3.1 The Parameters of Motion
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1 CHAPTER 3 3. The Parameer of Moion In analying a moion, we wih o find informaion abou he following parameer of he moion: he ime oer which he moion of he body i being udied; he diplacemen and/or he diance raelled by he body during hi ime ineral; he iniial elociy of he body (i.e. he elociy of he body a he beginning of he ime ineral); he final elociy of he body (i.e. he elociy of he body a he end of he ime ineral); and he acceleraion of he body during he ime ineral. Time Thi refer o he duraion of he conan acceleraion, i.e. he ime ineral oer which he conan acceleraion ac. The commonly ued ymbol for hi are or, and he andard inernaional (SI) uni for ime i he econd (). Diplacemen and Diance Traelled The diplacemen of he body refer o i final diplacemen from i iniial poiion during he ime ineral in which he conan acceleraion ac. Of coure, if he body i ju raelling in one direcion along a line, he diance raelled ha he ame numerical alue a he diplacemen. Howeer, if he body reere i direcion of rael during he moion, hen he diplacemen and he diance raelled will hae differen alue. Conider he iuaion in Figure 3., which how a body moing along a number line, a depiced. The number line i marked in one-mere ineral. The body ar a poin A and, in he fir hree econd, i moe o B. Then i reere direcion and, in he nex four econd, i moe o C. If we are conidering he moion in he fir hree econd, he diplacemen of he body i 6m o he righ and he diance raelled i 6m. If we are conidering he moion oer he full een econd, he diplacemen i m o he lef and he oal diance raelled i 4m C A B Fig. 3. The uual ymbol for hee quaniie are or x. Thu, diplacemen i repreened by () or x (x) and diance i repreened by or x. The SI uni for diplacemen and diance raelled i he mere (m). Velociy and Speed The aerage elociy oer ome ime ineral i defined a he rae of change of diplacemen wih repec o ime, age or age. The aerage peed of a body oer a gien ime ineral equal he diance raelled, diided by he ime, diance raelled age. The inananeou elociy of a body i he limiing alue of he aerage elociy a he ime ineral ge ery mall (approache zero). The inananeou peed of a body i ju he magniude of he aerage elociy. 54 Eenial Tex Book
2 Uniformly Acceleraed Moion in One Dimenion In our calculaion, we will be mainly inereed in he elociy of he body a he beginning of he ime ineral oer which he acceleraion ac (i.e. he iniial elociy i or i ), and he elociy of he body a he end of hi ime ineral (he final elociy or ). To aoid exceie ue of f f ubcrip, i i a common pracice o refer o iniial elociy a u (or u); and o he final elociy a (or ). The SI uni of elociy and peed i mere per econd (m/ or m ). Acceleraion The aerage acceleraion of a body oer a ime ineral i calculaed by he rae of change of elociy wih repec o ime, f i aage. If acceleraion i conan, hi formula alo gie u he alue of he inananeou acceleraion a any ime during hi ime ineral. The common ymbol for acceleraion i a (or a). The SI uni for acceleraion i mere per econd per econd (m// or m ). 3. Moion wih Conan Velociy Thi i a paricular cae of moion wih conan acceleraion, wih acceleraion being equal o zero. Thi i a ery imple moion. The iniial elociy, he final elociy, he aerage elociy oer any ime ineral, and he inananeou elociy a any ime are all he ame. The elociy of he body and i diplacemen oer any ime ineral are conneced by he relaionhip: or. In hi paricular cae he elociy i in one direcion only, and hu he diance raelled by he body i he ame a i diplacemen during he ime ineral. Since he only relean direcion in hi moion i he direcion in which he body i raelling, hi equaion i ofen more imply wrien a where i he peed of he body and i he diance raelled during he ime ineral. Example () An ahlee run a diance of 800m in min, 47 6 a conan peed. A wha peed i he running? () A hip i ailing a a conan peed of 5 kno. How far will i ail in 4h? ( kno 0 539m ) Soluion () 800m; m, m () 5 kno m 8475m 4h ,0,04m 0km Problem-Soling Mehod If elociy i conan, here i only one formula ha i applicable o he moion:, or i rearrangemen or. Noe ha in boh of he aboe example, one of he parameer i gien in non-andard uni. To aoid poenial problem in your calculaion, alway coner all gien daa o SI uni, immediaely. Eenial Tex Book 55
3 CHAPTER Moion wih Conan Acceleraion Aerage Velociy For all moion, wheher acceleraion i conan or no, he aerage elociy i gien by diplacemen. age ime Howeer, conider a moion wih conan acceleraion, a depiced in Figure 3.. The body ar wih an iniial elociy u and accelerae for econd, finihing wih a final elociy. The area of he haded rapezium gie he diplacemen in hi ime period. u Remember ha he area of a rapezium i gien by a + b A h 0 0 where a and b are he lengh of he parallel ide and h i he diance beween he wo parallel ide. u + Therefore, he diplaceme n u + and age. Thu, when acceleraion i conan, you can deermine he aerage elociy in any ime ineral ju by finding he aerage of he iniial and he final elociie in ha ime. Newon Equaion of Moion In he lae 7 h cenury, Sir Iaac Newon, in hi udy of he moion of a body moing in a raigh line wih conan acceleraion, deried hree equaion ha relaed he fie parameer of hi moion. They are u + a u + u a where: a he acceleraion of he body; he ime ineral for which hi acceleraion ac; u he elociy of he body a he beginning of hi ime ineral, i.e. he iniial elociy of he body; he elociy of he body a he end of hi ime ineral, i.e. he final elociy of he body; he diplacemen of he body in hi ime ineral. I i poible o ole eery problem inoling he moion of a body in a raigh line wih conan acceleraion by uing hee hree equaion, ogeher wih he wo formulae inoling aerage elociy gien aboe, i.e. u + age and age. a elociy Fig 3. ime 56 Eenial Tex Book
4 Uniformly Acceleraed Moion in One Dimenion Deriaion of equaion From he definiion of acceleraion: i.e. herefore i.e. a u a a u u + a Deriaion of equaion age u + u + u + a ( u + a) u + a hi i rue for all moion acceleraion i conan, u + a (from equaion ) age u + Deriaion of equaion 3 age + u u + u a ( u)( + u) a u a u a a u a cancel he ' Problem-Soling Uing Newon Equaion of Moion Example A car i moing wih a peed of 3 0m and hen peed up uniformly for 8 0 wih an acceleraion of 4 5m. Find () he peed of he car a he end of hi ime; () he diance ha he car rael in hi ime. () u 3 0m, a 4 5m, 8 0,? u + a m () u 3 0m, a 4 5m, 8 0,? u m a Eenial Tex Book 57
5 CHAPTER 3 Noe he following. () The imple approach o hee problem i o: li all he daa gien and alo he quaniy ha you are aked o find; apply a formula ha link he unknown parameer wih he gien parameer. () If he queion ae ha he body i peeding up uniformly, i mean ha he acceleraion i conan, and hence all he equaion gien in hi ecion are applicable. (3) Thi example i a imple and raighforward applicaion in he direc ue of he equaion of moion. In hi cae he diagram doe no conribue ignificanly o he iualiaion and oluion of he problem. Howeer, in more difficul problem, he diagram ignificanly help you o iualie he problem. Ge ino he habi of keching diagram, een for imple problem. Example 3 An aeroplane i landing on a runway wih a peed of 75 0m. The lengh of he runway i 0km. The aeroplane come o a hal ju before he end of he runway. Find () i acceleraion if i low down uniformly; () he ime aken for he aeroplane o op. () u 75 0m, a?, 0, km 00m u 0 75 a. a a 400 a 34m () u 75 0m, 0, 00m, a 34375m,? u + a Noe he following. () In hi example we hae ecor in wo direcion. By aing ha he iniial elociy u 75 0m, we are aying ha he direcion of he moion of he aircraf i he poiie direcion (o he lef on he diagram). The acceleraion of he aircraf i in he oppoie direcion o i elociy (a he aircraf i lowing down). Thu, he acceleraion of he aircraf mu be negaie. The negaie ign ju mean ha he direcion of he acceleraion i oppoie o he direcion of he elociy (o he righ on he diagram). () The queion ae ha he body i lowing down uniformly. Thi mean ha he acceleraion i conan, and hence all he equaion gien in hi ecion are applicable. (3) The daa in he queion i all gien o an accuracy of hree ignifican figure; herefore he anwer hould only be gien o hree ignifican figure. Any more figure han hi implie a higher accuracy han can poibly be juified from he daa. Howeer, if you calculae a alue uch a acceleraion in hi example ha you are going o ue laer in he queion, hen keep a record of hi alue o one or wo ignifican figure more han you are uing in your anwer, and ue hi alue in furher calculaion 58 Eenial Tex Book
6 Uniformly Acceleraed Moion in One Dimenion Example 4 A car ar from re and accelerae uniformly for 0 0, achieing a peed of 0m. I hen rael a conan elociy for 0min and hen brake o a hal in 3 0. Wha i he oal diance raelled by he car? () u 0, 0, 0m age ( u + ) ( 0 + 0) 00m 0 () 0, 0m con m Thu, he oal diance raelled by hi car m (3) u 0m, 0, 3 age ( u + ) (0 + 0) 3 30m Noe: The equaion of moion only apply for moion wih conan acceleraion. Thu we hae o rea hi moion a he combinaion of hree eparae moion a follow (auming he car i moing from lef o righ): a moion wih iniial elociy of 0, and poiie acceleraion for 0 ; a moion wih conan elociy of 0m for 0; a moion wih iniial elociy of 0m and final elociy of 0, for Moion in he Earh Graiaional Field The ancien Greek philoopher Ariole (384 3 BC) augh ha heaier objec dropped faer han ligher one. Thi wa belieed o be rue for almo wo millennia, unil abou he year 600. A ha ime, he grea Ialian phyici and aronomer Galileo Galilei (564 64) proed ha all objec dropped near he urface of he Earh fall a he ame rae, regardle of heir ma. He howed ha all objec experience he ame conan downward acceleraion near he urface of he Earh. The meaure of hi acceleraion i 9 80m. I i uually repreened by he ymbol g. Any objec dropped from ome heigh near he urface of he Earh experience an acceleraion of 9 80m downward, in he abence of air reiance. Thu i coninue o accelerae Galileo Galilei (564 64) down, i peed increaing by 9 80m in eery econd unil i hi he Earh. Any objec projeced erically upward near he urface of he Earh experience a erical acceleraion of 9 80m downward. Thu i will low down, i upward peed decreaing by 9 80m in eery econd unil i ha reduced o zero, a which age he body ha come o a momenary hal, and hen he body reurn o Earh wih a downward acceleraion of 9 80m. Any body moing in he Earh graiaional field, which only experience he graiaional acceleraion g 9 80m, i aid o be in free-fall. I doe no maer wheher he body i moing up or down; i i ill decribed a free-falling. Graiaional acceleraion g i ofen decribed a free-fall acceleraion. In all free-fall calculaion, we neglec he effec of air reiance and aume ha he acceleraion i conan for mall heigh aboe he urface of he Earh. Thu he moion can be analyed a one-dimenional moion wih conan acceleraion, and all he equaion of moion, preiouly deried, will apply. Eenial Tex Book 59
7 CHAPTER 3 Example 5 An objec i dropped from a heigh of 40m. () How long doe i ake o reach he ground? () Wih wha peed doe i hi he ground? 40m a g 9.8m - () u 0, 40m, a 9 80m,? u a 9 8 () 857, u 0, 40m, a 9 80m,? u a m Noe: The ue of he word dropped here implie ha he iniial elociy of he body i zero. Becaue all he moion i downward and he acceleraion i downward, we ake hi direcion o be poiie. Thu elociy a any ime, and diplacemen and acceleraion, are all poiie. In par (), we could hae ued u + a o deermine he final peed of he body. Example 6 A firework kyrocke i projeced erically upward from ground leel wih a peed of 49m. () How high will i rie? () How long before i body fall back o he ground? (3) Wha will be i peed when i reurn o ground? a g 9.8m - Noe: In hi cae he moion i iniially upward and he acceleraion i downward. We uually ake he upward direcion o be poiie and he downward direcion o be negaie. Thu acceleraion i negaie hroughou he moion, while elociy i poiie on he upward pah and negaie when he body i falling. A he opmo poin of i pah, he elociy of he body i momenarily zero. () Mehod. Find he diplacemen when i elociy i equal o zero. u 49m, a 9 80m, 0,? u 0 49 a m () Alernaie approach. Find he ime aken for i o reach he maximum heigh and hen find i diplacemen a ha ime. u + a u + a m 60 Eenial Tex Book
8 Uniformly Acceleraed Moion in One Dimenion () Mehod. Find he ime when he diplacemen of he body (from i original poiion) i equal o zero. u 49m, a 9 80m, 0,? ( ) u or or 0 ime aken o reurn o ground 0 (3) Mehod. The body i free-falling from a heigh of 5m, wih an iniial elociy 0. Thu, ake he downward direcion a poiie. u 0,.5, 5, a 9 80m,? u + a 49m a () Alernaie approach. Find he ime aken for he body o rie o i maximum heigh (a aboe) and hen ju double i. The ime for i o fall from i opmo poin back o ground i he ame a he ime aken o rie o i maximum heigh, a hown below. u 0,.5m, a 9 8m,? u Noe ha for hi calculaion all he moion i downward, and o we ake he downward direcion o be poiie. (3) Alernaie approach. u a 40 49m a Noe ha in he abence of air reiance, he peed wih which i reurn o ground i he ame a he peed wih which i i projeced. 3.5 Applicaion: Traffic Flow & Acciden Ineigaion 3.5. Traffic Flow Car moing along a road acceleraing, deceleraing, opping ec. are an example of moion in one dimenion. Traffic engineer need o enure ha raffic run a moohly a poible wih a minimum of hold-up. They need o reearch and make deciion on he following hing. Paern of raffic flow How many car can we expec o hae moing along a paricular road in a gien ime? Timing of raffic ligh A a paricular inerecion, for how long gien he widh of he road and he expeced raffic deniy for how long mu he ligh remain green, in order o clear he raffic efficienly? Alo, for how long mu he amber ligh be on, in order o enable car o clear he inerecion before he red ligh come on? Sequencing of raffic ligh Speed limi in differen iuaion If a group of car i opped a a red ligh a he beginning of a raffic ligh equence, and hen moe off when he green ligh come on, when mu he nex e of ligh change o green, in order for hee car o be able o keep on moing wihou haing o op? Wha compulory or adiory peed limi are uiable for he road condiion a ariou place, aking ino accoun he harpne of bend? Eenial Tex Book 6
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