Phys 221 Fall 2014 Chaper 2 Moion in One Dimension 2014, 2005 A. Dzyubenko 2004 Brooks/Cole 1
Kinemaics Kinemaics, a par of classical mechanics: Describes moion in erms of space and ime Ignores he agen ha caused ha moion Describe he moing objec as a paricle A paricle is a poin-like objec ha is, an objec wih mass bu haing infiniesimal size 2
Posiion The moion of a paricle is compleely known if he paricle s posiion in space is known a all ime A paricle s posiion is he locaion of he paricle wih respec o a chosen reference poin Time-posiion graph 3
Vecor Quaniies Vecor quaniies are compleely described by: magniude (size) direcion Represened by an arrow Head of he arrow represens he direcion Lengh represens is magniude Same or differen? Use an arrow! Generally prined in boldface ype, b b 4
Scalar Quaniies Scalar quaniies are compleely described by magniude only 5
Displacemen Defines he change in posiion f i a final posiion an iniial posiion Δ f i Represened as Δ (if horizonal) or Δy (if erical) Vecor quaniy + or is generally sufficien o indicae direcion for one-dimensional moion Unis are meers (m) in SI, cenimeers (cm) in cgs or fee (f) in US Cusomary 6
Displacemens Δ > 0 f i < 0 7
Disance Disance is he lengh of a pah followed by a paricle Gray-blue line shows he disance Red line shows he displacemen 8
Velociy I akes ime for an objec o undergo a displacemen The aerage elociy of a paricle is he paricle s displacemen Δ diided by he ime ineral Δ during which ha displacemen occurs: Δ Δ f Δ i 9
Velociy, con Direcion will be he same as he direcion of he displacemen (ime ineral is always posiie, Δ >0) + or - is sufficien Unis of elociy are m/s (SI), cm/s (cgs) or f/s (US Cus.) Oher unis may be gien in a problem, bu generally will need o be conered o hese 10
Speed Speed is a scalar quaniy The aerage speed of a paricle is he oal disance raeled diided by he oal ime ineral required o rael ha disance: Aerage speed oal disance/oal ime Same unis as elociy May be, bu is no necessarily, he magniude of he elociy 11
Quick Quiz (a) (b) (c) A fooball player receies a kickoff a his own goal, runs downfield o wihin inches of a ouchdown, and hen reerses direcion o race backward unil he is ackled a he eac locaion where he firs caugh he ball. During his run, wha is oal disance he raels his displacemen his aerage elociy in he direcion? 12
Insananeous Velociy The insananeous elociy equals he limiing alue of he raio Δ/Δ as Δ approaches zero: Δ lim Δ Δ 0 d d This limi is called he deriaie of wih respec o The insananeous elociy can be posiie, negaie or zero 13
Graphical Inerpreaion of Velociy Velociy can be deermined from a posiion-ime graph Aerage elociy equals he slope of he line joining he iniial and final posiions 14
Graphical Inerpreaion of Velociy Insananeous elociy is he slope of he angen o he cure a he ime of ineres insananeous elociy Insananeous speed is he magniude of he insananeous elociy aerage elociy 15
Acceleraion Changing elociy means he acceleraion is presen The aerage acceleraion of he paricle is defined as a change in elociy diided by he ime ineral during which ha change occurs: a Δ Δ f f i i Dimension: [L/T 2 ] Unis: m/s 2 (SI), cm/s 2 (cgm), f/s 2 (US Cusomary) 16
Aerage Acceleraion Vecor quaniy When he sign of he elociy and acceleraion are he same (eiher posiie or negaie), hen he speed is increasing When he sign of he elociy and acceleraion are in he opposie direcions, he speed is decreasing 17
Insananeous Acceleraion The insananeous acceleraion is he limi of he aerage acceleraion as Δ approaches zero: Δ a lim Δ Δ 0 d Insananeous acceleraion: he deriaie of he elociy wih respec o ime d d d d a d d d The acceleraion equals he second deriaie of wih respec o ime d 2 d 2 18
Graphical Inerpreaion of Aerage acceleraion: he slope of he line Acceleraion connecing he iniial and final elociies Insananeous acceleraion: he slope of he angen o he cure 19
Graphical Relaionship beween and a A each insan, he acceleraion a? The slope of he line angen o he s! 20
Graphical Relaionships beween, and a The posiion of an objec moing along he ais aries wih he ime as in Fig. (a) ()? a ()?? 21
Moion Diagrams: Relaionship beween Velociy and Acceleraion Uniform elociy (shown by red arrows mainaining he same size) Acceleraion equals zero 22
Moion Diagrams: Relaionship beween Velociy and Acceleraion Velociy and acceleraion are in he same direcion Acceleraion is uniform (blue arrows mainain he same lengh) Velociy is increasing (red arrows are geing longer) 23
Moion Diagrams: Relaionship beween Velociy and Acceleraion Acceleraion and elociy are in opposie direcions Acceleraion is uniform (blue arrows mainain he same lengh) Velociy is decreasing (red arrows are geing shorer) 24
Quick Quiz Which of he following is rue? (a) If a car is raeling easward, is acceleraion is easward (b) If a car is slowing down, is acceleraion mus be negaie (c) A paricle wih consan acceleraion can neer sop 25
One-Dimensional Moion wih Consan Acceleraion a f i + a o f i Deermine he objec s elociy a any ime Graph () is a sraigh line: posiie slope -- posiie acceleraion, negaie slope negaie acceleraion 26
a cons i + 2 f The aerage elociy is he arihmeic mean of he iniial elociy and final elociy Δ f i i + 2 f Gies displacemen as a funcion of elociy and ime 27
a cons 1 + + f i i 2 ( ) f + a f i + + f i i 1 2 a 2 Gies he final posiion in erms of he elociy and acceleraion A posiion-ime graph for moion a consan acceleraion is a parabola 28
a cons f i + 1 2 ( ) + f i i f a i + f 2 2a 2 i f 2 2 i + 2a ( ) f i Proides he final elociy in erms of he acceleraion and he displacemen 29
One-Dimensional Uniform Moion a 0 f i + f i Velociy is consan and posiion changes linearly wih ime 30
Quick Quiz Mach each elociyime graph on he lef wih he acceleraionime graphs on he righ ha bes describes he moion 31
32 Fig. 2.Table 2, p.38
Free Fall All objecs moing under he influence of only graiy are said o be in free fall All objecs falling near he earh s surface fall wih a consan acceleraion Galileo originaed our presen ideas abou free fall from his inclined planes The acceleraion is called he acceleraion due o graiy, and indicaed by g Galileo Galilei 1564-1642 33
Acceleraion due o Graiy Symbolized by g g 9.8 m/s² g is always direced downward, oward he cener of he Earh 34
Freely Falling Objecs A freely falling objec is any objec moing freely under he influence of graiy, regardless of is iniial moion Objecs hrown upward or downward and hose released from he res are all falling freely once hey are released Any freely falling objec eperiences an acceleraion direced downward, regardless of is iniial moion All equaions for objecs moing wih consan acceleraion can be applied 35
Quick Quiz Which alues represen he ball s erical elociy and acceleraion a poins A, C and E? ( a) ( b) ( c) ( d ) y y y y 0, 0, 0, a a a y y y 9.80m 9.80m 9.80m 0 s 2, a y s 2 s 2 0 36
Kinemaic Equaions Deried Find he posiion if he elociy is known as a funcion of ime from Calculus Δ Δ n i f n lim Δ 0 Δ n n n ( ) d Δ n Displacemen area under he - graph 37
Uniform Moion i consan Δ i f f ( ) d i d Δ The displacemen of he paricle during he ime ineral Δ is equal o he area of he recangle 38
a consan Moion wih Consan Acceleraion Δ i f ( ) d i f a d a i f d 1 2 a Δ 2 The displacemen of he paricle during he ime ineral Δ is equal o he area of he riangle 39
Kinemaic Equaions a d d f i 0 a d For case in which acceleraion is consan: ( ) a a d a 0 f i 0 40
41 Kinemaic Equaions, con Kinemaic Equaions, con Because i f d d d 0 ( ) 2 2 0 0 0 2 1 0 2 0) ( a a d a d d a i i i i i f + + + + a i +
Problem-Soling Sraegy 42
Soling Problem Read he problem idenify ype of problem, principle inoled Draw a diagram include appropriae alues and coordinae sysem some ypes of problems require ery specific ypes of diagrams 43
Soling Problem, con Visualize he problem Idenify informaion idenify he principle inoled lis he daa (gien informaion) indicae he unknown (wha you are looking for) 44
Soling Problem, con Choose equaion(s) based on he principle, choose an equaion or se of equaions o apply o he problem sole for he unknown Sole he equaion(s) subsiue he daa ino he equaion! include unis! 45
Soling Problem, final Ealuae he answer find he numerical resul! deermine he unis of he resul! Check he answer are he unis correc for he quaniy being found? does he answer seem reasonable? check order of magniude are signs appropriae and meaningful? 46