Kinematics: Motion in One Dimension

Transcription

1 Kinemaics: Moion in One Dimension When yo drie, yo are spposed o follow he hree-second ailgaing rle. When he car in fron of yo passes a sign a he side of he road, yor car shold be far enogh behind i ha i akes yo 3 s o reach he same sign. In his chaper, we will learn he physics behind he hree-second rle. CIENTIT OFTEN AK qesions abo hings ha mos people accep as js he way i is. For eample, in he norhern hemisphere, here are more hors of dayligh in Jne han in December, whereas in he sohern h emisphere, he opposie is re. cieniss wan eplanaions for sch simple phenomena. In his chaper, we will learn o describe a phenomenon ha we enconer eery day b rarely qesion moion. Wha is a safe following disance beween yor car and he car in fron of yo? Can yo be moing and no moing a he same ime? Why do physiciss say ha an objec hrown pward is falling? BE URE YOU KNOW HOW TO: Decide when an objec can be modeled as a poin-like objec (ecion 1.). Use significan digis in calclaions (ecion 1.3). 13 M_ETKI183 E_C.indd 13 /9/17 1:55 PM

2 14 CHAPTER Kinemaics: Moion in One Dimension.1 Wha is moion? Imagine yo are a passenger in a car. Yo see yor friend sanding on he sidewalk and wae a her, rning yor head o look a her while driing pas her. he oo rns her head as she waes back. We rn or heads o follow a moing objec, so i is no srprising ha yor friend rns her head yo are in a moing car. B why do yo rn yor head? Afer all, yor friend is sanding sill on he sidewalk. Howeer, from yor perspecie, she is moing away from yo. This eample helps s ndersand a ery imporan idea: differen obserers see he same moion differenly. In oher words, moion is relaie o he obserer, he person who is describing he moion. o who is moing in his scenario, yo or yor friend? The answer is ha i depends on who is he obserer. The drier of yor car, for eample, sees yo (a passenger in he car) as no moing. This eample shows s ha when describing moion we need o specify boh he objec whose moion we are describing (he objec of ineres) and he person who is doing he describing (he obserer). FIGURE.1 Moion is relaie. Two obserers eplain he moion of he n relaie o Earh differenly. Moion is a change in an objec s posiion relaie o a gien obserer dring a cerain ime ineral. Wiho idenifying he obserer, i is impossible o say wheher he objec of ineres moed. Physiciss say moion is relaie, meaning ha he moion of any objec of ineres depends on he poin of iew of he obserer. (a) An obserer on Earh sees he n moe in an arc across he sky. (b) An obserer in a spaceship sees he person on Earh as roaing nder a saionary n. Wha makes he idea of relaie moion confsing a firs is ha people iniiely se Earh as he objec of reference he objec wih respec o which hey describe moion. If an objec does no moe wih respec o Earh, many people wold say ha he objec is no moing. Tha is why i ook scieniss hosands of years o ndersand why Earh has days and nighs. An obserer on Earh ses Earh as he objec of reference and sees he n moing in an arc across he sky (Figre.1a). An obserer on a disan spaceship, howeer, sees Earh roaing on is ais so ha differen pars of is srface face he n a differen imes (Figre.1b). Reference frames pecifying he obserer before describing he moion of an objec of ineres is an eremely imporan par of consrcing wha physiciss call a reference frame. A reference frame incldes an objec of reference, a coordinae sysem wih a scale for measring disances, and a clock o measre ime. If he objec of reference is large and canno be considered a poin-like objec, i is imporan o specify where on he objec of reference he origin of he coordinae sysem is placed. For eample, if yo wan o describe he moion of a bicyclis and choose yor objec of reference o be Earh, yo place he origin of he coordinae sysem a Earh s srface, no is cener. Modeling moion When we model objecs, we make simplified assmpions in order o analyze complicaed siaions. Js as we simplified an objec o model i as a poin-like objec in Chaper 1, we can also simplify a process inoling moion. Wha is he simples way an objec can moe? Imagine ha yo haen ridden a bike in a while. Yo probably woldn sar by aemping a rn; insead, yo wold ride in a sraigh line. This kind of moion is called linear moion or one-dimensional moion.

3 . A concepal descripion of moion 15 Linear moion is a model of moion ha assmes ha an objec, considered as a poin-like objec, moes along a sraigh line. For eample, sppose we wan o model a car s moion along a sraigh srech of highway. We can assme ha he car is a poin-like objec (i is small compared o he lengh of he highway) and he moion is linear moion (he highway is long and sraigh). REVIEW QUETION.1 Wha does he saemen Moion is relaie mean?. A concepal descripion of moion To describe linear moion more precisely, we sar by deising a isal represenaion. Consider Obseraional Eperimen Table.1, in which a bowling ball rolls on a smooh floor. OBERVATIONAL EXPERIMENT TABLE.1 Using dos o represen moion VIDEO OET.1 Obseraional eperimen Eperimen 1. Yo psh a bowling ball (he objec of ineres) and le i roll on a smooh linolem floor. Each second, yo place a beanbag beside he bowling ball. The beanbags are eenly spaced. Analysis The dos in his diagram represen he posiions of he beanbags yo placed each second as he bowling ball slowly rolled on he floor. Eperimen. Yo repea Eperimen 1, b yo psh he ball harder before yo le i roll. The beanbags are farher apar b are sill eenly spaced. The dos in his diagram represen he posiions of he beanbags, which are sill eenly spaced b separaed by a greaer disance han he bags in Eperimen 1. Eperimen 3. Yo psh he bowling ball and le i roll on a carpeed floor insead of a linolem floor. The disance beween he beanbags decreases as he ball rolls. The dos in his diagram represen he decreasing disance beween he beanbags as he ball rolls on he carpe. Eperimen 4. Yo roll he ball on he linolem floor and genly and coninally psh on i wih a board. The disance beween he beanbags increases as he pshed ball rolls. The dos in his diagram represen he increasing disance beween beanbags as he ball is coninally pshed across he linolem floor. Paern The spacing of he dos allows s o isalize he moion of he objec of ineres. When he objec slows down, he dos ge closer ogeher. When he objec raels wiho speeding p or slowing down, he dos are eenly spaced. When he objec moes faser and faser, he dos ge farher apar. Moion diagrams In he eperimens in Table.1, we sed beanbags as an approimae record of where he ball was locaed as ime passed o help s isalize he moion of he ball. Le s now se blinking LEDs o help s more precisely isalize moion, his ime of

4 16 CHAPTER Kinemaics: Moion in One Dimension FIGURE. Long-eposre phoographs of a moing car wih a blinking LED. (a) (b) (c) (d) a car. Figre.a shows a car wih an aached blinking LED (on for 31 ms, hen off for 31 ms). Figre.b is a long-eposre phoo of he car as i is moing on a horizonal srface a seady speed, similar o he siaion in Eperimen 1 of Table.1. In Figre.c, he car is moing faser. Ne, in Figre.d, he moing car is being pshed in he direcion opposie o is moion nil i sops. In Figre.e, he car is being pshed from res and moes faser and faser. As yo can see, he ligh races in he for phoos clearly illsrae he differen ypes of moion of he car. The same-lengh ligh races in Figre.b indicae ha he car moed he same disance each ime he LED ligh was on (ha is, was raeling a a seady speed), while he longer races of Figre.c show ha he car coered a longer disance dring each ligh blink. In Figre.d, he races ge shorer as he car was slowing down and rn ino a poin when he car sopped. In Figre.e, he races ge longer as he car was speeding p. These LED phoos help s consrc a new represenaion of moion ha has more informaion han he do diagrams: his represenaion is called a moion diagram. We draw dos o represen he posiions of he car and, insead of he races of ligh, we draw arrows o indicae he car s direcion of moion. The lengh of each arrow shows how fas he car was moing a he locaion of ha do. These arrows are called elociy arrows. Figre.3 shows differen ypes of moion represened wih dos and elociy arrows. Noice he small arrow aboe each leer (for elociy) in he figres. The arrow indicaes ha his characerisic of moion has a direcion as well as a magnide. Remember from Chaper 1 ha sch qaniies are called ecor qaniies. (e) Velociy change arrows FIGURE.3 Moion diagrams represen he ypes of moion shown in Table.1. (a) The elociy arrows represen how fas he ball is moing and is direcion. The dos represen he posiions of he ball a reglar imes. s 1 s s 3 s (b) This ball is moing a a consan b faser speed han he ball in (a). s (c) 1s s The ball is slowing down. In Eperimen 4 in Table.1 he bowling ball was moing increasingly faser while being pshed. The elociy arrows in he moion diagram in Figre.3d hs are increas ingly longer. We can represen his change by drawing a elociy change (D ) arrow. The D (dela) means a change in whaeer qaniy follows he D, a change in in his case. The D doesn ell s he eac increase or decrease in he elociy; i only indicaes a qaliaie difference beween he elociies a wo adjacen poins in he diagram. In Figre.4a we hae redrawn he diagram shown in Figre.3d. For illsraion prposes only, we nmber he arrows for each posiion: 1,, 3, ec. To draw he elociy change arrow as he ball moes from posiion o posiion 3 in Figre.4a, we place he arrow 3 direcly aboe he arrow, as shown. The 3 arrow is longer han he arrow. This ells s ha he objec was moing faser a posiion 3 han a posiion. To isalize he change in elociy, we need o hink abo how arrow can be rned ino 3. We can do i by placing he ail of a elociy change arrow D 3 a he head of so ha he head of D 3 makes + D 3 he same lengh as 3 (Figre.4b). ince hey are he same lengh and poin in he same direcion, he wo ecors + D 3 and 3 are eqal: (d) The ball is speeding p. + D 3 = 3 FIGURE.4 Deermining he magnide and direcion of he elociy change arrow in a moion diagram. (a) M_ETKI183 E_C.indd 16 (b) The ball is speeding p. 3 4 How can we represen he change in elociy from o 3? 1 D3 5 3 D3 5 3 D3 Add he D3 arrow o o ge 3. /9/17 1:55 PM

5 . A concepal descripion of moion 17 Noe ha if we moe o he oher side of he eqaion, hen D 3 = 3 - Ths, D 3 is he difference of he hird elociy arrow and he second elociy arrow he change in elociy beween posiion and posiion 3. Making a complee moion diagram We now place he D arrows aboe and beween he dos in or diagrams where he elociy change occrred (see Figre.5a). The dos in hese more deailed moion diagrams indicae he objec s posiion a eqal ime inerals; elociy arrows and elociy change arrows are also inclded. A D arrow poins in he same direcion as he arrows when he objec is speeding p; he D arrow poins in he opposie direcion of he arrows when he objec is slowing down. When elociy changes by he same amon dring each consecie ime ineral, he D arrows for each ineral are he same lengh. In sch cases we need only one D arrow for he enire moion diagram (see Figre.5b). Physics Tool Bo.1 smmarizes he procedre for consrcing a moion diagram. Noice ha in he eperimen represened in his diagram, he objec is moing from righ o lef and slowing down. PHYIC TOOL BOX.1 Consrcing a moion diagram FIGURE.5 Two complee moion diagrams, inclding posiion dos, arrows, and D arrows. (a) A moion diagram showing he elociy change for each consecie posiion change D 1 D 3 D (b) When he elociy change is consan from ime ineral o ime ineral, we need only one D for he diagram. D 1. Draw dos o represen he posiion of he objec wih respec o he obserer a eqal ime inerals.. Poin elociy arrows in he direcion of moion and draw heir relaie lenghs o indicae approimaely how fas he objec is moing. D 3. Draw a elociy change arrow o indicae how he elociy arrows are changing beween adjacen posiions. Concepal Eercise.1 will help yo pracice represening moion wih moion diagrams. When working on Concepal Eercises, firs isalize he siaion, hen draw a skech, and finally consrc a physics represenaion (in his case, a moion diagram). When drawing a moion diagram, TIP always specify he posiion of he obserer. CONCEPTUAL EXERCIE.1 Driing in he ciy A car a res a a raffic ligh sars moing faser and faser when he ligh rns green. The car reaches he speed limi in 4 s, conines a he speed limi for 3 s, hen, approaching a second, red, raffic ligh, slows down and sops in s. A he ligh, he car wais for 1 s nil i rns green. Meanwhile, a cyclis approaching he firs green ligh keeps moing wiho slowing down or speeding p. he reaches he second sopligh js as i rns green. Draw a moion diagram for he car and anoher for he bicycle as seen by an obserer on he grond. If yo place one diagram below he oher, i will be easier o compare hem. kech and ranslae Visalize he moion for he car and for he bicycle as seen by he obserer on he grond. The car and he bicycle are or objecs of ineres. The moion of he car has for disinc pars: 1. saring a res and moing faser and faser for 4 s;. moing a a consan rae for 3 s; 3. slowing down o a sop for s; and 4. siing a res for 1 s. (conined)

6 18 CHAPTER Kinemaics: Moion in One Dimension The bicycle moes a a consan rae wih respec o he grond for he enire ime. implify and diagram We model he car and he bicycle as poin-like objecs (dos). In each moion diagram, here are 11 dos, one for each second of ime (inclding one for ime ). The las wo dos for he car are on op of each oher since he car is a res from ime = 9 s o ime = 1 s. The dos for he bicycle are eenly spaced. peeding p Consan speed lowing down Consan speed Try i yorself Two bowling balls are rolling along a linolem floor. One of hem is moing wice as fas as he oher. A ime, hey are ne o each oher on he floor. Consrc moion diagrams for each ball dring a ime ineral of 4 s, as seen by an obserer on he grond. Indicae on he diagrams he locaions a which he balls were ne o each oher a he same ime. Wha are he possible misakes ha a sden cold make when doing his eercise? Answer ee he figre a lef. The balls are side by side only a ime he firs do for each ball. I looks like hey are side by side when a he -m posiion, b he slower ball is a he -m posiion a s and he faser ball is here a 1 s. imilar reasoning applies for he 4-m posiions he balls reach ha poin a differen imes. REVIEW QUETION. Is he following saemen re? If an objec s elociy change arrows poin o he lef, i is slowing down. ppor yor answer wih eamples. FIGURE.6 Using arrows o represen ecors. (a) Tail 7 Head 8 (b) (c) A A 1 B A B A B B.3 Operaions wih ecors Physics Tool Bo.1 ells yo o deermine he direcion of a D arrow on a moion diagram. To do his, yo need o know eiher how o add ecors (he operaion + D 3 = 3 in Figre.4) or how o sbrac hem (he operaion D 3 = 3 - in Figre.4). Here we discss hese operaions in more deail. To sar, we firs consider more simple ecor operaions. Careflly eamine Figre.6, which shows eigh arrows of differen lenghs and direcions represening ecors. The lengh of each arrow is he ecor s magnide, in any nis yo choose, and he orienaion of each ecor is deermined by he direcion of he arrow. Each ecor has a ail (he poin where i originaes) and a head (he ip of he arrow). Which arrows represen he same ecor? Yo see ha arrows 1, 3, and 4 hae he same lengh and he same direcion: hey herefore represen he same ecor, A. Arrows, 5, and 7 represen a differen ecor, ecor B. From his eercise we see ha moing he ecor parallel o iself will no change he ecor. Wha abo arrows 6 and 8? They hae he same lengh as A and B, respeciely, b hey poin in he opposie direcions. In his case, we say ha ecor 6 is -A, he negaie of ecor A, and ecor 8 is -B. The mins sign is sed o mean ha A has he same lengh as A b poins in he opposie direcion. Now ha we know how o moe ecors arond and draw he negaie of a ecor, we can eplore how o add and sbrac ecors. Wha rles of ecor addiion and sbracion can yo infer by analyzing he eamples shown in Figre.6b (addiion, A + B) and Figre.6c (sbracion, A - B)? pend a few momens hinking abo his before reading he following eplanaion of he rles.

7 .3 Operaions wih ecors 19 Adding ecors ppose we wan o graphically add he wo ecors K and L shown in Figre.7a. We can moe a ecor from one locaion o anoher as long as we do no change is magnide or direcion; herefore, we can place he ail of ecor L a he head of K, as shown in Figre.7b, wiho changing L. Ne, we draw anoher ecor R from he ail of K o he head of L, as shown in Figre.7c, o represen he resl of he addiion of he ecors K and L. We can wrie he reslan ecor (ecor sm) as a mahemaical eqaion: R = K + L. As yo can see in Figre.7c, he magnide of he ecor sm is no in general eqal o he sm of he magnides of K and L. Howeer, here are wo special cases, as Figre.7d shows: When wo ecors are parallel and poin in he same direcion, he addiion is mch easier: he magnide of he reslan ecor eqals he sm of he magnides of he ecors being added. When wo ecors are parallel b poin in opposie direcions (called aniparallel), he magnide of he reslan ecor eqals he absole ale of he difference in he magnides of he ecors being added. If yo need o add more ecors, follow he same procedre of ail o head for all of hem, and hen draw he reslan ecor. For accrae resls in graphical addiion, draw careflly sing a rler and a proracor. FIGURE.7 Graphical ecor addiion. FIGURE.8 Graphical ecor sbracion. (a) K L To add ecors K and L (a) M P To sbrac ecor P from ecor M (b) (c) (d) L L B K K R 5 K 1 L A 1 B A L Adding parallel and aniparallel ecors place he ail of L a he head of K and draw a ecor from he ail of K o he head of L. D A A 1 D (b) (c) P (d) P B A Q 5 M P (e) bracing parallel and aniparallel ecors D M M M A B P P A D firs draw P. Then place he ail of P a he head of M and draw a ecor from he ail of M o he head of P. A bracing ecors ppose we now wan o sbrac ecor P from ecor M, shown in Figre.8a. Yo can inerpre he operaion of sbracing ecor P from ecor M as he addiion of ecor - P and ecor M. The ecor eqaion is Q = M - P = M + 1- P. As Figre.8 shows, we herefore need o draw he ecor - P firs, and hen proceed wih graphical ecor addiion of - P and M. Figre.8e shows he sbracion of parallel and aniparallel ecors. This par is relean o deermining he direcion and magnide of he D ecor on a moion diagram.

8 CHAPTER Kinemaics: Moion in One Dimension CONCEPTUAL EXERCIE. Mliple ways o bild ecor X Vecor X is 3 cm long and poins wes. (a) Draw hree differen combinaions of wo ecors whose addiion gies ecor X. One of he combinaions shold hae aniparallel ecors. (b) Draw hree differen combinaions of wo ecors whose sbracion gies ecor X. One of he combinaions shold hae parallel ecors. kech and ranslae (a) We draw ecor X and hink of possible combinaions of wo ecors ha we can add o ge X. Using he mehod for graphical addiion, we know ha he sm X ms span from he ail of one ecor o he head of he oher when he wo ecors are placed head o ail. The figre below shows hree possible combinaions. H and K are aniparallel ecors whose sm is X. draw any ecor for he firs ecor and hen add X o i o ge he second ecor. The figre below shows hree possible combinaions. L and M are parallel ecors whose difference is X. X L R N M P Q X X5 R Q X5 N P X5 L M E D X F G X Try i yorself Eamine Figre.4. Where in his figre do we se ecor addiion and where do we se ecor sbracion? H K X5 E 1 D X5 F 1 G X5 H 1 K (b) If he sbracion of one ecor from anoher gies X, hen we know ha he firs ecor pls X gies he second ecor. We can herefore Answer The elociy arrow 3 is a ecor ha is he sm of he wo ecors and D 3: 3 = + D 3; he elociy change arrow is a ecor D 3 ha is he resl of sbracion of ecor from ecor 3: D 3 = 3 -. FIGURE.9 Mliplying a ecor by a scalar. (a) (b) C 3C C D 1 D 1 3 D (c) C C C 3C Mliplying a ecor by a scalar In addiion o ecor addiion and sbracion, here is anoher operaion wih ecors ha yo will need when sdying physics. I inoles mliplying a ecor by a posiie or a negaie nmber (a scalar qaniy) for eample, finding ecor C when ecor C is known. Figre.9a shows he resl of his operaion. Vecor C is wice as long as C, or eqal o he sm of wo C ecors, and poins in he same direcion. Vecor 3 C is hree imes as long, or eqal o he sm of hree C ecors, and poins in he same direcion. Figre.9b shows he resl of diiding ecor D by and by 3. We see ha he resl is eqal o mliplying ecor D by 1> and 1>3, respeciely (he ecor is one-half or one-hird of he lengh of ecor D). Finally, if yo need o mliply ecor C by a negaie nmber, for eample - or -3, yo need o firs draw ecor - C, as we learned aboe, and hen mliply i by or 3 (Figre.9c). In general, when yo mliply a ecor by a scalar, he resl is a ecor parallel or aniparallel o he original ecor whose magnide eqals he prodc of he magnide of he original ecor and he magnide of he scalar. [Noe ha if he scalar in his operaion has nis, he resling ecor has nis ha are he prodc of he original ecor nis and he nis of he scalar.] REVIEW QUETION.3 Egenia says ha o find he D arrow on a moion diagram yo need o sbrac ecors. Alan says ha yo need o add ecors. Boh are righ. How can his be? Eplain and show eamples.

9 .4 Qaniies for describing moion 1.4 Qaniies for describing moion A moion diagram helps represen moion qaliaiely. To analyze siaions more precisely, for eample, o deermine how far a car will rael afer he brakes are applied, we need o describe moion qaniaiely. In his secion, we deise some of he qaniies we need o describe linear moion. Time and ime ineral People se he word ime o mean he reading on a clock and how long a process akes. Physiciss disingish beween hese wo meanings wih differen erms: Time and ime ineral Time (clock reading) is he reading on a clock or some oher ime-measring insrmen. Time ineral 1-1 or D is he difference beween wo imes. In he I sysem (meric nis), he ni of boh ime and ime ineral is he second. Oher nis are mines, hors, days, and years. Time and ime ineral are boh scalar qaniies. FIGURE.1 Posiion, displacemen, disance, and pah lengh for a shor car rip. (a) Posiions i and f Car backs p, moing in negaie direcion oward origin. Car hen moes forward o f. Posiion, displacemen, disance, and pah lengh We also need o precisely define for qaniies ha describe he locaion and moion of an objec: Posiion, displacemen, disance, and pah lengh The posiion of an objec is is locaion wih respec o a pariclar coordinae sysem (sally indicaed by or y). The displacemen of an objec, sally indicaed by d, is a ecor ha sars from an objec s iniial posiion and ends a is final posiion. The magnide (lengh) of he displacemen ecor is called he disance d. The pah lengh l is how far he objec moed as i raeled from is iniial posiion o is final posiion. Imagine laying a sring along he pah he objec ook. The lengh of he sring is he pah lengh. i f Origin Iniial (b) Displacemen d and disance d d Final i f d 5 ) f i ) (c) Pah lengh l Car moed from i o and hen o f. i f l 5 ) i ) 1 ) f ) Figre.1a shows a car s iniial posiion i a iniial ime i. The car firs backs p (moing in he negaie direcion) oward he origin of he coordinae sysem a =. The car sops and hen moes in he posiie -direcion o is final posiion f. Noice ha he iniial posiion and he origin of a coordinae sysem are no necessarily he same poin! The displacemen d for he whole rip is a ecor ha poins from he saring posiion a i o he final posiion a f (Figre.1b). The disance for he rip is he magnide of he displacemen (always a posiie ale). The pah lengh l is he disance from i o pls he disance from o f (Figre.1c). Noe ha he pah lengh does no eqal he disance. calar componen of displacemen for moion along one ais To describe linear moion qaniaiely we firs specify a reference frame. For simpliciy we can poin one coordinae ais eiher parallel or aniparallel o he objec s direcion of moion. For linear moion, we need only one coordinae ais o describe he objec s changing posiion. In he eample of he car rip, a he iniial ime i he car is a posiion 1 i, and a he final ime f he car is a 1 f (Figre.11). The noaion 1 omeimes we se he sbscrips TIP 1,, and 3 for imes and he corresponding posiions o commnicae a seqence of differen and disingishable sages in any process, and someimes we se he sbscrips i (iniial) and f (final) o commnicae he seqence. bscrips i and f mean he beginning and end of or obseraions, no he beginning and end of he process. FIGURE.11 Indicaing an objec s posiion a a pariclar ime, for eample, 1. ( i ) ( f ) Iniial posiion Final posiion

10 CHAPTER Kinemaics: Moion in One Dimension FIGURE.1 The -componen of displacemen is (a) posiie; (b) negaie. (a) Posiie displacemen when he person moes in he posiie direcion i, Ai m d A 5 1. m (m) f, Af m means he posiion is a fncion of clock reading (spoken of ), no mliplied by. When we need o noe a specific ale of posiion a a specific clock reading 1, insead of wriing 1 1, we will wrie 1, keeping in mind ha 1 is a shorc for ime 1. The same applies o i, f, ec. The ecor ha poins from he iniial posiion i o he final posiion f is he displacemen. The qaniy ha we deermine hrogh he operaion f - i is called he -scalar componen of he displacemen ecor and is abbreiaed d (sally we drop he erm scalar and js call his he -componen of he displacemen). Figre.1a shows ha he iniial posiion of person A is Ai = +3. m and he final posiion is Af = +5. m; hs he -componen of he person s displacemen is d A = Af - Ai = 1+5. m m = +. m The displacemen is posiie since he person moed in he posiie -direcion. In Figre.1b, person B moed in he negaie direcion from he iniial posiion of +5. m o he final posiion of +3. m; hs he -componen of displacemen of he person is negaie: (b) Negaie displacemen when he person moes in he negaie direcion d B 5. m d B = Bf - Bi = 1+3. m m = -. m Disance is always posiie, as i eqals he absole ale of he displacemen f - i. In he eample aboe, he displacemens for A and B are differen, b he disances are boh +. m f, Bf m (m) i, Bi m FIGURE.13 Consrcing a kinemaics posiion-erss-ime graph. ignifican digis Noe ha in Figre.1 he posiions were wrien as +3. m and +5. m. Cold we hae wrien hem insead as +3 m and +5 m, or as +3. m and +5. m? The hickness of a hman body from he back o he fron is abo. m 1 cm. Ths, we shold be able o measre he person s locaion a one insan of ime o wihin abo.1 m b no o.1 m (1 cm). Ths, he locaions can reasonably be gien as +3. m, which implies an accracy of {.1 m. (For more on significan digis, see Chaper 1.) (a) s s 4 s 6 s (b) (m) 1 8 (m) REVIEW QUETION.4 Jade wen hiking beween wo camps ha were separaed by abo 1 km. he hiked approimaely 16 km o ge from one camp o he oher. Translae 1 km and 16 km ino he langage of physical qaniies Represening moion wih daa ables and graphs (c) (m) Trendline (s) (s) o far, we hae learned how o represen linear moion wih moion diagrams. In his secion, we will learn o represen linear moion wih daa ables and graphs. Imagine yor friend (he objec of ineres) walking across he fron of yor classroom. To record her posiion, yo drop a beanbag on he floor a her posiion each second (Figre.13a). The floor is he objec of reference. The origin of he coordinae sysem is 1. m from he firs beanbag, and he posiion ais poins in he direcion of moion. Table. shows each bag s posiion and he corresponding clock reading. Do yo see a paern in he able s daa? One way o deermine if here is a paern is o plo he daa on a graph (Figre.13b). This graph is called a kinemaics posiion-erss-ime graph. In physics, he word kinemaics means descripion of moion. Kinemaics graphs conain more precise informaion abo an objec s moion han moion diagrams can.

11 .5 Represening moion wih daa ables and graphs 3 Time is sally considered o be he independen ariable, as ime progresses een if here is no moion, so he horizonal ais will be he -ais. Posiion is he dependen ariable (posiion changes wih ime), so he erical ais will be he -ais. Plo he daa in each row in Table. on he aes. Each poin on he horizonal ais represens a ime (clock reading). Each poin on he erical ais represens he posiion of a beanbag. When we draw lines hrogh hese poins and perpendiclar o he aes, hey inersec a a single locaion a do on he graph ha simlaneosly represens a ime and he corresponding posiion of he objec. This do is no a locaion in real space b raher a represenaion of he posiion of he beanbag a a specific ime. Is here a rend in he locaions of he dos on he graph? We see ha he posiion increases as he ime increases. This makes sense. We can draw a smooh bes-fi cre ha passes as close as possible o he daa poins a rendline (Figre.13c). I looks like a sraigh line in his pariclar case he posiion is linearly dependen on ime. Correspondence beween a moion diagram and posiion-erss-ime graph To ndersand how graphs relae o moion diagrams, consider he moion represened by he daa in Table. and in Figre.13c. Figre.14 shows a modified moion diagram for he daa in Table. (he do imes are shown and he D arrows hae been remoed for simpliciy) and he corresponding posiion-erss-ime graph. The posiion of each do on he moion diagram corresponds o a poin on he posiion ais. The graph line combines he informaion abo he posiion of an objec and he clock reading when his posiion occrred. Noe, for eample, ha he = 4. s do on he moion diagram a posiion = 7.6 m is a 7.6 m on he posiion ais. The corresponding do on he graph is a he inersecion of he erical line passing hrogh 4. s and he horizonal line passing hogh 7.6 m. The role of a reference frame Always keep in mind ha represenaions of moion (moion diagrams, ables, kinemaics graphs, eqaions, ec.) depend on he reference frame chosen. Le s look a he represenaions of he moion of a cyclis sing wo differen reference frames. TABLE. Time-posiion daa for linear moion Clock reading (ime) Posiion =. s = 1. m 1 = 1. s 1 =.4 m =. s = 4.13 m 3 = 3. s 3 = 5.5 m 4 = 4. s 4 = 7.6 m 5 = 5. s 5 = 8.41 m 6 = 6. s 6 = 1. m The qaniy ha appears on TIP he erical ais of a graph can represen he posiion of an objec whose acal posiion is changing along any ais. The posiion on he erical ais does no mean he objec is moing in he erical direcion. FIGURE.14 Correspondence beween a moion diagram and he posiion-erss-ime graph (m) s s 4 s 6 s Each do on he moion diagram becomes a poin on he graph A 5 4 s he objec s posiion m. (s) (m) CONCEPTUAL EXERCIE.3 Effec of reference frame on moion descripion Two obserers each se differen reference frames o record he changing posiion of a bicycle rider. Boh reference frames se Earh as he objec of reference, b he origins of he coordinae sysems and he direcions of he -aes are differen. The daa for he cyclis s rip are TABLE.3 Time-posiion daa for cyclis when sing reference frame 1 Clock reading (ime) Posiion =. s = 4. m 1 = 1. s 1 = 3. m =. s =. m 3 = 3. s 3 = 1. m 4 = 4. s 4 =. m presened in Table.3 for obserer 1 and in Table.4 for obserer. kech a moion diagram and a posiion-erss-ime graph for he moion when sing each reference frame. TABLE.4 Time-posiion daa for cyclis when sing reference frame Clock reading (ime) Posiion =. s =. m 1 = 1. s 1 = 1. m =. s =. m 3 = 3. s 3 = 3. m 4 = 4. s 4 = 4. m (conined)

12 4 CHAPTER Kinemaics: Moion in One Dimension kech and ranslae According o Table.3, he obserer in reference frame 1 sees he cyclis (he objec of ineres) a ime =. s a posiion = 4. m and a 4 = 4. s a posiion 4 =. m. Ths, he cyclis is moing in he negaie direcion relaie o he coordinae ais in reference frame 1. Meanwhile, according o Table.4, he obserer in reference frame sees he cyclis a ime =. s a posiion =. m and a ime 4 = 4. s a posiion 4 = 4. m. Ths, he cyclis is moing in he posiie direcion relaie o reference frame. Posiion-erss-ime graph for obserer 1 Posiion-erss-ime graph for obserer Reference frame 1 Try i yorself A hird obserer recorded he ales (in Table.5) for he ime and posiion of he same cyclis. Describe he reference frame of his obserer. Reference frame implify and diagram ince he size of he cyclis is small compared o he disance he is raeling, we can represen him as a poin-like objec. The moion diagram for he cyclis is he same for boh obserers, as hey are sing he same objec of reference. Using he daa in he ables, we plo kinemaics posiion-erss-ime graphs for each obserer a op righ. Alhogh he graph for obserer 1 looks ery differen from he graph for obserer, hey represen he same moion. The graphs look differen becase he reference frames are differen. TABLE.5 Daa colleced by he hird obserer Answer Clock reading (ime) Posiion =. s =. m 1 = 1. s 1 = -. m =. s = -4. m 3 = 3. s 3 = -6. m 4 = 4. s 4 = -8. m The poin of reference cold be anoher cyclis moing in he opposie posiie direcion from he direcion in which he firs cyclis is raeling, wih each coering he same disance relaie o he grond dring he same ime ineral. REVIEW QUETION.5 A posiionerss-ime graph represening a moing objec is shown in Figre.15. Wha are he posiions of he objec a clock readings.6 s and 5.8 s? FIGURE.15 A posiion-erss-ime graph represening a moing objec. (m) (s) FIGURE.16 Posiions of cars A and B a s and 5. s. 5. s A s A Finish line A 5 1. m A m (m) 5. s s B B (m) B 5 1. m B m.6 Consan elociy linear moion In he las secion we deised a graphical represenaion of moion. Here we will connec graphs o mahemaical represenaions sing he eample of wo moorized oy cars racing oward a finish line. A ime hey are ne o each oher, b car B is moing faser han car A and reaches he finish line firs (Figre.16). The daa ha we collec are shown in Obseraional Eperimen Table.6. Earh is he objec of reference. The origin of he coordinae sysem (he poin of reference) is 1. m o he lef of he posiion of he cars a =. The posiie -direcion poins righ in he direcion of he cars moions. Now le s se he daa o find a paern.

13 .6 Consan elociy linear moion 5 OBERVATIONAL EXPERIMENT TABLE.6 Graphing he moion of cars Obseraional eperimen Daa for car A Daa for car B We graph he daa wih he goal of finding a paern. The =. s = 1. m =. s = 1. m rendlines for boh cars are 1 = 1. s 1 = 1.4 m 1 = 1. s 1 = 1.9 m sraigh lines. The line for =. s = 1.9 m =. s = 3. m car B has a bigger angle wih 3 = 3. s 3 =.5 m 3 = 3. s 3 = 3.9 m he ime ais han he line 4 = 4. s 4 =.9 m 4 = 4. s 4 = 5. m for car A. 5 = 5. s 5 = 3.5 m 5 = 5. s 5 = 6. m Analysis (m) Car B 3 Car A (s) Paern I looks like a sraigh line is he simples reasonable choice for he bes-fi cre in boh cases (he daa poins do no hae o be eacly on he line). In Table.6, he slope of he line represening he moion of car B is greaer han he slope of he line represening he moion of car A. Wha is he physical meaning of his slope? In mahemaics he ale of a dependen ariable is sally wrien as y and depends on he ale of an independen ariable, sally wrien as. A fncion y1 = f 1 is an operaion ha one needs o do o as an inp o hae y as he op. For a sraigh line, he fncion y1 is y1 = m + b, where m is he slope and b is he y-inercep he ale of he y when =. In he case of he cars, he independen ariable is ime and he dependen ariable is posiion. The eqaion of a sraigh line becomes 1 = m + b, where b is he -inercep of he line, and m is he slope of he line. The -inercep is he -posiion when =, also called he iniial posiion of he car. Boh cars sared a he same locaion: A = B = 1. m. To find he slope m of a sraigh line, we can choose any wo poins on he line and diide he change in he erical qaniy (D in his case) by he change in he horizonal qaniy (D in his case): m = For eample, for car A he slope of he line is m A = The slope of he line for car B is m B = 3.5 m - 1. m 5. s -. s 6. m - 1. m 5. s -. s = D D = +.5 m>s = +1. m>s Now we hae all he informaion we need o wrie mahemaical eqaions ha describe he moion of each of he wo cars: Car A: A = 1+.5 m>s m Car B: B = 1+1. m>s m Noice ha he nis of he slope are meers per second. The slope indicaes how he objec s posiion changes wih respec o ime. The slope of he line conains more informaion han js how fas he car is going. I also ells s he direcion of moion relaie o he coordinae ais.

14 6 CHAPTER Kinemaics: Moion in One Dimension FIGURE.17 The sign of he slope indicaes he direcion of moion. (a) (b) (m) The posiionerss-ime graph for car D has negaie slope. Car C 18 m m C 5 4 s 5 1 m/s (8 m) m D 5 4 s 5 m/s Car D The posiionerss-ime graph for car C has posiie slope. (s) Consider he moions represened graphically in Figre.17a. The slope of he posiion-erss-ime graph for car C is + m>s, b he slope of he posiion- erssime graph for car D is - m>s. Wha is he significance of he mins sign? Car C is moing in he posiie direcion, b car D is moing in he negaie direcion (check he moion diagrams in Figre.17b). The magnides of he slopes of heir posiion-erss-ime graphs are he same, b he signs are differen. Ths, in addiion o he informaion abo how fas he car is raeling (is speed), he slope ells in wha direcion i is raeling. Togeher, speed and direcion are called elociy, and his is wha he slope of a posiion-erss-ime graph represens. Yo are already familiar wih he erm elociy arrow sed on moion diagrams. Now yo hae a formal definiion for elociy as a physical qaniy. Velociy and speed for consan elociy linear moion For consan elociy linear moion, he componen of elociy along he ais of moion can be fond as he slope of he posiion-erss-ime graph or he raio of he componen of he displacemen of an objec - 1 dring any ime ineral - 1 : = = D D where he sbscrip ne o he symbol indicaes he direcion of he ais and 1 and are he wo clock readings when he posiions 1 and were measred. Eamples of nis of elociy are m>s, km>h, and mi>h (which is ofen wrien as mph). peed is he magnide of he elociy and is always a posiie nmber. (.1) 8 18 (m) Eqaion (.1) allows yo o se TIP any change in posiion diided by he ime ineral dring which ha change occrred o obain he same nmber as long as he posiion-erss-ime graph is a sraigh line (he objec is moing a consan elociy). Laer in he chaper, yo will learn how o modify his eqaion for cases in which he elociy is no consan. Noe ha elociy is a ecor qaniy. In ecor form, moion a consan elociy is = d > D, where d is he displacemen ecor and D is he ime ineral. Here we diide a ecor by a scalar. Yo learned his operaion in ecion.3. The elociy ecor has he same direcion as he displacemen ecor (direcion of moion), b he magnide is differen as well he nis. The magnide of he elociy ecor is speed. The displacemen ecor has nis of lengh (m), and he ime ineral has nis of ime (s). Ths he elociy ecor has he nis of m/s. ince i is difficl o operae mahemaically wih ecors, we will work wih componens. When he elociy ecor poins in he posiie direcion of he -ais, he -componen of elociy is eqal o he magnide of he elociy ecor. When he elociy ecor poins in he negaie direcion of he ais, he -componen of elociy is eqal o he magnide of he elociy ecor wih he negaie sign. Eqaion of moion for consan elociy linear moion We can rearrange Eq. (.1) ino a form ha allows s o deermine he posiion of an objec a ime knowing only is posiion a ime 1 and he -componen of is elociy: = If we apply his eqaion for ime 1 = when he iniial posiion is, hen he posiion a any laer posiion ime can be wrien as follows: Posiion eqaion for consan elociy linear moion = + (.) where is he fncion 1, posiion is he posiion of he objec a ime = wih respec o a pariclar reference frame, and he (consan) -componen of he elociy of he objec is he slope of he posiion-erss-ime graph.

15 .6 Consan elociy linear moion 7 Below yo see a new ype of ask a Qaniaie Eercise. Qaniaie Eercises inclde wo seps of he problem-soling process: Represen mahemaically and ole and ealae. Their prpose is o help yo pracice sing new eqaions righ away. QUANTITATIVE EXERCIE.4 A cyclis In Concepal Eercise.3, yo consrced graphs for he moion of a cyclis sing wo differen reference frames. Now consrc mahemaical represenaions (eqaions) for he cyclis s moion for each of he wo graphs. Do he eqaions indicae he same posiion for he cyclis a ime = 6. s? Represen mahemaically The cyclis moes a consan elociy; hs he general mahemaical descripion of his moion is = +, where = ole and ealae Using he graph for reference frame 1 in Concepal Eercise.3, we see ha he cyclis s iniial posiion is = +4 m. The elociy along he -ais (he slope of he graph) is = m - 4 m = -1 m>s 4 s - s The mins sign indicaes ha he elociy poins in he negaie -direcion (oward he lef) relaie o ha ais. The moion of he bike wih respec o reference frame 1 is described by he eqaion = + = 4 m m>s Using he graph for reference frame, we see ha he cyclis s iniial posiion is = m. The -componen of he elociy along he ais of moion is = 4 m - m 4 s - s = +1 m>s The posiie sign indicaes ha he elociy poins in he posiie -direcion (oward he lef). The moion of he bike relaie o reference frame is described by he eqaion = + = m + 11 m>s The posiion of he cyclis a ime 1 = 6 s wih respec o reference frame 1 is = 4 m m>s16 s = - m Wih respec o reference frame : = m m>s16 s = +6 m How can he posiion of he cyclis be boh - m and +6 m? Remember ha he descripion of moion of an objec depends on he reference frame. If yo p a do on coordinae ais 1 a he - m posiion and a do on coordinae ais a he +6 m posiion, yo find ha he dos are in fac a he same locaion, een hogh ha locaion corresponds o a differen posiion in each reference frame. Boh descripions of he moion are correc and consisen, b each one is meaningfl only wih respec o he corresponding reference frame. (m) Reference frame Reference frame (m) Try i yorself Use he daa for he moion of he cyclis as seen by he hird obserer in he Try i yorself par of Concepal Eercise.3 o wrie he eqaion of moion. Why is he magnide of he cyclis s elociy differen han he 1 m>s in he eample aboe? Answer = 1 m + 1- m>s. The obserer is moing wih respec o Earh a he same speed in he direcion opposie o he cyclis. Below yo see anoher new ype of ask a worked eample. The worked eamples inclde all for seps of he problem-soling sraegy we se in his book. (ee Chaper 1 for descripions of hese seps.) EXAMPLE.5 Yo chase yor siser Yor yong siser is rnning a. m>s oward a md pddle ha is 6. m in fron of her. Yo are 1. m behind her rnning a 5. m>s o cach her before she jmps ino he md. Will she need a bah? kech and ranslae We sar by drawing a skech of wha is happening. Yor siser and yo are wo objecs of ineres. Ne, we choose a reference frame wih Earh as he objec of reference. The origin of he coordinae sysem is yor iniial posiion and he posiie direcion is oward he righ, in he direcion ha yo boh rn, as shown a righ. Noice he sbscrips for he gien qaniies. The firs sbscrip indicaes he objec, and he second indicaes he ime. For eample, Y means yor posiion a he zero clock reading. (conined)

16 8 CHAPTER Kinemaics: Moion in One Dimension We can now mahemaically describe he posiions and elociies of yo and yor siser a he beginning of he process. The iniial clock reading is zero a he momen ha yo are a he origin. Noe ha yo were boh rnning before ime zero; his js happened o be he ime when we sared analyzing he process. We wan o know he ime when yo and yor siser are a he same posiion. This will be he posiion where yo cach p o her. implify and diagram We assme ha yo and yor siser are poinlike objecs. To skech graphs of he moions, find yor siser s posiion a 1 s by mliplying her speed by 1 s and adding i o her iniial posiion. Do his for and 3 s as well. Plo hese ales on a graph for he corresponding clock readings (1 s, s, 3 s, ec.) and draw a line ha eends hrogh hese poins. Repea his process for yorself. Rearrange he aboe o deermine he ime when yo are boh a he same posiion: 1. m>s m>s = 1. m m 1-3. m>s = -11. m = s The s nmber prodced by or calclaor has many more significan digis han he giens. hold we rond i o hae he same nmber of significan digis as he gien qaniies? The rle of hmb is ha if i is he final resl, yo need o rond his nmber o 3.3, as he answer canno be more precise han he gien informaion. Howeer, we do no rond he resl of an inermediae calclaion. We se he resl as is o calclae he ne qaniy needed o ge he final answer and hen rond he final resl. iser: 1 = 11. m + 1. m>s s = 16.7 m Yo: Y 1 = 1. m m>s s = 16.7 m Represen mahemaically Use Eq. (.) o consrc mahemaical represenaions of moion. The form of he eqaion is he same for boh 1 = + ; howeer, he ales for he iniial posiions and he componens of he elociies along he ais for yor siser, = +, and for yo, Y = Y + Y, are differen. Now we can se he ales of he qaniies from he skech. iser: = 11. m + 1. m>s Yo: Y = 1. m m>s From he graphs, we see ha he disance beween yo and yor siser is shrinking wih ime. Do he eqaions ell he same sory? For eample, a ime =. s, yor siser is a posiion and yo are a 1 s = 11. m + 1. m>s1. s = 14. m Y 1 s = 1. m+15. m>s1. s = 1. m Yo are caching p o yor siser. ole and ealae The ime a which he wo of yo are a he same posiion can be fond by seing 1 = Y 1: 11. m + 1. m>s = 1. m m>s Noe ha if yo sed he ronded nmber 3.3 s, yo wold ge 16.6 m for yor siser and 16.5 m for yo. These nmbers are slighly smaller han he resl calclaed aboe. Howeer, for or prposes i does no maer, as he goal of his eample was o decide if yo cold cach yor siser before she reaches he pddle. ince yo cagh her a a posiion of abo 16.7 m, wih he ncerainy of abo.1 m, his posiion is slighly greaer han he 16.-m disance o he pddle. Therefore, yor siser reaches he pddle before yo and will indeed need a bah. This answer seems consisen wih he graphical represenaion of he moion shown aboe. Noe ha we sed he symbol o denoe he specific clock reading when yo cach yor siser. Ofen, he same symbol is sed o denoe boh a ariable and some specific ale. Always ask yorself wha a symbol means when yo read i. Try i yorself Describe he problem siaion sing a reference frame wih yor siser (no Earh) as he objec and poin of reference and he posiie direcion poining oward he pddle. Answer Wih respec o his reference frame, yor siser is a posiion and a res; yo are iniially a -1. m and moing oward yor siser wih elociy +3. m>s; and he md pddle is iniially a +6. m and moing oward yor siser wih elociy -. m>s. TIP In he reference frame we chose in Eample.5, he posiions of yo and yor siser are always posiie, as are he componens of yor elociies. Also, yor iniial posiion is zero. Ths he calclaions are he easies. Ofen he descripion of he moion of objec(s) will be simples in one pariclar reference frame. There is no one general rle for finding sch a frame; howeer, i is helpfl o consider which frame will resl in posiie ales for he mos qaniies.

17 .6 Consan elociy linear moion 9 Graphing elociy o far, we hae learned o make posiion-erss-ime graphs. We cold also consrc a graph of an objec s elociy as a fncion of ime. Consider Eample.5, in which yo chase yor siser. Again we will se Earh as he objec of reference. Yo are moing a a consan elociy whose -componen is = +5. m>s. For yor siser, he -componen of elociy is +. m>s. We place clock readings on he horizonal ais and he -componen of yor and yor siser s elociies on he erical ais; hen we plo poins for hese elociies a each ime (see Figre.18a). The bes-fi cre for each person is a horizonal sraigh line, which makes sense since neiher elociy is changing. For yo, he eqaion of he bes-fi line is Y 1 = +5. m>s, and for yor siser, i is 1 = +. m>s, where 1 represens he -componen of elociy as a fncion of ime. If insead we choose yor siser as he objec of reference, her elociy wih respec o herself is zero, so he bes-fi cre is again a horizonal line b a a ale of. m>s insead of +. m>s; yor elociy is +3. m>s (see Figre.18b); and he md s elociy is -. m>s. The mins sign indicaes ha from yor siser s poin of iew, he md is moing in he negaie direcion oward her a speed +. m>s. Finding displacemen from a elociy graph We hae js learned o consrc a elociy-erss-ime graph. Can we ge anyhing more o of sch graphs besides being able o represen elociy graphically? As yo know, for consan elociy linear moion, he posiion of an objec changes wih ime according o = +. Rearranging his eqaion a bi, we ge - =, where is he ime ineral dring which he moion occrred, assming ha he moion sared a zero clock reading 1 =. The lef side is he displacemen of he objec from ime zero o ime. Now look a he righ side: is he erical heigh of he elociy-erss-ime graph line and is he horizonal widh from ime zero o ime (see Figre.19). We can inerpre he righ side as he shaded area beween he elociy-erss-ime graph line and he ime ais. In eqaion - =, his area (he righ side) eqals he displacemen of he objec from ime zero o ime on he lef side of he eqaion. Here he displacemen is a posiie nmber becase we chose an eample wih posiie elociy. Le s eend his reasoning o more general cases: an objec iniially a posiion 1 a ime 1 and laer a posiion a ime and moing in eiher he posiie or negaie direcion. FIGURE.18 Velociy-erss-ime graphs. (a) -erss- graph lines wih Earh as objec of reference (m/s) Yo iser 1 Md 1 3 (b) -erss- graph lines wih iser as objec of reference (m/s) Yo iser 1 3 Md (s) (s) Noice ha a horizonal line on a TIP posiion-erss-ime graph means ha he objec is a res (he posiion is consan wih ime). A horizonal line on a elociy-erss-ime graph means ha he objec is moing a consan elociy (is elociy does no change wih ime). FIGURE.19 Using he -erss- graph o deermine displacemen -. (m/s) Displacemen is he area beween a elociy-erss-ime graph line and he ime ais For moion wih consan elociy, he magnide of he displacemen - 1 (he disance raeled) of an objec dring a ime ineral from 1 o is he area beween a elociy-erss-ime graph line and he ime ais beween hose wo clock readings. The displacemen is he area wih a pls sign when he elociy is posiie and he area wih a negaie sign when elociy is negaie. Area 5 3 ( ) 5 (s) 5 An objec s displacemen beween 5 and ime ineral is he area beween he -erss- cre and he ais. QUANTITATIVE EXERCIE.6 Displacemen of yo and yor siser Use he elociy-erss-ime graphs shown in Figre.18a for yo and yor siser (see Eample.5) o find yor displacemens wih respec o Earh for he ime ineral from o 3. s. The elociy componens relaie o Earh are +. m>s for yor siser and +5. m>s for yo. Represen mahemaically For consan elociy moion, he objec s displacemen - is he area of a recangle whose erical side eqals he objec s elociy and whose horizonal side eqals he ime ineral - = - dring which he moion occrred. ole and ealae The displacemen of yor siser heading oward he md pddle dring he 3.@s ime ineral is he prodc of her elociy and he ime ineral: d = 1 - = 1+. m>s13. s = +6. m (conined)

18 3 CHAPTER Kinemaics: Moion in One Dimension he was originally a posiion 1. m, so she is now a posiion m = +16. m. Yor displacemen dring ha same 3.@s ime ineral is he prodc of yor elociy and he ime ineral: d Y = 1 - Y = 1+5. m>s13. s = +15. m Yo were originally a. m, so yo are now a +15. m, 1. m behind yor siser. Try i yorself Deermine he magnides of he displacemens of yo and yor siser from ime o ime. s and yor posiions a ha ime. Yor iniial posiion is zero and yor siser s is 1 m. Answer Yor siser s ales are d = 4. m and = 14. m, and yor ales are dy = 1. m and Y = 1. m. he is 4. m ahead of yo a ha ime. REVIEW QUETION.6 Why is he following saemen re? Displacemen is eqal o he area beween a elociy-erss-ime graph line and he ime ais wih a posiie or negaie sign. FIGURE. Velociy-erss-ime graph for moion wih changing elociy. 1 Velociy a ime 1 () Velociy a ime 1 omeimes we se aerage speed TIP insead of aerage elociy. To find he aerage speed diide he oal pah lengh by he ime of rael. If he objec is no changing is direcion of moion, he aerage speed is eqal o he magnide of he aerage elociy. FIGURE.1 Dropping a meal ball ono a foam block from differen heighs..7 Moion a consan acceleraion In he las secion, he fncion 1 was a horizonal line on he elociy-erss-ime graph becase he elociy was consan. How wold he graph look if he elociy were changing? One eample of sch a graph is shown in Figre.. A poin on he cre indicaes he elociy of he objec shown on he erical ais a a pariclar ime shown on he horizonal ais. In his case, he elociy is coninally changing and is posiie. Insananeos elociy and aerage elociy The elociy of an objec a a pariclar ime is called he insananeos elociy. Figre. shows a elociy-erss-ime graph for moion wih coninally changing insananeos elociy. When an objec s elociy is changing, we canno se Eq. (.1) o deermine is insananeos elociy sing any ime ineral and a corresponding posiion change, becase he raio = = D D is no he same for differen ime inerals he way i was when he objec was moing a consan elociy. We can sill se his eqaion o deermine he aerage elociy, which is he raio of he change in posiion and he ime ineral dring which his change occrred. For moion a consan elociy, he insananeos and aerage elociy are eqal; for moion wih changing elociy, hey are no. Howeer, if he wo poins chosen are ery close ogeher, D and D are boh ery small, b he raio D> D is no necessarily large or small. In fac, wihin a anishingly small ime ineral, we can se D> D as he definiion of he -componen of insananeos elociy. When an objec moes wih changing elociy, is elociy can change qickly or slowly. To characerize he rae a which he elociy of an objec is changing, we need a new physical qaniy. Acceleraion Le s sar wih a simple eperimen. We ake a meal ball and drop i ono a foam block ha deforms when oched. The phoos in Figre.1 show ha he den in he block becomes deeper when he ball is dropped from a larger heigh. Can his resl be de o a change in he elociy of he ball as i is falling? Tha is, does a longer fall mean a

19 .7 Moion a consan acceleraion 31 greaer elociy and hs a bigger den? We need o collec daa o answer his qesion. The firs wo rows of Table.7 show posiion and ime daa for he falling ball: TABLE.7 Posiion and ime daa for a falling ball 1s y 1m * 1s a 1m,s From he able yo can see ha he disance ha he ball raels dring each sccessie ime ineral increases; Figre. shows he posiion of he ball eery.1 s. If he disance ha he ball raels dring each sccessie ineral increases, he elociy ms increase, oo. To deermine he aerage elociy dring each ime ineral, we calclae he displacemen of he ball beween consecie imes and hen diide by he ime ineral. For eample, he aerage elociy beween.1 s and. s is m -.49 m>1. s -.1 s = 1.47 m>s; i is posiie becase we poined he ais down. We can hink of his aerage elociy as he insananeos elociy ha he ball had in he middle of he ime ineral in his case, he.15 clock reading. We can now p hese calclaed elociies a and he associaed clock readings * in he middle of each ime ineral in he wo boom rows of he able. These daa now allow s o plo a elociy-erss-ime graph (Figre.3). Noice ha he bes-fi cre for hese daa is a sraigh line. This occrs when he elociy of he objec increases or decreases a a consan rae, ha is, by he same amon dring he same ime ineral. In or case, he elociy increases by 9.8 m>s eery second in he downward direcion. FIGURE.3 A elociy-erss-ime graph for a falling ball. y (m/s) FIGURE. The posiion of a falling ball eery.1 s.. m.49 m.196 m y.441 m.784 m 1.5 m D y 1 lope: m/s D (s) The physical qaniy ha characerizes how fas insananeos elociy is changing is eqal o he slope of he elociy-erss-ime graph and is called acceleraion a. The ecor symbol aboe his qaniy indicaes ha i has direcion, commnicaed by he sign of he slope of he elociy-erss-ime graph for one-dimensional moion. imilar o elociy, he acceleraion of an objec moing in a sraigh line along an -coordinae ais is defined as a = = D D We see ha he nis of acceleraion based on he definiion of acceleraion are m>s = m s s # = m s s. In he case of a small falling ball, we fond ha is acceleraion is abo 9.8 m>s, which means ha is elociy increases by 9.8 m>s eery second. To beer ndersand acceleraion, hink of a cyclis and a car riding along a road a increasing speeds (Figres.4a and b, on he ne page). The acceleraions of boh objecs are consan (as seen from he analysis of he slopes of heir elociy-erss-ime graphs) b no he same; he car has a greaer acceleraion. If he car were slowing down,

20 3 CHAPTER Kinemaics: Moion in One Dimension FIGURE.4 Velociy-erss-ime graph when he elociy is changing a a consan rae. (a) A moing cyclis The elociy in he posiie direcion is increasing a a consan rae. D 1 D 1 D 5 D 1 D 5 D 1 (b) A moing car The elociy in he posiie direcion is increasing a a greaer rae han in (a). D 1 D 1 D (c) A car slowing down The elociy is in he posiie direcion (. and is aboe he ais). The acceleraion is negaie (a, ) since is decreasing and is () graph has a negaie slope. D is elociy- erss-ime graph wold insead hae a negaie slope, which corresponds o a decreasing speed and a negaie acceleraion, as is smaller han 1 (Figre.4c). Howeer, i is no necessary for an objec moing wih negaie acceleraion o be slowing down. Eamine Figre.4d. The car has a negaie acceleraion and i is speeding p! This happens when he objec is moing in he negaie direcion and has a negaie componen of elociy, b is speed in he negaie direcion is increasing in magnide. Becase elociy is a ecor qaniy and he acceleraion shows how qickly he elociy changes as ime progresses, acceleraion is also a ecor qaniy. We can define acceleraion in a more general way. The aerage acceleraion of an objec dring a ime ineral is he following: a = = D D To deermine he acceleraion, we need o deermine he elociy change ecor D = - 1. This eqaion inoles he sbracion of ecors, b yo can hink of i in erms of addiion by rearranging i o be 1 + D =. Noe ha D is he ecor ha we add o 1 o ge (Figre.5). We did his when making moion diagrams in ecion.3, only hen we were no concerned wih he eac lenghs of he ecors. The acceleraion ecor a = D > D is in he same direcion as he elociy change ecor D, as he ime ineral D is a scalar qaniy. The nis of he acceleraion ecor are deermined by diiding he nis of he elociy change ecor by he nis of ime. Acceleraion An objec s aerage acceleraion dring a ime ineral D is he change in is elociy D diided by ha ime ineral: a = = D D (.3) If D is ery small, hen he acceleraion gien by his eqaion is he insananeos acceleraion of he objec. For one-dimensional moion, he componen of he aerage acceleraion along a pariclar ais (for eample, for he -ais) is (d) A car speeding p The elociy is in he negaie direcion (below he ais). The acceleraion is negaie (a, ) since is increasing in he negaie direcion and is () graph has a negaie slope. FIGURE.5 How o deermine he change in elociy D. 1 We add D o 1 o ge. D 1 D 3 a = The nis of acceleraion are 1m>s>s = m>s. = D D Noe ha if an objec has an acceleraion of +6 m>s, i means ha is elociy changes by +6 m>s in 1 s, or by +1 m>s in s 31+1 m>s>1 s = +6 m>s 4. I is possible for an objec o hae a zero elociy and nonzero acceleraion for eample, a he momen when an objec sars moing from res. An objec can also hae a nonzero elociy and zero acceleraion for eample, an objec moing a consan elociy. Noe ha he acceleraion of an objec depends on he obserer. For eample, a car is acceleraing for an obserer on he grond b is no acceleraing for he drier of ha car. If i is difficl for yo o hink abo elociy and acceleraion in absrac TIP erms, ry calclaing he acceleraion for simple ineger elociies. Deermining elociy change from acceleraion If, a = for linear moion, he -componen of he elociy of some objec is and is acceleraion a is consan, hen is elociy a a laer ime can be deermined by sbsiing hese qaniies ino Eq. (.4): a = - - (.4)

21 .7 Moion a consan acceleraion 33 Rearranging, we ge an epression for he changing insananeos elociy of he objec as a fncion of ime: = + a (.5) Here he ale of he elociy a any clock reading is he insananeos elociy. For one-dimensional moion, he direcions of he ecor componens a,, and are indicaed by heir signs relaie o he ais of moion posiie if in he posiie -direcion and negaie if in he negaie -direcion. EXAMPLE.7 Bicycle ride ppose ha yo are siing on a bench waching someone riding a bicycle on a fla, sraigh road. In a.-s ime ineral, he elociy of he bicycle changes from -4. m>s o -7. m>s. Describe he moion of he bicycle as flly as possible. kech and ranslae We can skech he process as shown below. The bicycle (he objec of ineres) is moing in he negaie direcion wih respec o he chosen reference frame. The componens of he bicycle s elociy along he ais of moion are negaie: = -4. m>s a ime =. s and = -7. m>s a =. s. The speed of he bicycle (he magnide of is elociy) increases. I is moing faser in he negaie direcion. We can deermine he acceleraion of he bicycle and describe he changes in is elociy. Represen mahemaically We apply Eq. (.4) o deermine he acceleraion: a = - - ole and ealae bsiing he gien elociies and imes, we ge 1-7. m>s m>s a = = -1.5 m>s. s -. s The bicycle s -componen of elociy a ime was -4. m>s. Is elociy was changing by -1.5 m>s each second. o 1 s laer, is elociy was 1s = + D = 1-4. m>s m>s = -5.5 m>s Dring he second 1-s ime ineral, he elociy changed by anoher -1.5 m>s and was hen m>s m>s = 1-7. m>s. In his eample, he bicycle was speeding p by 1.5 m>s each second in he negaie direcion. implify and diagram Or skech of he moion diagram for he bicycle is shown below. Noe ha D poins in he negaie -direcion. Try i yorself A car s acceleraion is -3. m>s. A ime is elociy is +14 m>s. Wha happens o he elociy of he car? Wha is is elociy afer 3 s? Answer The car s elociy in he posiie -direcion is decreasing. Afer 3 s, = 5. m>s. I is possible for an objec o hae a posiie acceleraion and be slowing TIP down as well as o hae a negaie acceleraion and be speeding p. When an objec is speeding p, he acceleraion ecor is in he same direcion as he elociy ecor, and he elociy and acceleraion componens along he same ais hae he same sign. When an objec is slowing down, he acceleraion is in he opposie direcion relaie o he elociy; heir componens hae opposie signs. FIGURE.6 A feaher and an apple fall side by side in a acm. Free fall Le s rern o he eample ha led s o he discoery of acceleraion: dropping a meal ball ono a foam block (Figre.1). Galileo Galilei ( ) did eperimens similar o ors in which he dropped objecs from differen heighs and eamined he resling dens in sand. Howeer, he also noiced ha objecs of differen shapes fall differenly for eample, a piece of paper falls more slowly han a heay brick. He hypohesized ha his difference was de o air resisance. If his hypohesis were correc, hen in a acm (space deoid of air), all objecs wold fall wih he same acceleraion. Indeed, in a large acm chamber, when a ligh feaher and a heay apple fall, hey are always ne o each oher, meaning ha hey hae he same elociy and acceleraion a eery insan (Figre.6). Yo can perform his esing eperimen,

22 34 CHAPTER Kinemaics: Moion in One Dimension FIGURE.7 Moion for an pward hrown ball. (a) y Going p Going down (b) lope of angen line is y (m) a he highes poin. lope of angen line 5 y (c) y (m/s) D D Velociy is zero a he highes poin. (s) (s) oo: ake a heay ball and a ennis ball, drop hem a he same ime from a heigh of abo m, and ideoape he process. Yo will see hem moing side by side! Earlier in his secion we fond ha he acceleraion of he falling meal ball was abo 9.8 m>s. A he same locaion, wiho air resisance, all falling objecs will hae his same acceleraion; howeer, a anoher locaion on Earh, he ale of acceleraion migh be slighly differen. We will learn why in Chaper 5. If we ideoape a small objec hrown pward and hen se he daa o consrc a elociy-erss-ime graph, we find ha is acceleraion is sill he same a all clock readings. For an pward-poining ais, he objec s acceleraion is -9.8 m>s on he way p, -9.8 m>s on he way down, and een -9.8 m>s a he insan when he objec is momenarily a res a he highes poin of is moion. A all imes dring he objec s fligh, is elociy is changing a a rae of -9.8 m>s each second. A moion diagram and he graphs represening he posiion-erss-ime, elociy-erss-ime, and acceleraion-erss-ime are shown in Figre.7. The posiie direcion is p. Noice ha when he objec is a is maimm heigh on he posiion-erss-ime graph, he elociy is insananeosly zero (he slope of he posiion-erss-ime graph is zero). The acceleraion is neer zero, een a he momen when he elociy of he objec is zero. I migh be emping o hink ha a he insan an objec is no moing, is acceleraion ms be zero. This is only re for an objec ha is a res and remains a res. In he case of an objec hrown pward, if is acceleraion a he op of he fligh is zero, i wold neer descend (i wold remain a res a is highes poin). The moion described aboe is called a free fall. Free fall is an eample of a simplified model of moion in which we assme ha here is no air or any oher medim affecing he moion of a dropped or hrown (pward or downward) objec. All objecs in free fall hae he same acceleraion. We will learn he eplanaion for his phenomenon in Chaper 3. Noe ha physiciss se he erm free fall een when he objec is moing p. (d) a y (m/s ) 9.8 (s) Acceleraion is 9.8 m/s a he highes poin. REVIEW QUETION.7 (a) Gie an eample in which an objec wih negaie acceleraion is speeding p. (b) Gie an eample in which an objec wih posiie acceleraion is slowing down. (c) Gie an eample in which an objec wih zero elociy is acceleraing..8 Displacemen of an objec moing a consan acceleraion Eqaion (.) allows s o predic he locaion of an objec moing a consan elociy a any ime if we know is elociy and iniial posiion. Can we consrc a similar eqaion for an objec moing a consan acceleraion? ppose ha a ime zero an objec is passing he mark on he ais of moion raeling a elociy and acceleraion a. Where will he objec be afer he ime ineral - =? To answer his qesion, we can se he concep of aerage elociy. If we know he objec s aerage elociy and iniial posiion, and we assme ha he objec is moing a his elociy dring he ime ineral of ineres, we can predic he posiion a any ime sing Eq. (.), = + a. The aerage elociy is a = + where, according o Eq. (.5), insananeos elociy is = + a. Ths we can wrie a = + = + a + = + a

23 Therefore, he posiion of he acceleraing objec a any ime is = + a = + a + a b 1 = a (The symbol 1 means ha his eqaion follows from he preios eqaion.) Does he aboe resl make sense? Consider a limiing case, for eample, when he objec is raeling a a consan elociy (when a = ). In his case he eqaion shold redce o he resl from or inesigaion of linear moion wih consan elociy 1 = +. I does. We can also check he nis of each erm in his eqaion for consisency (when erms in an eqaion are added or sbraced, each of hose erms ms hae he same nis). Each erm has nis of meers, so he nis also check..8 Displacemen of an objec moing a consan acceleraion 35 Yo can only add or sbrac physical TIP qaniies ha hae he same nis. The resl has he same nis as hose wo qaniies. When yo mliply or diide qaniies, yo make a new physical qaniy whose nis are he resl of mliplicaion or diision of he original qaniies. FIGURE.8 (a) A posiion-erss-ime graph for consan acceleraion moion. (b) The insananeos elociy is he slope of he posiion-erss-ime graph a a pariclar clock reading. (a) () Posiion of an objec dring linear moion wih consan acceleraion For any iniial posiion a clock reading =, we can deermine he posiion of an objec a any laer ime, proided we also know he iniial elociy of he objec and is consan acceleraion a : Parabolic cre = a (.6) ince appears in Eq. (.6), he posiion-erss-ime graph for his moion is no a sraigh line b a parabola (a parabola is a graph for a qadraic fncion) (Figre.8a). Unlike he posiion-erss-ime graph for consan elociy moion where D> D is he same for any ime ineral, he posiion-erss-ime graph for acceleraed moion is no a sraigh line; i does no hae a consan slope. A differen imes, he change in posiion D has a differen ale for he same ime ineral D (Figre.8b). The line angen o he posiion-erss-ime graph a a pariclar ime has a slope D> D ha eqals he elociy of he objec a ha ime. The slopes of he angen lines a differen imes for he graph in Figre.8b differ hey are greaer when he posiion changes more dring he same ime ineral. The ne eample inoles erical moion. Ths, we will se a erical y-coordinae ais and apply he eqaions of moion. (b) () The slope a is greaer han he slope a 1. D D D 1 1, D 5 D D 1 D D EXAMPLE.8 Esimaing acceleraion of a falling person A woman jmps off a large bolder. he is moing a a speed of abo 5 m>s when she reaches he grond. Esimae her acceleraion dring landing. Indicae any oher qaniies yo se in he esimae. kech and ranslae To sole he problem, we assme ha he woman s knees bend while landing so ha her body raels.4 m closer o he grond compared o he sanding prigh posiion. Tha is,.4 m is he sopping disance for he main par of her body. A righ we hae skeched he iniial and final siaions dring he landing. We choose he cenral par of her body as he objec of ineres. We se a erical reference wih a y-ais poining down and he final locaion of he cenral par of her body as he origin of he coordinae ais. The iniial ales of her moion a he ime = when she firs oches he grond are y = -.4 m (he cenral par of her body is.4 m in he negaie direcion aboe her final posiion) and y = +5 m>s (she is moing downward, he posiie direcion relaie o he y-ais). The final ales a some nknown ime a he insan she sops are y = and y =. Noe: I is imporan o hae a coordinae ais and he iniial and final ales wih appropriae signs. (conined)

24 36 CHAPTER Kinemaics: Moion in One Dimension implify and diagram The moion diagram a righ represens her moion while sopping, assming consan acceleraion. We canno model he woman as a poin-like objec in his siaion, so we focs on he moion of her midsecion. Represen mahemaically The challenge in represening his siaion mahemaically is ha here are wo nknowns: he magnide of her acceleraion a y (he nknown we wish o deermine) and he ime ineral beween when she firs conacs he grond and when she comes o res. Howeer, boh Eqs. (.5) and (.6) describe linear moion wih consan acceleraion and hae a y and in hem. ince we hae wo eqaions and wo nknowns, we can handle his challenge by soling Eq. (.5) for he ime = y - y a y and sbsie i ino rearranged Eq. (.6): The resl is: y = y + y + 1 a y y - y = y a y - y a y b + a y1 y - y a y Using algebra, we can simplify he aboe eqaion: ole and ealae Now we can se he preios eqaion o find her acceleraion: a y = y - y 1y - y = 1-15 m>s m4 = -31 m>s < -3 m>s Is he answer reasonable? The sign is negaie. This means ha he acceleraion poins pward, as does he elociy change arrow in he moion diagram. This is correc. The nis for acceleraion are correc. We canno jdge ye if he magnide is reasonable. We will learn laer ha i is. The answer has one significan digi, as i shold he same as he informaion gien in he problem saemen. Try i yorself Using he epression a y = y - y 1y - y decide how he acceleraion wold change if (a) he sopping disance dobles and (b) he iniial speed dobles. Noe ha he final elociy is y =. 1 a y 1y - y = y 1 y - y + 1 y - y 1 a y 1y - y = 1 y y - y +1 y - y y + y Answer (a) ay wold be half he magnide; (b) ay wold be for imes he magnide. 1 a y 1y - y = y - y 1 a y = y - y 1y - y In he Represen mahemaically sep aboe, we deeloped a new mahemaical relaion ha is sefl when yo do no know he ime of rael. Yo do no need o memorize i, b i can come in handy when yo sole problems. A sefl eqaion for linear moion wih consan acceleraion for siaions in which yo do no know he ime ineral dring which he changes in posiion and elociy occrred: a 1 - = - (.7) Figre.9 shows a differen way o derie Eq. (.6): sing he elociy- erssime graph. Eamine he for pars of he figre careflly. Yo can se he idea of he displacemen as he area beween he consan elociy graph and he ime ais as he saring poin for his deriaion (Figre.9a). Breaking he ime ais ino small inerals allows s o consider he elociy for hose ime inerals as consan and o apply he mehod we sed in Figre.19 (Figre.9b). The resl is ha he displacemen is he area of a rapezim (Figre.9c), which can be broken p ino wo easy-o-calclae areas (Figre.9d).

25 .9 kills for analyzing siaions inoling moion 37 (a) () The displacemen D dring a shor ime ineral D is he area of he shaded recangle. (b) () Displacemen beween and is he sm of he areas of he narrow recangles. FIGURE.9 How o deermine he displacemen of an objec moing a consan acceleraion sing is elociy-erss-ime graph. D 5 3 D (heigh)(widh) D (c) () Displacemen beween and is he area beween he -erss- graph line and he ais. (d) () Addiional displacemen becase of he acceleraion 1 Area 5 ( ) Area 5 Area 5 5 Displacemen for consan elociy moion REVIEW QUETION.8 Eplain qaliaiely, wiho algebra, why he displacemen of an objec moing a consan acceleraion is proporional o ime sqared, no o ime o he power of 1 as i is for moion a consan elociy..9 kills for analyzing siaions inoling moion To help analyze physical processes inoling moion, we will represen processes in mliple ways: he words in he problem saemen, a skech, one or more diagrams, possibly a graph, and a mahemaical descripion. Differen represenaions hae o agree wih each oher; in oher words, hey need o be consisen. Moion a consan elociy EXAMPLE.9 Two walking friends Yo sand on a sidewalk and obsere wo friends walking a consan elociy. A ime Jim is 4. m eas of yo and walking away from yo a speed. m>s. Also a ime, arah is 1. m eas of yo and walking oward yo a speed 1.5 m>s. Represen heir moions wih an iniial skech, wih moion diagrams, and mahemaically. kech and ranslae We choose Earh as he objec of reference wih yor posiion as he reference poin. The posiie direcion poins o he eas. We hae wo objecs of ineres here: Jim and arah. Jim s iniial posiion is = +4. m and his consan elociy is = +. m>s. arah s iniial posiion is X = +1. m and her consan elociy is V = -1.5 m>s (he elociy is negaie since she is moing wesward). In or skech we are sing capial leers o represen arah and lowercase leers o represen Jim. (conined)

26 38 CHAPTER Kinemaics: Moion in One Dimension implify and diagram We can model boh friends as poin-like objecs since he disances hey moe are somewha greaer han heir own sizes. The moion diagrams below represen heir moions. ole and ealae We were no asked o sole for any qaniy. We will do i in he Try i yorself eercise. Try i yorself Deermine he ime when Jim and arah are a he same posiion, and where ha posiion is. Represen mahemaically Now consrc eqaions o represen Jim s and arah s moion: Jim: arah: = +4. m + 1. m>s X = +1. m m>s Answer They are a he same posiion when = 1.7 s and when = X = 7.4 m. Eqaion Jeopardy problems Learning o read he mahemaical langage of physics wih ndersanding is an imporan skill. To help deelop his skill, his e incldes Jeopardy-syle problems. In his ype of problem, yo hae o work backwards: yo are gien one or more eqaions and are asked o se hem o consrc a consisen skech of a process. Yo hen coner he skech ino a diagram of a process ha is consisen wih he eqaions and skech. Finally, yo inen a word problem ha he eqaions cold be sed o sole. Noe ha here are ofen many possible word problems for a pariclar mahemaical descripion. CONCEPTUAL EXERCIE.1 Eqaion Jeopardy The following eqaion describes an objec s moion: = 15. m m>s Consrc a skech, a moion diagram, kinemaics graphs, and a erbal descripion of a siaion ha is consisen wih his eqaion. There are many possible siaions ha he eqaion describes eqally well. kech and ranslae This eqaion looks like a specific eample of or general eqaion for he linear moion of an objec wih consan elociy: = +. The mins sign in fron of he 3. m>s indicaes ha he objec is moing in he negaie -direcion. A ime, he objec is locaed a posiion = +5. m wih respec o some chosen objec of reference and is already moing. Le s imagine ha his chosen objec of reference is a rnning person (he obserer) and he eqaion represens he moion of a person (he objec of ineres) siing on a bench as seen by he rnner. The skech below illsraes his possible scenario. The posiie ais poins from he obserer (he rnner) oward he person on he bench, and a ime he person siing on i is 5. m in fron of he rnner and coming closer o he rnner as ime elapses. The rnner is he obserer (objec of reference). The person on he bench is he objec of ineres moing oward he rnner. implify and diagram Model he objec of ineres as a poin-like objec. A moion diagram for he siaion is shown below. The eqal spacing of he dos and he eqal lenghs of elociy arrows boh indicae ha he objec of ineres is moing a consan elociy wih respec o he obserer. Moion diagram for bench relaie o rnner Posiion-erss-ime and elociy-erss-ime kinemaics graphs of he process are shown below. The posiion-erss-ime graph has a consan -3. m>s slope and a +5. m inercep wih he erical (posiion) ais. The elociy-erss-ime graph has a consan ale 1-3. m>s and a zero slope (he elociy is no changing). The following erbal descripion describes his pariclar process: A jogger sees a person on Inercep m D lope 5 D 5 3. m/s 5 3. m/s

27 .9 kills for analyzing siaions inoling moion 39 a bench in he park 5. m in fron of him. The bench is approaching a a speed of 3. m>s as seen in he jogger s reference frame. The direcion poining from he jogger o he bench is posiie. Answer Moion diagram for rnner relaie o person on bench Try i yorself ppose we swich he roles of obserer and objec of reference. Now he person on he bench is he objec of reference and obseres he rnner. We choose o describe he process by he same eqaion as in he eample: = 15. m m>s Consrc an iniial skech and a moion diagram ha are consisen wih he eqaion and wih he new obserer and new objec of reference. The person on he bench is he obserer (objec of reference). A rnner is he objec of ineres moing oward a person on a bench. An iniial skech for his process and a consisen moion diagram are shown below. Moion a consan nonzero acceleraion Now le s apply some represenaion echniqes o linear moion wih consan (nonzero) acceleraion. EXAMPLE.11 Eqaion Jeopardy A process is represened mahemaically by he following eqaion: = 1-6 m + 11 m>s m>s Use he eqaion o consrc an iniial skech, a moion diagram, and a erbal descripion of a process ha is consisen wih his eqaion. kech and ranslae The aboe eqaion appears o be an applicaion of Eq. (.6), which we consrced o describe linear moion wih consan acceleraion, if we assme ha he 1. m>s in fron of is he resl of diiding. m>s by : car s elociy and acceleraion are boh posiie. Ths, he car s elociy in he posiie -direcion is increasing as i moes oward he an (oward he origin). Below is a moion diagram for he car s moion as seen from he an. The sccessie dos in he diagram are spaced increasingly farher apar as he elociy increases; he elociy arrows are drawn increasingly longer. The elociy change arrow (and he acceleraion) poin in he posiie -direcion, ha is, in same direcion as he elociy arrows. = a = 1-6 m + 11 m>s m>s I looks like he iniial posiion of he objec of ineres is = -6 m, is iniial elociy is = +1 m>s, and is acceleraion is a = +. m>s. Le s imagine ha his eqaion describes he moion of a car on a sraigh highway passing a an in which yo, he obserer, are riding. The car is 6 m behind yo and moing 1 m>s faser han yor an. The car speeds p a a rae of. m>s wih respec o he an. The objec of reference is yo in he an; he posiie direcion is he direcion in which he car and an are moing. A skech of he siaion follows. The car is he objec of ineres caching p o he an. Yo in he an are he obserer (he objec of reference). Represen mahemaically The mahemaical represenaion of he siaion appears a he sar of he Eqaion Jeopardy eample. ole and ealae To ealae wha we hae done, we can check he consisency (agreemen) of he differen represenaions. For eample, we can check wheher he iniial posiion and elociy in he eqaion, he skech, and he moion diagram are consisen. In his case, hey are. Try i yorself Describe a differen scenario for he same mahemaical represenaion. implify and diagram The car can be considered a poin-like objec mch smaller han he dimensions of he pah i raels. The Answer This mahemaical represenaion cold also describe he moion of a cyclis moing on a sraigh pah as seen by a person sanding on a sidewalk 6 m in fron of he cyclis. The posiie direcion is in he direcion he cyclis is raeling. When he person sars obsering he cyclis, she is moing a an iniial elociy of = +1 m>s and speeding p wih acceleraion a = +. m>s.

28 4 CHAPTER Kinemaics: Moion in One Dimension The problem-soling sraegy in his e ses mliple represenaions, an approach ha was fond o be eremely sefl in helping sole physics problems. The complee problem-soling sraegy for a pariclar ype of problem will be presened along wih an eample a he side. Below, he problem-soling seps for kinemaics (moion) problems are presened on he lef side and illsraed on he righ side. The conen of his eample is also ery imporan: i shows why i is necessary o remain a safe disance behind he car in fron of yo when driing. PROBLEM-OLVING TRATEGY.1 Kinemaics EXAMPLE.1 An acciden inoling ailgaing A car follows abo wo car lenghs (1. m) behind a an. A firs, boh ehicles are raeling a a conseraie speed of 5 m>s (56 mi>h). The drier of he an sddenly slams on he brakes o aoid an acciden, slowing down a 9. m>s. The car drier s reacion ime is.8 s and he car s maimm acceleraion while slowing down is also 9. m>s. Will he car be able o sop before hiing he an? kech and ranslae kech he siaion described in he problem. Choose he objec of ineres. Inclde an objec of reference and a coordinae sysem. Indicae he origin and he posiie direcion. Label he skech wih relean informaion. Idenify he nknown ha yo need o find. Label i wih a qesion mark on he skech. Below we represen his siaion for each ehicle (we hae wo objecs of ineres). We se capial leers o indicae qaniies referring o he an and lowercase leers for qaniies referring o he car. We se he coordinae sysem shown wih he origin of he coordinaes a he iniial posiion of he car s fron bmper. The posiie direcion is in he direcion of moion. The process sars when he an sars braking. The an moes a consan negaie acceleraion hrogho he enire process. We separae he moion of he car ino wo pars: (1) is moion before he drier applies he brakes (consan posiie elociy) and () is moion afer he drier sars braking (consan negaie acceleraion). implify and diagram Decide how yo will model each moing objec (for eample, as a poin-like objec). Can yo model he moion as consan elociy or consan acceleraion? Draw moion diagrams and kinemaics graphs if needed. We model each ehicle as a poin-like objec, b since we are rying o deermine if hey collide, we need o be more specific abo heir posiions. The posiion of he car is he posiion of is fron bmper. The posiion of he an is he posiion of is rear bmper. We look a he moion of each ehicle separaely. If he car s final posiion is greaer han he an s final posiion, hen a collision has occrred a some poin dring heir moion. Assme ha he ehicles hae consan acceleraion so ha we can apply or model of moion wih consan acceleraion. A elociy-erss-ime graph line for each ehicle is shown a righ.

29 .9 kills for analyzing siaions inoling moion 41 Represen mahemaically Use he skech(es), moion diagram(s), and kinemaics graph(s) o consrc a mahemaical represenaion (eqaions) of he process. Be sre o consider he sign of each qaniy. Eqaion (.7) can be sed o deermine he disance he an raels while sopping: A 1X - X = V - V 1 X = V - V A + X The car Par 1 ince he car is iniially raeling a consan elociy, we se Eq. (.). The sbscrip indicaes he momen he drier sees he an sar slowing down. The sbscrip 1 indicaes he momen he car drier sars braking. 1 = + 1 The car Par Afer applying he brakes, he car has an acceleraion of -9. m>s. The sbscrip indicaes he momen he car sops moing. We represen his par of he moion sing Eq. (.7): a 1-1 = = - 1 a = - 1 a The las sep came from insering he resl from Par 1 for 1. ole and ealae ole he eqaions o find he answer o he qesion yo are inesigaing. Ealae he resls o see if hey are reasonable. Check he nis and decide if he calclaed qaniies hae reasonable ales (sign, magnide). Check limiing cases: eamine wheher he final eqaion leads o a reasonable resl if one of he qaniies is zero or infiniy. This sraegy applies when yo derie a new eqaion while soling a problem. The an s iniial elociy is V = +5 m>s, is final elociy is V =, and is acceleraion is A = -9. m>s. Is iniial posiion is wo car lenghs in fron of he fron of he car, so X = * 5. m = 1 m. The final posiion of he an is X = V - V + X A = - 15 m>s 1-9. m>s + 1 m = 45 m The car s iniial posiion is =, is iniial elociy is = 5 m>s, is final elociy is =, and is acceleraion when braking is a = -9. m>s. The car s final posiion is = a + 1 = - 15 m>s 1-9. m>s + 3 m + 15 m>s1.8 s4 = 55 m The car will sop abo 1 m beyond where he an will sop. There will be a collision beween he wo ehicles. This analysis illsraes why ailgaing is sch a big problem. The car raeled a a 5@m>s consan elociy dring he relaiely shor.8-s reacion ime. Dring he same.8 s, he an s elociy decreased by 1.8 s1-9. m>s = 7. m>s from 5 m>s o abo 18 m>s. o he an was moing somewha slower han he car when he car finally sared o brake. ince hey were boh slowing down a abo he same rae, he ailgaing ehicle s elociy was always greaer han ha of he ehicle in fron nil hey hi. Try i yorself Two cars, one behind he oher, are raeling a 3 m>s (67 mi>h). The fron car his he brakes and slows down a he rae of 1 m>s. The drier of he second car has a 1.-s reacion ime. The fron car s speed has decreased o m/s dring ha 1. s. The rear car raeling a 3 m>s sars braking, slowing down a he same rae of 1 m>s. How far behind he fron car shold he rear car be so i does no hi he fron car? Answer The rear car shold be a leas 3 m behind he fron car. REVIEW QUETION.9 A car s moion wih respec o he grond is described by he following fncion: = 1-48 m + 11 m>s m>s Mike says ha is original posiion is 1-48 m and is acceleraion is 1-. m>s. Do yo agree? If yes, eplain why; if no, eplain how o correc his answer.

30 4 CHAPTER Kinemaics: Moion in One Dimension mmary A reference frame consiss of an objec of reference, a poin of reference on ha objec, a coordinae sysem whose origin is a he poin of reference, and a clock. (ecion.1) Objec of reference Clock Poin of reference (origin) Objec of ineres Coordinae ais Time or clock reading (a scalar qaniy) is he reading on a clock or anoher ime measring insrmen. (ecion.4) Time ineral D (a scalar qaniy) is he difference of wo imes. (ecion.4) 1 Time: Time ineral: D = - 1 Posiion (a scalar qaniy) is he locaion of an objec relaie o he chosen origin. (ecion.4) Displacemen d is a ecor drawn from he iniial posiion of an objec o is final posiion. The -componen of he displacemen d is he change in posiion of he objec along he -ais. (ecion.4) d d = - 7 if d poins in he posiie direcion of -ais. d = - 6 if d poins in he negaie direcion. d = - Disance d (a scalar qaniy) is he magnide of he displacemen and is always posiie. (ecion.4) Pah lengh l is he lengh of a sring laid along he pah he objec ook. (ecion.4) Velociy (a ecor qaniy) is he raio of he displacemen of an objec dring a ime ineral o ha ime ineral. The elociy is insananeos if he ime ineral is ery small and aerage if he ime ineral is longer. (ecion.6) peed (a scalar qaniy) is he magnide of he elociy. (ecion.6) For consan elociy linear moion, 1 = d > D 1 = D D = Eq. (.1) D = ` D ` Acceleraion a (a ecor qaniy) is he raio of he change in an objec s elociy D dring a ime ineral D o he ime ineral. The acceleraion is insananeos if he ime ineral is ery small and aerage if he ime ineral is longer. (ecion.7) (rearranged) a = a = D D = D D Eq. (.4) = - - Eq. (.4) Moion wih consan elociy or consan acceleraion can be represened wih a skech, a moion diagram, kinemaics graphs, and mahemaically. (ecions.6.9) D 5 D = + a Eq. (.5) = a Eq. (.6) a 1 - = - Eq. (.7) (rearranged) - = - Eq. (.7) a

31 Qesions Mliple Choice Qesions 1. Mach he general elemens of physics knowledge (lef) wih he appropriae eamples (righ). Model of a process Free fall Model of an objec Acceleraion Physical qaniy Rolling ball Physical phenomenon Poin-like objec (a) Model of a process Acceleraion; Model of an objec Poin-like objec; Physical qaniy Free fall; Physical phenomenon Rolling ball. (b) Model of a process Rolling ball; Model of an objec Poin-like objec; Physical qaniy Acceleraion; Physical phenomenon Free fall. (c) Model of a process Free fall; Model of an objec Poin-like objec; Physical qaniy Acceleraion; Physical phenomenon Rolling ball.. Which grop of qaniies below consiss only of scalar qaniies? (a) Aerage speed, displacemen, ime ineral (b) Aerage speed, pah lengh, clock reading (c) Temperare, acceleraion, posiion 3. Which of he following are eamples of ime ineral? (1) I woke p a 7 a.m. () The lesson lased 45 mines. (3) elana was born on Noember 6. (4) An asrona orbied Earh in 4 hors. (a) 1,, 3, and 4 (b) and 4 (c) (d) 4 (e) 3 4. A sden said, The displacemen beween my dorm and he lecre hall is 1 kilomeer. Is he sing he correc physical qaniy for he informaion proided? Wha shold he hae called he 1 kilomeer? (a) Disance (b) Pah lengh (c) Posiion (d) Boh a and b are correc. 5. An objec moes so ha is posiion depends on ime as = +1 m - 14 m>s + 11 m>s. Which saemen below is no re? (a) The objec is acceleraing. (b) The speed of he objec is always decreasing. (c) The objec firs moes in he negaie direcion and hen in he posiie direcion. (d) The acceleraion of he objec is + m>s. (e) The objec sops for an insan a. s. 6. Choose he correc approimae elociy-erss-ime graph for he following hypoheical moion: a car moes a consan elociy, and hen slows o a sop and wiho a pase moes in he opposie direcion wih he same acceleraion (Figre Q.6). FIGURE Q.6 (a) (b) (c) Qesions 43 (a) A 1 he elociy of he do was zero. (b) A 1 he speed of he do was greaes. (c) A 1 he do was a posiion D. (d) A he do was a posiion A. (e) A he elociy of he do was posiie. 8. Olier akes wo idenical marbles and drops he firs one from a cerain heigh. A shor ime laer, he drops he second one from he same heigh. How does he disance beween he marbles change while hey are falling o he grond? Choose he bes answer and eplanaion. (a) The disance increases becase he firs marble moes faser han he second marble a eery insan of ime. (b) The disance increases becase he firs marble moes wih greaer acceleraion han he second marble. (c) The disance says he same becase boh marbles are falling wih he same acceleraion. (d) The disance says he same becase he elociies of he marbles are eqal a eery insan of ime. 9. Yor car is raeling wes a 1 m>s. A sopligh (he origin of he coordinae ais) o he wes of yo rns yellow when yo are m from he edge of he inersecion (see Figre Q.9). Yo apply he brakes and yor car s speed decreases. Yor car sops before i reaches he sopligh. Wha are he signs for he componens of kinemaics qaniies? FIGURE Q.9 5 s 5 } m 5 } 1 m/s a 5 } 6. m/s a (a) (b) (c) (d) (e) Which elociy-erss-ime graph in Figre Q.1 bes describes he moion of he car in he preios problem (see Figre Q.9) as i approaches he sopligh? FIGURE Q.1 (a) (b) (c) 7. Figre Q.7b shows he posiion-erss-ime graph for a small red do ha was moing along he -ais from poin C a =, as shown in Figre Q.7a. elec he wo answers ha correcly describe he moion of he do. (d) (e) No graph represens he moion. FIGURE Q.7 (a) (b) D C B A C 1

32 44 CHAPTER Kinemaics: Moion in One Dimension 11. Azra wans o deermine he aerage speed of he high-speed rain ha operaes beween Paris and Lyon. Before she boards he rain, she measres he lengh of he rain L and he disance d beween he elecric line poles ha are placed along he racks. Which of he following procedres shold she se? (a) Diide he disance beween he adjacen poles by 1 imes he ime needed o pass 1 poles. (b) Diide he lengh of he rain by 1 1 of he ime needed o pass 1 poles. (c) Con how many poles he rain passed in 1 s and diide his nmber by d # 1 s. (d) Con how many poles he rain passed in 1 s and mliply his d nmber by 1 s. 1. A sandbag hangs from a rope aached o a rising ho air balloon. The rope connecing he bag o he balloon is c. How will wo obserers see he moion of he sandbag? Obserer 1 is in he ho air balloon and obserer is on he grond? (a) Boh 1 and will see i go down. (b) 1 will see i go down and will see i go p. (c) 1 will see i go down and will see i go p and hen down. 13. An apple falls from a ree. I his he grond a a speed of abo 5. m>s. Wha is he approimae heigh of he ree? (a).5 m (b) 1.3 m (c) 1. m (d).4 m 14. Yo hae wo small meal balls. Yo drop he firs ball and hrow he oher one in he downward direcion. Choose he saemens ha are no correc. (a) The second ball will spend less ime in fligh. (b) The firs ball will hae a slower final speed when i reaches he grond. (c) The second ball will hae larger acceleraion. (d) Boh balls will hae he same acceleraion. 15. Which of he graphs in Figre Q.15 represen he moion of an objec ha sars from res and hen, afer ndergoing some moion, rerns o is iniial posiion? Mliple answers cold be correc. FIGURE Q.15 (a) y (b) y (c) y Concepal Qesions 17. Figre Q.17 shows ecors E, F, and G. Draw he following ecors sing rles for graphical addiion and sbracion: (a) E + F, (b) E - F, (c) E + G, (d) -G + F, (e) E + F + G, and (f) E - F + G. 18. Peer is cycling along an 8-m sraigh srech of a rack. His speed is 13 m>s. Choose all of he graphical represenaions of moion from Figre Q.18 ha correcly describe Peer s moion. FIGURE Q.18 (a) (d) (b) (e) FIGURE Q In wha reasonable ways can yo represen or describe he moion of a car raeling from one sopligh o he ne? Consrc each represenaion for he moing car.. Wha is he difference beween speed and elociy? Beween pah lengh and disance? Beween disance and displacemen? Gie an eample of each. 1. Wha physical qaniies do we se o describe moion? Wha does each qaniy characerize? Wha are heir I nis?. Deise sories describing each of he moions shown in each of he graphs in Figre Q.. pecify he objec of reference. FIGURE Q. (c) (f) E G F (a) (m/s) 8 6 (d) y (e) y (s) (b) y (m/s) 16. Yo hrow a small ball pward and noice he ime i akes o come back. If yo hen hrow he same ball so ha i akes wice as mch ime o come back, wha is re abo he moion of he ball he second ime? (a) Is iniial speed was wice he speed in he firs eperimen. (b) I raeled an pward disance ha is wice he disance of he original oss. (c) I had wice as mch acceleraion on he way p as i did he firs ime. (d) The ball sopped a he highes poin and had zero acceleraion a ha poin. 4 4 (c) y (m/s) (s) (s)

33 Problems For each of he posiion-erss-ime graphs in Figre Q.3, draw elociy-erss-ime graphs and acceleraion-erss-ime graphs. FIGURE Q.3 (a) (b) Figre Q.4 shows elociy-erssime graphs for wo objecs, A and B. Draw moion diagrams ha correspond FIGURE Q.4 y o he moion of hese wo objecs. 5. Can an objec hae a nonzero elociy A B and zero acceleraion? If so, gie an eample. 6. Can an objec a one insan of ime hae zero elociy and nonzero acceleraion? If so, gie an eample. 7. Yor lile siser has a baery-powered oy rck. When he rck is moing, how can yo deermine wheher i has consan elociy, consan speed, consan acceleraion, or changing acceleraion? Eplain in deail. 8. Yo hrow a ball pward. Yor friend says ha a he op of is fligh he ball has zero elociy and zero acceleraion. Do yo agree or disagree? If yo agree, eplain why. If yo disagree, how wold yo conince yor friend of yor opinion? Problems Below, indicaes a problem wih a biological or medical focs. Problems labeled ask yo o esimae he answer o a qaniaie problem raher han derie a specific answer. Aserisks indicae he leel of difficly of he problem. Problems wih no * are considered o be he leas difficl. A single * marks moderaely difficl problems. Two ** indicae more difficl problems.. A concepal descripion of moion 1. A car sars a res from a sopligh and speeds p. I hen moes a consan speed for a while. Then i slows down nil reaching he ne sopligh. Represen he moion wih a moion diagram as seen by he obserer on he grond.. * Yo are an obserer on he grond. (a) Draw wo moion diagrams represening he moions of wo rnners moing a he same consan speeds in opposie direcions oward yo. Rnner 1, coming from he eas, reaches yo in 5 s, and rnner reaches yo in 3 s. (b) Draw a moion diagram for he second rnner as seen by he firs rnner. 3. * A car is moing a consan speed on a highway. A second car caches p and passes he firs car 5 s afer i sars o speed p. Represen he siaion wih a moion diagram. pecify he obserer wih respec o whom yo drew he diagram. 4. * A ha falls off a man s head and lands in he snow. Draw a moion diagram represening he moion of he ha as seen by he man..3 Operaions wih ecors 5. Figre P.5 shows seeral displacemen ecors ha are all in he 1, y plane. (a) Lis all displacemen ecors wih a scalar componen eqal o - nis. (b) Lis all displacemen ecors wih a scalar componen eqal o nis. (c) Lis all displacemen ecors ha represen he disance raeled of nis. (d) Find hree ecors FIGURE P.5 D G A F 5 ha are relaed as X + Y = Z and hree ha are relaed as X - Y = Z. Q y L B E H M 3 K 5 6. Figre P.6 shows an incomplee moion diagram for an objec. (a) For each pair of adjacen elociies, draw a corresponding D ecor. Use he grid o deermine heir lenghs accraely. (b) Eplain how o consrc a D ecor by sing he rle for addiion of ecors and how o do i sing he rle for sbracion of ecors. FIGURE P TABLE.8.4 and.5 Qaniies for describing moion and Represening moion wih daa ables and graphs 7. * Yo drie 1 km eas, do some sighseeing, and hen rn arond and drie 5 km wes, where yo sop for lnch. (a) Represen yor rip wih a displacemen ecor. Choose an objec of reference and coordinae ais so ha he scalar componen of his ecor is (b) posiie; (c) negaie; (d) zero. 8. * Choose an objec of reference and a se of coordinae aes associaed wih i. how how wo people can sar and end heir rips a differen locaions b sill hae he same displacemen ecors in his reference frame. 9. The scalar -componen of a displacemen ecor for a rip is -7 km. Represen he rip sing a coordinae ais and an objec of reference. Then change he ais so ha he displacemen componen becomes +7 km. 1. * Yo recorded yor posiion wih respec o he fron door of yor hose as yo walked o he mailbo. Eamine he daa presened in Table.8 and answer he following qesions: (a) Wha insrmens migh yo hae sed o collec daa? (b) Represen yor moion sing a posiion-erss-ime graph. (c) Tell he sory of yor moion in words. (d) how on he graph he displacemen, disance, and pah lengh. (s) (seps)

34 46 CHAPTER Kinemaics: Moion in One Dimension.6 Consan elociy linear moion 11. * Yo need o deermine he ime ineral (in seconds) needed for ligh o pass an aomic ncles. Wha informaion do yo need? How will yo se i? Wha simplifying assmpions abo he objecs and processes do yo need o make? Wha approimaely is ha ime ineral? 1. A speedomeer reads 65 mi/h. (a) Use as many differen nis as possible o represen he speed of he car. (b) If he speedomeer reads 1 km>h, wha is he car s speed in mi/h? 13. Coner he following record speeds so ha hey are in mi/h, km>h, and m>s. (a) Asralian dragonfly 36 mi/h; (b) he diing peregrine falcon 349 km>h; and (c) he Lockheed R-71 je aircraf 98 m>s (abo hree imes he speed of sond). 14. Hair growh speed Esimae he rae ha yor hair grows in meers per second. Indicae any assmpions yo made. 15. * A kidnapped banker looking hrogh a sli in a an window cons her hearbeas and obseres ha wo highway eis pass in 8 hearbeas. he knows ha he disance beween he eis is 1.6 km (1 mile). (a) Esimae he an s speed. (b) Choose and describe a reference frame and draw a posiion-erss-ime graph for he an. 16. * ome comper scanners scan docmens by moing he scanner head wih a consan speed across he docmen. A scanner can scan p o 3 pages per mine sing an aomaic paper feed ray. Esimae he maimm speed of he scanner head when scanning a sandard leer shee (1.59 cm * 7.94 cm). Indicae any assmpions ha yo made. 17. * Eqaion Jeopardy Two obserers obsere wo differen moing objecs. Howeer, hey describe heir moions mahemaically wih he same eqaion: 1 = 1 km - 14 km>h. (a) Wrie wo shor sories abo hese wo moions. pecify where each obserer is and wha she is doing. Wha is happening o he moing objec a =? (b) Use significan digis o deermine he ineral wihin which he iniial posiion is known. 18. * Yor friend s pedomeer shows ha he ook 17, seps in.5 h dring a hike. Deermine eeryhing yo can abo he hike. Wha assmpions did yo make? How cerain are yo in yor answer? How wold he answer change if he ime were gien as.5 h insead of.5 h? 19. Dring a hike, wo friends were cagh in a hndersorm. For seconds afer seeing lighning from a disan clod, hey heard hnder. How far away was he clod (in kilomeers)? Wrie yor answer as an ineral sing significan digis as yor gide. ond raels in air a abo 34 m>s.. Ligh raels a a speed of 3. * 1 8 m>s in a acm. The approimae disance beween Earh and he n is 15 * 1 6 km. How long does i ake ligh o rael from he n o Earh? Wha are he margins wihin which yo know he answer? 1. Proima Cenari is 4. {.1 ligh-years from Earh. Deermine he lengh of 1 ligh-year and coner he disance o he sar ino meers. Wha is he ncerainy in he answer?. * paceships raeling o oher planes in he solar sysem moe a an aerage speed of 1.1 * 1 4 m>s. I ook Voyager abo 1 years o reach he orbi of Urans. Wha can yo learn abo he solar sysem sing hese daa? Wha assmpion did yo make? How did his assmpion affec he resls? 3. ** Figre P.3 shows a elociy-erss-ime graph for he bicycle rips of wo friends wih respec o he parking lo where hey sared. (a) Deermine heir displacemens in s. (b) If Xena s posiion a ime zero is and Gabriele s posiion is 6 m, wha ime ineral is needed for Xena o cach Gabriele? (c) Use he informaion from (b) o wrie he fncion () for Gabriele wih respec o Xena. FIGURE P.3 (m/s) 1 Xena 8 Gabriele (s) 4. * Table.9 shows posiion and ime daa for yor walk along a sraigh pah. (a) Tell eeryhing yo can abo he walk. pecify he objec of reference. (b) Draw a moion diagram, draw a graph 1, and wrie a fncion 1 ha is consisen wih he daa and he chosen reference frame. TABLE.9 Time (s) Posiion (m) TABLE.1 Time (s) Posiion (m) * Table.1 shows posiion and ime daa for yor friend s bicycle ride along a sraigh bike pah. (a) Tell eeryhing yo can abo his ride. pecify he obserer. (b) Draw a moion diagram, draw a graph 1, and wrie a fncion 1 ha is consisen wih he ride. 6. * Yo are walking o yor physics class a speed 1. m>s wih respec o he grond. Yor friend leaes. min afer yo and is walking a speed 1.3 m>s in he same direcion. How fas is she walking wih respec o yo? How far does yor friend rael before she caches p wih yo? Indicae he ncerainy in yor answers. Describe any assmpions ha yo made. 7. * Gabriele eners an eas wes sraigh bike pah a he 3.-km mark and rides wes a a consan speed of 8. m>s. A he same ime, Xena rides eas from he 1.-km mark a a consan speed of 6. m>s. (a) Wrie fncions 1 ha describe heir posiions as a fncion of ime wih respec o Earh. (b) Where do hey mee each oher? In how many differen ways can yo sole his problem? (c) Wrie a fncion 1 ha describes Xena s moion wih respec o Gabriele. 8. * Jim is driing his car a 3 m>s (7 mi>h) along a highway where he speed limi is 5 m>s (55 mi/h). A highway parol car obseres him pass and qickly reaches a speed of 36 m>s. A ha poin, Jim is 3 m ahead of he parol car. How far does he parol car rael before caching Jim? 9. * Yo hike wo-hirds of he way o he op of a hill a a speed of 3. mi>h and rn he final hird a a speed of 6. mi>h. Wha was yor aerage speed? 3. * Olympic champion swimmer Michael Phelps swam a an aerage speed of.1 m>s dring he firs half of he ime needed o complee a race. Wha was his aerage swimming speed dring he second half of he race if he ied he record, which was a an aerage speed of.5 m>s? 31. * A car makes a 1-km rip. I raels he firs 5 km a an aerage speed of 5 km>h. How fas ms i rael he second 5 km so ha is aerage speed is 1 km>h? 3. * Jane and Bob see each oher when 1 m apar. They are moing oward each oher, Jane a consan speed 4. m>s and Bob a consan speed 3. m>s wih respec o he grond. Wha can yo deermine abo his siaion sing hese daa? 33. * The graph in Figre P.33 represens for differen moions. (a) Wrie a fncion 1 for each moion. (b) Use he informaion in he graph o deermine as many qaniies relaed o he moion of hese objecs as possible. (c) Ac o hese moions wih wo friends. (Hin: Think of wha each objec was doing a =.) FIGURE P (m) (s)

35 Problems 47.7 and.8 Moion a consan acceleraion and Displacemen of an objec moing a consan acceleraion 34. A car sars from res and reaches he speed of 1 m>s in 3 s. Wha can yo deermine abo he moion of he car sing his informaion? 35. A rck is raeling eas a +16 m>s. (a) The drier sees ha he road is empy and acceleraes a +1. m>s for 5. s. Wha can yo deermine abo he rck s moion sing hese daa? (b) The drier hen sees a red ligh ahead and deceleraes a -. m>s for 3. s. Wha can yo deermine abo he rck s moion sing hese daa? (c) Deermine he ales of he qaniies yo lised in (a) and (b). 36. Bmper car collision On a bmper car ride, friends smash heir cars ino each oher (head-on), and each has a speed change of 3. m>s. If he magnides of acceleraion of each car dring he collision aeraged 8 m>s, deermine he ime ineral needed o sop and he sopping disance for each car while colliding. pecify yor reference frame. 37. A bs leaes an inersecion acceleraing a +. m>s. Where is he bs afer 5. s? Wha assmpion did yo make? If his assmpion is no alid, wold he bs be closer or farher away from he inersecion compared o yor original answer? Eplain. 38. A jogger is rnning a +4. m>s when a bs passes her. The bs is acceleraing from +16. m>s o +. m>s in 8. s. The jogger speeds p wih he same acceleraion. Wha can yo deermine abo he jogger s moion sing hese daa? 39. * The moion of a person as seen by anoher person is described by he eqaion = -3. m>s m>s. (a) Represen his moion wih a moion diagram and posiion-, elociy-, and acceleraion-erss-ime graphs. (b) ay eeryhing yo can abo his moion and describe wha happens o he person when his speed becomes zero. 4. While cycling a a speed of 1 m>s, a cyclis sars going downhill wih an acceleraion of magnide 1. m>s. The descen akes 1. s. Wha can yo deermine abo he cyclis s moion sing hese daa? Wha assmpions did yo make? 41. * To his srprise, Daniel fond ha an egg did no break when he accidenally dropped i from a heigh of.4 m ono his floor, coered wih.-cm-hick carpe. Esimae he minimm acceleraion of he egg while i was slowing down afer oching he carpe. Indicae any assmpions yo made. FIGURE P qid proplsion Lolligncla breis sqid se a form of je proplsion o swim hey ejec waer o of jes ha can poin in differen direcions, allowing hem o change direcion qickly. When swimming a a speed of.15 m>s or greaer, hey can accelerae a 1. m>s. (a) Deermine he ime ineral needed for a sqid o increase is speed from.15 m>s o.45 m>s. (b) Wha oher qesions can yo answer sing he daa? 43. Dragser record on he deser In 1977, Kiy O Neil droe a hydrogen peroide powered rocke dragser for a record ime ineral (3. s) and final speed (663 km>h) on a 4-m-long Mojae Deser rack. Deermine her aerage acceleraion dring he race and he acceleraion while sopping (i ook abo s o sop). Wha assmpions did yo make? 44. * Imagine ha a spriner acceleraes from res o a maimm speed of 1.8 m>s in 1.8 s. In wha ime ineral will he finish he 1-m race if he keeps his speed consan a 1.8 m>s for he las par of he race? Wha assmpions did yo make? 45. ** Two rnners are rnning ne o each oher when one decides o speed p a consan acceleraion a. The second rnner noices he acceleraion afer a shor ime ineral D when he disance beween he rnners is d. The second rnner acceleraes a he same acceleraion. Represen heir moions wih a moion diagram and posiion-erss-ime graph (boh graph lines on he same se of aes). Use any of he represenaions o predic wha will happen o he disance beween he rnners will i say d, increase, or decrease? Assme ha he rnners conine o hae he same acceleraion for he draion of he problem. 46. * Meeorie his car In 199, a 14-kg meeorie srck a car in Peekskill, NY, leaing a -cm-deep den in he rnk. (a) If he meeorie was moing a 5 m>s before sriking he car, wha was he magnide of is acceleraion while sopping? Indicae any assmpions yo made. (b) Wha oher qesions can yo answer sing he daa in he problem? 47. Froghopper jmp A spilebg called he froghopper (Philaens spmaris) is belieed o be he bes jmper in he animal world. I pshes off wih msclar rear legs for.1 s, reaching a speed of 4. m>s. Deermine is acceleraion dring his lanch and he disance ha he froghopper moes while is legs are pshing. 48. Tennis sere The fases serer in women s ennis is abine Lisicki, who recorded a sere of 131 mi>h 111 km>h in 14. If her racke pshed on he ball for a disance of.1 m, wha was he aerage acceleraion of he ball dring her sere? Wha was he ime ineral for he racke-ball conac? 49. * ho from a cannon In 1998, Daid Cannonball mih se he disance record for being sho from a cannon (56.64 m). Dring a lanch in he cannon s barrel, his speed increased from zero o 8 km>h in.4 s. While he was being sopped by he caching ne, his speed decreased from 8 km>h o zero wih an aerage acceleraion of 18 m>s. Wha can yo deermine abo mih s fligh sing his informaion? 5. Col. John app s final sled rn Col. John app led he U.. Air Force Aero Medical Laboraory s research ino he effecs of higher acceleraions. On app s final sled rn, he sled reached a speed of 8.5 m>s (63 mi/h) and hen sopped wih he aid of waer brakes in 1.4 s. app was barely conscios and los his ision for seeral days b recoered. Deermine his acceleraion while sopping and he disance he raeled while sopping. 51. * priner Usain Bol reached a maimm speed of 11. m>s in. s while rnning he 1-m dash. (a) Wha was his acceleraion? (b) Wha disance did he rael dring his firs. s of he race? (c) Wha assmpions did yo make? (d) Wha ime ineral was needed o complee he race, assming ha he ran he las par of he race a his maimm speed? (e) Wha is he oal ime for he race? How cerain are yo of he nmber yo calclaed? 5. ** Imagine ha Usain Bol can reach his maimm speed in 1.7 s. Wha shold be his maimm speed in order o ie he s record for he -m dash? 53. * A bs is moing a a speed of 36 km>h. How far from a bs sop shold he bs sar o slow down so ha he passengers feel comforable (a comforable acceleraion is 1. m>s )? 54. * Yo wan o esimae how fas yor car acceleraes. Wha informaion can yo collec o answer his qesion? Wha assmpions do yo need o make o do he calclaion sing he informaion? 55. * In yor car, yo coered. m dring he firs 1. s, 4. m dring he second 1. s, 6. m dring he hird 1. s, and so forh. Was his moion a consan acceleraion? Eplain. 56. (a) Deermine he acceleraion of a car in which he elociy changes from -1 m>s o - m>s in 4. s. (b) Deermine he car s acceleraion if is elociy changes from - m>s o -18 m>s in. s. (c) Eplain why he sign of he acceleraion is differen in (a) and (b). 57. Yo accidenally drop an eraser o he window of an aparmen 15 m aboe he grond. (a) How long will i ake for he eraser o reach he grond? (b) Wha speed will i hae js before i reaches he grond? (c) If yo mliply he ime ineral answer from (a) and he speed answer from (b), why is he resl mch more han 15 m? 58. * Wha is he aerage speed of he eraser in he preios problem from he insan i is released o he insan i reaches he grond? 59. Yo hrow a ennis ball sraigh pward. The iniial speed is abo 1 m>s. ay eeryhing yo can abo he moion of he ball. Is 1 m>s a realisic speed for an objec ha yo can hrow wih yor hands? 6. While skydiing, yor parache opens and yo slow from 5. m>s o 8. m>s in.8 s. Deermine he disance yo fall while he parache is opening. ome people fain if hey eperience acceleraion greaer han 5g 15 imes 9.8 m>s. Will yo feel fain? Eplain and discss simplifying assmpions inheren in yor eplanaion.

36 48 CHAPTER Kinemaics: Moion in One Dimension 61. * Afer landing from yor skydiing eperience, yo are so ecied ha yo hrow yor helme pward. The helme rises 5. m aboe yor hands. Wha was he iniial speed of he helme when i lef yor hands? How long was i moing from he ime i lef yor hands nil i rerned? 6. * Yo are sanding on he rim of a canyon. Yo drop a rock and in 7. s hear he sond of i hiing he boom. How deep is he canyon? Wha assmpions did yo make? Eamine how each assmpion affecs he answer. Does i lead o a larger or smaller deph han he calclaed deph? (The speed of sond in air is abo 34 m>s.) 63. * Yo are doing an eperimen o deermine yor reacion ime. Yor friend holds a rler. Yo place yor fingers near he sides of he lower par of he rler wiho oching i. The friend drops he rler wiho warning yo. Yo cach he rler afer i falls 1. cm. Wha was yor reacion ime? 64. Cliff diers Diers in Acaplco fall 36 m from a cliff ino he waer. Esimae heir speed when hey ener he waer and he ime ineral needed o reach he waer. Wha assmpion did yo make? Does his assmpion make he calclaed speed larger or smaller han acal speed? 65. * Galileo dropped a ligh rock and a heay rock from he Leaning Tower of Pisa, which is abo 55 m high. ppose ha Galileo dropped one rock.5 s before he second rock. Wih wha iniial elociy shold he drop he second rock so ha i reaches he grond a he same ime as he firs rock? 66. * A person holding a lnch bag is moing pward in a ho air balloon a a consan speed of 7. m>s. When he balloon is 4 m aboe he grond, she accidenally releases he bag. Wha is he speed of he bag js before i reaches he grond? 67. * A parachis falling erically a a consan speed of 1 m>s drops a penknife when m aboe he grond. Wha is he speed of he knife js before i reaches he grond?.9 kills for analyzing siaions inoling moion 68. A diagram represening he moion of wo cars is shown in Figre P.68. The nmber near each do indicaes he clock reading in seconds when he car passes ha locaion. (a) Indicae imes when he cars hae he same speed. (b) Indicae imes when hey hae he same posiion. FIGURE P Car Car 69. Use he elociy-erss-ime graph lines in Figre P.69 o deermine he change in he posiion of each car from s o 6 s. Represen he moion of each car mahemaically as a fncion 1. Their iniial posiions are A ( m) and B 1- m. FIGURE P.69 (m/s) B * An objec moes so ha is posiion changes in he following way: = 1 m - 14 m>s. (a) Describe all of he known qaniies for his moion. (b) Inen a sory for he moion. (c) Draw a posiion-erss-ime graph, and se he graph o deermine when he objec reaches he origin of he reference frame. (e) Ac o he moion. A 5 (s) * While babysiing heir yonger broher, Chrisso and Dein are playing wih oys. They noice ha he sqishy Pigle slows down in a repeaable way when hey psh i along he smooh wooden floor. They propose a hypohesis ha he oy slows down wih a consan acceleraion, which does no depend on he oy s iniial elociy. For each of fie differen iniial speeds, hey measre he disance raeled by he oy from he ime hey sop pshing i o he ime he oy sops moing, and hey measre he corresponding ime ineral. Their daa are presened below. Do he daa sppor heir hypohesis? Eplain. If yes, deermine he aerage acceleraion of Pigle and he maimm speed wih which Chrisso and Dein psh Pigle. Eperimen # Disance (m) Time (s) ** An objec moes so ha is posiion changes in he following way: 1 = -1 m + 13 m>s m>s. (a) Wha kind of moion is his (consan elociy, consan acceleraion, or changing acceleraion)? (b) Describe all of he known qaniies for his moion. (c) Inen a sory for he moion. (d) Draw a elociy-erss-ime graph, and se i o deermine when he objec sops. (e) Use eqaions o deermine when and where i sops. Did yo ge he same answer sing graphs and eqaions? 73. * The posiions of objecs A and B wih respec o Earh depend on ime as follows: 1 A = 1. m m>s; 1 B = -1 m + 16 m>s. Represen heir moions on a moion diagram and graphically (posiionerss-ime and elociy-erss-ime graphs). Use he graphical represenaions o find where and when hey will mee. Confirm he resl wih mahemaics. 74. * Two cars on a sraigh road a ime zero are beside each oher. The firs car, raeling a speed 3 m>s, is passing he second car, which is raeling a 4 m>s. eeing a cow on he road ahead, he drier of each car sars o slow down a 6. m>s. Represen he moions of he cars mahemaically and on a elociy-erss-ime graph from he poin of iew of a pedesrian. Where is each car when i sops? 75. * Olier drops a ennis ball from a cerain heigh aboe a concree floor. Figre P.75 shows he elociy-erss-ime graph of he ball s moion from he momen he ball is released o he momen he ball reaches is maimm heigh afer boncing p from he floor. (a) How is he y-ais direced: p or down? (b) Deermine he iniial heigh from which he ball is released and he final heigh o which he ball bonces. (c) Deermine he aerage speed of he ball dring he downward moion. FIGURE P.75 y (m/s) (s)

37 Problems * Waer sriders Waer sriders are insecs ha propel hemseles on he srface of ponds by creaing orices in he waer shed by heir driing legs. The elociy-erss-ime graph of a 17-mm-long waer srider ha moed in a sraigh line was creaed from a ideo (Figre P.76). The insec sared from res, sped p by aking wo srides, and hen slowed down nil i sopped. Esimae (a) he maimm speed (in m>s), (b) he maimm acceleraion (in m>s ), and (c) he oal displacemen (in m) of he waer srider. Noe ha he elociy on he graph is gien in nis of lengh of waer srider body per second. FIGURE P.76 (lengh/s) Yo are raeling in yor car a m>s a disance of m behind a car raeling a he same speed. The drier of he oher car slams on he brakes o sop for a pedesrian who is crossing he sree. Will yo hi he car? Yor reacion ime is.6 s. The maimm acceleraion of each car is 9. m>s. 78. * Yo are driing a car behind anoher car. Boh cars are moing a speed 8 km>h. Wha minimm disance behind he car in fron shold yo drie so ha yo do no crash ino he car s rear end if he drier of ha car slams on he brakes? Indicae any assmpions yo made. 79. * A drier wih a.8-s reacion ime applies he brakes, casing he car o hae 7.@m>s acceleraion opposie he direcion of moion. If he car is iniially raeling a 1 m>s, how far does he car rael dring he reacion ime? How far does he car rael afer he brakes are applied and while skidding o a sop? 8. ** ome people in a hoel are dropping waer balloons from heir open window ono he grond below. The balloons ake.15 s o pass yor 1.6-m-all window. Where shold secriy look for he racos hoel gess? Indicae any assmpions ha yo made in yor solion. 81. ** Aoiding injry from hockey pck Hockey players wear proecie helmes wih facemasks. Why? Becase he bone in he pper par of he cheek (he zygomaic bone) can fracre if he acceleraion of a hockey pck de o is ineracion wih he bone eceeds 9g for a ime lasing 6. ms or longer. ppose a player was no wearing a facemask. Is i likely ha he acceleraion of a hockey pck when hiing he bone wold eceed hese nmbers? Use some reasonable nmbers of yor choice and esimae he pck s acceleraion if hiing an nproeced zygomaic bone. 8. ** A bole rocke brns for 1.6 s. Afer i sops brning, i conines moing p o a maimm heigh of 8 m aboe he place where i sopped brning. Esimae he acceleraion of he rocke dring lanch. Indicae any assmpions made dring yor solion. Eamine heir effec. (s) 83. * Daa from sae drier s manal The sae drier s manal liss he reacion disances, braking disances, and oal sopping disances for aomobiles raeling a differen iniial speeds (Table.11). Use he daa o deermine he drier s reacion ime ineral and he acceleraion of he aomobile while braking. The nmbers assme dry srfaces for passenger ehicles. TABLE.11 Daa from drier s manal peed (mi>h) Reacion disance (m) 84. ** Esimae he ime ineral needed o pass a semi-railer rck on a highway. If yo are on a wo-lane highway, how far away from yo ms an approaching car be in order for yo o safely pass he rck wiho colliding wih he oncoming raffic? Indicae any assmpions sed in yor esimae. 85. * Car A is heading eas a 3 m>s and Car B is heading wes a m>s. ddenly, as hey approach each oher, hey see a one-way bridge ahead. They are 1 m apar when hey each apply he brakes. Car A s speed decreases a 7. m>s each second and car B decreases a 9. m>s each second. Do he cars collide? Proide wo solions: one sing eqaions and one sing graphs. Reading Passage Problems Braking disance (m) Toal sopping disance (m) Head injries in spors A research grop a Darmoh College has deeloped a Head Impac Telemery (HIT) ysem ha can be sed o collec daa abo head acceleraions dring impacs on he playing field. The researchers obsered 49,613 impacs from 43 fooball players a nine colleges and high schools and colleced collision daa from paricipans in oher spors. The acceleraions dring mos head impacs 1789% in helmeed spors cased head acceleraions less han a magnide of 4 m>s. Howeer, a oal of 11 concssions were diagnosed in players whose impacs cased acceleraions beween 6 and 18 m>s, wih mos of he 11 oer 1 m>s. 86. ppose ha he magnide of he head elociy change was 1 m>s. Which ime ineral for he collision wold be closes o prodcing a possible concssion wih an acceleraion of 1 m>s? (a) 1 s (b).1 s (c) 1 - s (d) 1-3 s (e) 1-4 s 87. Using nmbers from he preios problem, which answer is closes o he aerage speed of he head while sopping? (a) 5 m>s (b) 1 m>s (c) 5 m>s (d).5 m>s (e).1 m>s 88. ppose he aerage speed while sopping was 4 m>s (no necessarily he correc ale) and he collision lased.1 s. Which answer is closes o he head s sopping disance (he disance i moes while sopping)? (a).4 m (b).4 m (c) 4 m (d). m (e).4 m 89. Use Eq. (.7) and he nmbers from Problem 86 o deermine which sopping disance is closes o ha which wold lead o a 1 m>s head acceleraion. (a).5 m (b).5 m (c).1 m (d).1 m (e).5 m 9. Choose he changes in he head impacs ha wold redce he acceleraion dring he impac. 1. A shorer impac ime ineral. A longer impac ime ineral 3. A shorer sopping disance 4. A longer sopping disance 5. A smaller iniial speed 6. A larger iniial speed (a) 1, 4, 6 (b) 1, 3, 5 (c) 1, 4, 5 (d), 4, 5 (e), 4, 6

38 5 CHAPTER Kinemaics: Moion in One Dimension Aomaic sliding doors The firs aomaic sliding doors were described by Hero of Aleandria almos years ago. The doors were moed by hanging conainers ha were filled wih waer. Modern sliding doors open or close aomaically. They are eqipped wih sensors ha deec he proimiy of a person and an elecronic circi ha processes he signals from he sensors and dries he elecromoor-based sysem ha moes he doors. The sensors ypically emi plses of infrared ligh or lrasond and deec he refleced plses. By measring he delay beween emied and receied plses, he sysem can deermine he disance o he objec from which he plse was refleced. The whole sysem ms be careflly designed o ensre safe and accrae fncioning. Designers of sch doors ake ino accon seeral ariables sch as ypical walking speeds of people and heir dimensions. Le s ry o learn more abo aomaic sliding doors by analyzing he moion of a single-side aomaic sliding door when a person is walking hrogh he door. Figre.3 shows he posiion-erss-ime graph of he moion of he edge of he door (marked wih a red cross in he phoo) from he momen he door sars opening o when he door is closed while a person walks oward and hrogh he door. The doors are adjsed o sar opening when a person is. m away. FIGURE.3 (m) How long does i ake for he door o flly open? (a) 1.5 s (b) 3 s (c) 5.5 s (d) 11 s 9. How long does i ake for he door o close afer i is opened? (a) 11 s (b) 8 s (c) 5.5 s (d) 3 s 93. A person is walking a consan speed of 1.15 m>s oward and hrogh he sliding door. How far from he door is he person when he door sars closing? (a).3 m (b) 4.3 m (c) 6.3 m (d) 8.3 m 94. Wha is he aerage opening speed of he door? (a).1 m>s (b).3 m>s (c).6 m>s (d) 3. m>s 95. Wha is he maimm speed of he door? (a).1 m>s (b).3 m>s (c).6 m>s (d) 3. m>s 96. A 5-cm-wide person is walking oward he door. Wha is he maimm walking speed of he person ha will allow her o pass hrogh he door wiho hiing i (assme he person aims for he opening)? (a).6 m>s (b) 1. m>s (c) 1.7 m>s (d).5 m>s (s)