Lecure : Moion in One Dimension Chper : Moion in One Dimension In his lesson we will iscuss moion in one imension. The oles one o he Phsics subjecs is he Mechnics h inesiges he moion o bo. I els no onl wih oobll, bu lso wih he ph o spcecr h goes rom he Erh o he Mrs! The Mechnics cn be iie ino wo prs: Kinemics n Dnmics. The kinemics ims he moion. I is imporn or he kinemics h which he ph bo ollows. I nswers he quesions such h: Where he moion sre? Where he moion soppe? Wh ime hs ken or he complee o moion? Wh he eloci bo h?. The nmics els wih he eecs h cree he moion or chnge he moion or sop he moion. I kes ino ccoun he orces n he properies o he bo h cn ec he moion. Aer h poin, we will ener ino he worl o kinemics, irs. The One-Dimensionl Moion is he sring poin or he kinemics. We will inrouce some einiions like isplcemen, eloci n ccelerion, n erie equions o moion or boies moing in one-imension wih consn ccelerion. We will lso ppl hese equions o he siuion o bo moing uner he inluence o gri lone. As Leonro D Vinci si, To unersn moion is o unersn nure, we nee o unersn he moion b obsering n oing prcice (eperimen) on i, irs. The reson is quie simple. Those hings h re o ineres in science re he hings h unergo chnge. I is bsic principle h: o unersn how somehing works, ou he o see i in cion. The workings o he unierse inclue nhing in he unierse which eperiences chnge ccoring o some repeiie pern. I is c h chnge coul be in he orm o chemicl recion, n increse or ecrese in he populion o buerlies, ec. In nure, he esies chnges o obsere re hose o moion: An objec is moe rom one posiion in spce o noher. In c, he moion generll lees he objec isel unchnge n hus simpliies he obserion. We irs nee some einiions o ieni he moion. I begins b eining he chnge in posiion o pricle. We cll i isplcemen. Displcemen is eine o be he chnge in posiion or isnce h n objec hs moe n is gien b he equion; r r r (.) where r is he inl posiion n r is he iniil posiion. The rrow inices h isplcemen is ecor quni: i hs irecion n mgniue, s we menione beore. In c, r is in -imension n wrien s r iˆ j ˆ zkˆ. In our clculions, we will onl ke -componen o r ino ccoun. In -imension, here re onl wo possible irecions h cn be speciie wih eiher plus (+) or minus (-) sign. As we know, oher emples o ecors re eloci, ccelerion n orce. In conrs, sclr quniies he onl mgniue. Some emples o sclrs re spee, mss, emperure n energ.
Lecure : Moion in One Dimension I is no enough o eine he isplcemen or moion. In orer o be useul, we lso nee o speci somehing bou ime. Aer ll, moion h cuses isplcemen o meer cn be lrge or smll epening on wheher i ook secon or housn ers o o i. The inernionl snr uni o ime is he secon. Hence, ime n spce re inericbl linke in phsics since we nee boh o eplin moion n moion is unmenl o ll res o phsics. In he lnguge o mhemics, we escribe he chnges in posiion, ime s or i i (.) where he i n subscrips epic iniil n inl, respeciel. Generll, i = =. There is no oub h his use o smbols requires some pience i ou re no lre comorble wih i. The norml enenc o suens new o phsics is o immeiel replce smbols wih numbers s soon s possible. Eperience will show ou h his is generll no goo hing o o. Wh's mos imporn o ou is he er prcicl problem o miskes: ou en o mke more lgebr errors wih numbers hn wih smbols (espie wh our insincs m ell ou, his is lws rue!). Wh's mos imporn o scienis, howeer, is h he smbols represen he essence. The numbers re hrl eer he imporn poin in unersning wh's going on. Numbers cn be chnge b siuion, choice, or n number o unepline resons, bu he mhemicl escripion o wh hppens o he smbols is wh represens he unerling ruh. In oher wors, i ou erie n equion b mhemicl rules h correcl escribe he w nure works i is he equion h is lws rue. The numbers h go ino he equion cn r b lrge mouns, bu wheer heir lues, he mus lws sis he equion! Our hope or he course is o mke he lnguge o he equions secon nure o ou so h he essence o he science represene b hem is cler. No or now, bu lso or uure. I we coninue our rher legl-souning einiions, i is noe h here re wo oher specs o moion h re imporn. The irs is h we woul like o quni he moun o moion king plce. Thereore, we shoul eine he erge eloci or spee o n objec. As we si boe, he ie o quniing moion inoles boh he isnce rele b he bo n he ime he bo ook o rel i. For h reson, i mkes since o eine he erge eloci h is he isplcemen oer ol ime,, s ollows: Tol Displcemen i i (.) Elpse Time i This equion hs he righ chrcerisics or wh we esire: i he ime inerl is smll or he isplcemen is lrge, we he lrge lue or, i.e. lo o moion is king plce. On he oher hn, i he ime inerl is lrge or he isplcemen is smll, hen he erge eloci is smll, i.e. no much is hppening s r s moion is concerne. Noice h on he grphs like he ones we showe boe, he erge eloci is jus he slope o he s.. Noe: is lws > so he sign o epens onl on he sign o.
Lecure : Moion in One Dimension Grphicl inerpreion o eloci: Consier -imensionl moion rom poin A (wih coorines i, i ) o poin B (, ). We cn plo he rjecor on grph (see Figure.). slope= B i A i Figure.: Grphicl inerpreion o eloci Then rom Eq. (.) is jus he slope o he line joining A n B. We he o noice h we onl el wih he inl n iniil poin. We know nohing bou he moion o he bo h moes beween poin A n poin B. We o no know wh ph he bo ollowe OR wh he shpe o he bo is OR wh kins o orces re pplie on he bo OR he bo pplie on surrounings? I mens h ou wke up in he ormior n he gone o clss in he morning n reurne o he ormior er 8 hours. Since he ol isplcemen is ZERO, our erge eloci is ZERO. Le s ssume h he erge eloci o he pricle be ieren or ieren ime inerls. In h coniion, he pricle s eloci will ier or ech ime inerl n i will be necessr o clcule is eloci or gien cerin ime,. This les us o eine insnneous eloci. Since he pricle m no ollow srigh line on is ph, he isplcemen ecors o h pricle will ier rom ech one b irecion n mgniue. I he isplcemen occurs in ime er cerin ime, hen he isplcemen will be r. I he isplcemen is ininiesiml smll, hen he eloci will ke cerin lue or h ime inerl. In Mhemics, lim (.4) This is he insnneous eloci o he pricle n is mgniue is clle spee. Aerge ccelerion is he chnge in eloci oer he chnge in ime: i i (.5) i The irecion o he ccelerion is in he irecion o he ecor /., n is mgniue is
Lecure : Moion in One Dimension As we he ollowe beore, we cn in he Insnneous ccelerion o he bo; Insnneous ccelerion is clcule b king shorer n shorer ime inerls, i.e. king : lim (.6) I shoul be noe h: Accelerion is he re o chnge o eloci. When eloci n ccelerion re in he sme irecion, spee increses wih ime. When eloci n ccelerion re in opposie irecions, spee ecreses wih ime. Grphicl inerpreion o ccelerion: On grph o ersus, he erge ccelerion beween A n B is he slope o he line beween A n B, n he insnneous ccelerion A is he ngen o he cure A. From now on eloci n ccelerion will reer o he insnneous quniies. One Dimensionl Moion wih Consn Accelerion As i is unersoo rom he subile, consn ccelerion mens eloci increses or ecreses he sme re hroughou he moion. Emple: n objec lling ner he erh's surce (neglecing ir resisnce). Deriion o Kinemics Equions o Moion We choose i, i, i n,,. Since =consn, hen. Then Eq. (.5) cn be wrien s, or (.7) Since is consn, chnges uniorml n ( ). From Eq. (.), we know h. Now, we cn combine ( ) n cn use Eq. (.7) we ge: ( ) ( ) (.8) This is he equion o moion or pricle in -imension, wih consn ccelerion. I we rewrie Eq. (.7), we in ( ) /. Then subsiuing his resul ino he ls erm o Eq. (.8) we in 4
Lecure : Moion in One Dimension ( ) (.9) I shoul be noe h his equion oes no epen on ime,. Freel Flling Boies We cll h reel lling objec is n objec h moes uner he inluence o gri onl. B neglecing ir resisnce, ll objecs in ree ll in he erh's griionl iel he consn ccelerion h is irece owrs he erh's cener, or perpeniculr o he erh's surce, n o mgniue g 9.8m. I moion is srigh up n own n we cn choose coorine ssem wih he posiie -is poining up n perpeniculr o he erh's surce, hen we cn escribe he moion wih Eq. (.7), Eq. (.8), Eq. (.9) wih g,. (Negie sign rises becuse he coorine ssem is chnge n he ccelerion irecion is ownwr.) So h, equions o moion or he -imensionl ericl moion o n objec in ree-ll cn be wrien s ollowing: g g g (.) Noe: Since he ccelerion ue o gri is he sme or n objec, he objec oes no ll ser hn ligh objec. One Dimensionl Moion wih Vrible Accelerion The proceure is he sme s we he one in he preious secion. The eloci ecor is r iˆ iˆ ˆj j ˆ z kˆ zkˆ. (.) Then he ccelerion is gien s. (.) Since he ccelerion is rible, hen i cn be wrien s consn. We shoul wrie he equion wih respec o he eloci ecor componens, king eriion o hem. So h, 5
Lecure : Moion in One Dimension ( iˆ iˆ ˆj ˆj kˆ z kˆ) z (.) In he one-imensionl ssem, we cn wrie he equion; ˆ i (.4) Emples n Problems Quesion.: Assume h cr eceleres.m/s n comes o sop er reling 5m. ) Fin he spee o he cr he sr o he ecelerion n b) Fin he ime require o come o sop. Soluion.: We re gien:.m 5m? ) From m b) From = + we he 5s. ( )(5) Quesion.: Assume h cr reling consn spee o m/s psses police cr res. The policemn srs o moe he momen he speeer psses his cr n cceleres consn re o.m/s unil he pulls een wih he speeing cr. ) Fin he ime require or he policemn o cch he speeer n b) Fin he isnce rele uring he chse. Soluion.: We re gien, or he speeer: s s m/ s, consn spee, hen n or he policemn: p.m/ s s s ) The isnce rele b he speeer is gien s. Disnce rele b p p p s p policemn p. When he policemn cches he speeer or, 6
Lecure : Moion in One Dimension s s, Soling or we he n () s. The irs soluion ells us h he speeer n he policemn sre he sme poin, n he secon one ells us h i kes s or he policemn o cch up o he speeer. b) Subsiuing bck in boe we in he isnce h he speeer hs ken s () 6m An lso or he policemn p p p p ()() 6m Quesion.: A rocke moes upwr, sring rom res wih n ccelerion o 9.4m/s or 4 s. A he en o his ime, i runs ou o uel n coninues o moe upwr. How high oes i go oll? Soluion.: For he irs sge o he ligh we re gien: 9.4m or 4s This gies us he eloci n posiion he en o he irs sge o he ligh: 9.4(4) 7.6m n o (9.4)(4) 5. m For he secon sge o he ligh, he rocke will go upwr wih is eloci ill i sops (Th is he Newon s Moion Lw).So, we sr wih 7.6m g 9.8m An we en up wih. We wn o in he isnce rele in he secon sge ( ). We he, (7.6) ( ) 75. 6m. g ( 9.8) Thereore, he ol isnce ken b he rocke is 75.6m 5. 75.6 94. 8m Quesion.4: A rin wih consn spee o 6km/h goes es or 4min. Then i goes 45 norh-es or min. An inll i goes wes or 5min. Wh is he erge eloci o he rin? Soluion.4: We re gien, or he rin: 6km/ h km/ min 7
Lecure : Moion in One Dimension 4min, es min,45 norh 5min, wes es I we wrie he posiion ecors or he rin or ech moion: ( )ˆ i (*4)ˆ i 4ˆ i ( cos ( )( iˆ) )ˆ i ( sin 5ˆ i ) ˆj The erge eloci is gien b; erge erge erge n Tol Displcemen Elpse Time 4ˆ i ( ( ( ) / ) ) / ( ) 7.7 (ˆ i ˆ) j ( 5ˆ) i 4 5 (ˆ i ( ˆ) j,.km/ min )ˆ i 8.km/ h is he ngle beween he inl ecor n es irecion. ˆj km/ min Quesion.5: The moion o pricle is gien b, where is isnce in meer n is ime in sec. ) Fin he eloci o he pricle er sec. b) Fin lso ccelerion o he pricle. Se wheher ccelerion is uniorm o rible. Soluion.5: We re gien, or he pricle:,equion o moion or pricle in -D. To in he eloci o he pricle cerin ime, we shoul ke he iereniion o he equion o moion: [ ], We in he eloci equion o h pricle. Then, or he secon, ( s) () m, This is he eloci o he pricle ime =secons. To in he ccelerion o he pricle, we ierenie he eloci w.r.. ime, hen [ ], m/ s In inl equion, i is seen h he ccelerion oes no epen on ime n hs consn lue o m/ s. Thereore he ccelerion o he pricle is uniorm (consn) n no rible. 8