he Dickinson College Web sie a hp//physics.dickinson.edu. (a) Refer o he moion described in he firs verse of he Moion Deecor Rag; namely, you are moving for he same amoun of ime ha you are sanding. Skech a posiion vs. ime graph for his moion. Also, describe he shape of he graph in words. (b) Refer o he moion described in he second verse of he Moion Deecor Rag. In his verse, you are making a seep downslope, hen a genle up-slope, and las a fla line. You spend he same amoun of ime engaged in each of hese acions. Skech a posiion vs. ime graph of his moion. Also, describe wha you are doing in words. Tha is, are you sanding sill, moving away from he origin (or moion deecor), moving oward he origin (or moion deecor)? Which moion is he mos rapid, and so on? (c) Refer o he moion described in he hird verse of he Moion Deecor Rag. You sar from res and move away from he moion deecor a an acceleraion of. m/s for 5 seconds. Skech he acceleraion vs. ime graph o his moion. Skech he corresponding velociy vs. ime graph. Skech he shape of he corresponding posiion vs. ime graph. 64. Hockey Puck A ime, a hockey puck is sen sliding over a frozen lake, direcly ino a srong wind. Figure -4 gives he velociy v of he puck vs. ime, as he puck moves along a single axis. A 4 s, wha is is posiion relaive o is posiion a? v (m/s) 8 5 (s) FIGURE -4 Problem Describing One-Dimensional Velociy Changes In each of he following siuaions you will be asked o refer o he mahemaical definiions and he conceps associaed wih he number line. Noe ha being more posiive is he same as being less negaive, and so on. (a) Suppose an objec undergoes a change in velociy from m/s o 4 m/s. Is is velociy becoming more posiive or less posiive? Wha is mean by more posiive? Less posiive? Is he acceleraion posiive or negaive? (b) Suppose an objec undergoes a change in velociy from 4 m/s o m/s. Is is velociy becoming more posiive or less posiive? Wha is mean by more posiive? Less posiive? Is he acceleraion posiive or negaive? (c) Suppose an objec is urning around so ha i undergoes a change in velociy from m/s o m/s. Is is velociy becoming more posiive or less posiive han i was before? Wha is mean by more posiive? Less posiive? Is i undergoing an acceleraion while i is urning around? Is he acceleraion posiive or negaive? (d) Anoher objec is urning around so ha i undergoes a change in velociy from m/s o m/s. Is is velociy becoming more posiive or less posiive han i was before? Wha is mean by more posiive? Less posiive? Is i undergoing an acceleraion while i is urning around? Is he acceleraion posiive or negaive? 66. Bowling Ball Graph A bowling Average ball was se ino moion on a fairly Time (s) Disance (m) smooh level surface, and daa were.. colleced for he oal disance covered by he ball a each of four imes These daa are shown below (a) Plo he daa poins on a graph. (b) Use a ruler o draw a sraigh line ha passes as close as possible o he daa poins you have graphed. (c) Using mehods you were augh in algebra, calculae he value of he slope, m, and find he value of he inercep, b, of he line you have skeched hrough he daa. 67. Modeling Bowling Ball Moion A bowling ball is se ino moion on a smooh level surface, and daa were colleced for he oal disance covered by he ball a each of four imes. These daa are shown in he able in Problem 66. Your job is o learn o use a spreadshee program for example, Microsof Excel o creae a mahemaical model of he bowling ball moion daa shown. You are o find wha you hink is he bes value for he slope, m, and he y-inercep, b. Pracicing wih a uorial workshee eniled MODTUT.XLS will help you o learn abou he process of

30 54 CHAPTER Moion Along a Sraigh Line modeling for a linear relaionship. Ask your insrucor where o find his uorial workshee. Afer using he uorial, you can creae a model for he bowling ball daa given above. To do his (a) Open a new workshee and ener a ile for your bowling ball graph. (b) Se he y-label o Disance (m) and he x-label o Time (s). (c) Refer o he daa able above. Ener he measured imes for he bowling ball in he Time (s) column (formerly x-label). (d) Se he y-exp column o D-daa (m) and ener he measured disances for he bowling ball (probably somehing like. m,. m, 4. m, and 6. m.). (e) Place he symbol m (for slope) in he cell B. Place he symbol b (for y-inercep) in cell B. (f) Se he y-heory column o D-model (m) and hen pu he appropriae equaion for a sraigh line of he form Disance m*time b in cells C7 hrough C. Be sure o refer o cells C for slope and C for y-inercep as absolues; ha is, use $C$l and $C$ when referring o hem. (g) Use he spreadshee graphing feaure o creae a graph of he daa in he D-exp and D-heory columns as a funcion of he daa in he Time column. (h) Change he values in cells Cl and C unil your heoreical line maches as closely as possible your red experimenal daa poins in he graph window. (i) Discuss he meaning of he slope of a graph of disance vs. ime. Wha does i ell you abou he moion of he bowling ball? 68. A Srange Moion Afer doing a number of he exercises wih cars and fans on ramps, i is easy o draw he conclusion ha everyhing ha moves is moving a eiher a consan velociy or a consan acceleraion. Le s examine he horizonal moion of a riangular frame wih a pendulum a is cener ha has been given a push. I undergoes an unusual moion. You should deermine wheher or no i is moving a eiher a consan velociy or consan acceleraion. (Noe You may wan o look a he moion of he riangular frame by viewing he digial movie eniled PASCO7. This movie is included on he VideoPoin compac disk. If you are no using VideoPoin, your insrucor may make he movie available o you some oher way.) The images in Fig. -4 are aken from he 7h, 6h, and 5h frames of ha movie. Daa for he posiion of he cener of he horizonal bar of he riangle were aken every enh of a second during is firs second of moion. The origin was placed a he zero cenimeer mark of a fixed meer sick. These daa follow. (a) Examine he posiion vs. ime graph of he daa shown above. Does he riangle appear o have a consan velociy hroughou he firs second? A consan acceleraion? Why or why no? (b) Discuss he naure of he moion based on he shape of he graph. A approximaely wha ime, if any, is he riangle changing direcion? A approximaely wha ime does i have he greaes negaive velociy? The greaes posiive velociy? Explain he reasons for your answers. (c) Use he daa able and he definiion of average velociy o calculae he average velociy of he riangle a each of he imes beween. s and.9 s. In his case you should use he posiion jus before he indicaed ime and he posiion jus afer he indicaed ime in your calculaion. For example, o calculae he average velociy a. seconds, use x cm and x 5. cm along wih he differences of he imes a 3 and. Hin Use only imes and posiions in he gray boxes o ge a velociy in a gray box and use only imes and posiions in he whie boxes o ge a velociy in a whie box. (d) Since people usually refer o velociy as disance divided by ime, maybe we can calculae he average velociies as simply x /, x /, x 3 / 3, and so on. This would be easier. Is his an equivalen mehod for Origin =. (s) =.5 (s) =.8 (s) Fr# Pr# (s) x(cm) <v>(cm/s) no enry 38. no enry x(cm) Triangle Moion (s). FIGURE -4 Problem 68.

31 Addiional Problems 55 finding he velociies a he differen imes? Try using his mehod of calculaion if you are no sure. Give reasons for your answer. (e) Ofen, when an oddly shaped bu smooh graph is obained from daa i is possible o fi a polynomial o i. For example, a fourh-order polynomial ha fis he daa is x {( 376 cm/s 4 ) 4 (79 cm/s 3 ) 3 (347 cm/s ) (5.63 cm/s) 5. cm} Using his polynomial approximaion, find he insananeous velociy a.7 s. Commen on how your answer compares o he average velociy you calculaed a.7 s. Are he wo values close? Is ha wha you expec? 69. Cedar Poin A he Cedar Poin Amusemen Park in Ohio, a cage conaining people is moving a a high iniial velociy as he resul of a previous free fall. I changes direcion on a curved rack and hen coass in a horizonal direcion unil he brakes are applied. This siuaion is depiced in a digial movie eniled DSON. (Noe This movie is included on he VideoPoin compac disk. If you are no using VideoPoin, your insrucor may make he movie available o you some oher way.) (a) Use video analysis sofware o gaher daa for he horizonal posiions of he ail of he cage in meers as a funcion of ime. Don forge o use he scale on he ile screen of he movie so your resuls are in meers raher han pixels. Summarize his daa in a able or in a prinou aached o your homework. (b) Transfer your daa o a spreadshee and do a parabolic model o show ha wihin 5% or beer x ( 7.5 m/s ) (.5 m/s).38 m. Please aach a prinou of his model and graph wih your name on i o your submission as proof of compleion. (Noe Your judgmens abou he locaion of he cage ail may lead o slighly differen resuls.) (c) Use he equaion you found along wih is inerpreaion as embodied in he firs kinemaic equaion o deermine he horizonal acceleraion, a, of he cage as i slows down. Wha is is iniial horizonal velociy, v, a ime s? Wha is he iniial posiion, x,of he cage? (d) The movie ends before he cage comes o a complee sop. Use your knowledge of a, v, and x along wih kinemaic equaions o deermine he horizonal posiion of he cage when i comes o a complee sop so ha he final velociy of he cage is given by v v. m/s. 7. Three Digial Movies Three digial movies depicing he moions of four single objecs have been seleced for you o examine using a video-analysis program. They are as follows PASCO4 A car moves on an upper rack while anoher moves on a rack jus below. PASCO53 A meal ball aached o a sring swings genly. HRSY3 A boa wih people moves in a waer rough a Hershey Amusemen Park. Please examine he horizonal moion of each objec carefully by viewing he digial movies. In oher words, jus examine he moion in he x direcion (and ignore any sligh moions in he y direcion). You may use VideoPoin, VideoGraph, or World-in-Moion digial analysis sofware and a spreadshee o analyze he moion in more deail if needed. Based on wha you have learned so far, here is more han one analysis mehod ha can be used o answer he quesions ha follow. Noe Since we are ineresed only in he naure of hese moions (no exac values) you do no need o scale any of he movies. Working in pixel unis is fine. (a) Which of hese four objecs (upper car, lower car, meal ball, or boa), if any, move a a consan horizonal velociy? Cie he evidence for your conclusions. (b) Which of hese four objecs, if any, move a a consan horizonal acceleraion? Cie he evidence for your conclusions. (c) Which of hese four objecs, if any, move a neiher a consan horizonal velociy nor acceleraion? Cie he evidence for your conclusions. (d) The kinemaic equaions are very useful for describing moions. Which of he four moions, if any, canno be described using he kinemaic equaions? Explain he reasons for your answer. 7. Speeding Up or Slowing Down Figure -4 shows he velociy vs. ime graph for an objec consrained o move in one dimension. The posiive direcion is o he righ. Velodiy (m/s) Time (s) FIGURE -4 Problems (a) A wha imes, or during wha ime periods, is he objec speeding up? (b) A wha imes, or during wha ime periods, is he objec slowing down? (c) A wha imes, or during wha ime periods, does he objec have a consan velociy? (d) A wha imes, or during wha ime periods, is he objec a res? If here is no ime or ime period for which a given condiion exiss, sae ha explicily. 7. Righ or Lef Figure -4 shows he velociy vs. ime graph for an objec consrained o move along a line. The posiive direcion is o he righ. (a) A wha imes, or during wha ime periods, is he objec speeding up and moving o he righ? (b) A wha imes, or during wha ime periods, is he objec slowing down and moving o he righ? (c) A wha imes, or during wha ime periods, does he objec have a consan velociy o he righ? (d) A wha imes, or during wha ime periods, is he objec speeding up and moving o he lef? (e) A wha imes, or during wha ime periods, is he objec slowing down and moving o he lef? (f) A wha imes, or during wha ime periods, does he objec have a consan velociy o he lef? If here is no ime or ime period for which a given condiion exiss, sae ha explicily. 73. Consan Acceleraion Figure -4 shows he velociy vs. ime graph for an objec consrained o move along a line. The posiive direcion is o he righ. (a) A wha imes, or during wha ime periods, is he objec s acceleraion zero?

32 56 CHAPTER Moion Along a Sraigh Line (b) A wha imes, or during wha ime periods, is he objec s acceleraion consan? (c) A wha imes, or during wha ime periods, is he objec s acceleraion changing? If here is no ime or ime period for which a given condiion exiss, sae ha explicily. 74. Acceleraion o he Righ or Lef Figure -4 shows he velociy vs. ime graph for an objec consrained o move along a line. The posiive direcion is o he righ. (a) A wha imes, or during wha ime periods, is he objec s acceleraion increasing and direced o he righ? (b) A wha imes, or during wha ime periods, is he objec s acceleraion decreasing and direced o he righ? (c) A wha imes, or during wha ime periods, does he objec have a consan acceleraion o he righ? (d) A wha imes, or during wha ime periods, is he objec s acceleraion increasing and direced o he lef? (e) A wha imes, or during wha ime periods, is he objec s acceleraion decreasing and direced o he lef? (d) A wha imes, or during wha ime periods, does he objec have a consan acceleraion o he lef? If here is no ime or ime period for which a given condiion exiss, sae ha explicily.