Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle at x =.0 m s.0 m/s. What s the acceleraton durng ths tme nterval? A. 4.0 m/s B. 3. m/s C. 6.4 m/s D. 8.0 m/s E. 5.7 m/s Moton n Two Dmensons In two- or three-dmensonal knematcs, everythng s the same as n one-dmensonal moton except that we must now use full vector notaton Postve and negatve sgns are no longer suffcent to determne the drecton Poston and Dsplacement The poston of an object s descrbed by ts poston vector, r The dsplacement of the object s defned as the change n ts poston r r r f Average Velocty The average velocty s the rato of the dsplacement to the tme nterval for the dsplacement r vavg t The drecton of the average velocty s the drecton of the dsplacement vector The average velocty between ponts s ndependent of the path taken Ths s because t s dependent on the dsplacement, also ndependent of the path Instantaneous Velocty The nstantaneous velocty s the lmt of the average velocty as Δt approaches zero r dr v lm t 0 t dt As the tme nterval becomes smaller, the drecton of the dsplacement approaches that of the lne tangent to the curve 1
Instantaneous Velocty, cont The drecton of the nstantaneous velocty vector at any pont n a partcle s path s along a lne tangent to the path at that pont and n the drecton of moton The magntude of the nstantaneous velocty vector s the speed The speed s a scalar quantty Average Acceleraton The average acceleraton of a partcle as t moves s defned as the change n the nstantaneous velocty vector dvded by the tme nterval durng whch that change occurs. vf v v aavg t t t f Average Acceleraton, cont As a partcle moves, the drecton of the change n velocty s found by vector subtracton v vf v The average acceleraton s a vector quantty drected along v Instantaneous Acceleraton The nstantaneous acceleraton s the lmtng value of the rato v t as Δt approaches zero lm v dv a t 0 t dt The nstantaneous equals the dervatve of the velocty vector wth respect to tme Producng An Acceleraton Varous changes n a partcle s moton may produce an acceleraton The magntude of the velocty vector may change The drecton of the velocty vector may change Even f the magntude remans constant Both may change smultaneously Knematc Equatons for Two- Dmensonal Moton When the two-dmensonal moton has a constant acceleraton, a seres of equatons can be developed that descrbe the moton These equatons wll be smlar to those of onedmensonal knematcs Moton n two dmensons can be modeled as two ndependent motons n each of the two perpendcular drectons assocated wth the x and y axes Any nfluence n the y drecton does not affect the moton n the x drecton
Knematc Equatons, Poston vector for a partcle movng n the xy plane r xˆy ˆj The velocty vector can be found from the poston vector dr v v ˆ ˆ x vyj dt Snce acceleraton s constant, we can also fnd an expresson for the velocty as a functon of tme: v v a f t Knematc Equatons, 3 The poston vector can also be expressed as a functon of tme: r r vt 1 at f Ths ndcates that the poston vector s the sum of three other vectors: The ntal poston vector The dsplacement resultng from the ntal velocty The dsplacement resultng from the acceleraton Knematc Equatons, Graphcal Representaton of Fnal Velocty The velocty vector can be represented by ts components v f s generally not along the drecton of ether v or a Knematc Equatons, Graphcal Representaton of Fnal Poston The vector representaton of the poston vector r s generally not along f the same drecton as v or as a v f and r f are generally not n the same drecton Graphcal Representaton Summary Varous startng postons and ntal veloctes can be chosen Note the relatonshps between changes made n ether the poston or velocty and the resultng effect on the other Lecture Quz A boy on a skate board skates off a horzontal bench at a velocty of 10 m/s. One tenth of a second after he leaves the bench, to two sgnfcant fgures, the magntudes of hs velocty and acceleraton are: A. 10 m/s; 9.8 m/s. B. 9.0 m/s; 9.8 m/s. C. 9.0 m/s; 9.0 m/s. D. 1.0 m/s; 9.0 m/s. E. 1.0 m/s; 9.8 m/s. 3
Projectle Moton An object may move n both the x and y drectons smultaneously The form of two-dmensonal moton we wll deal wth s called projectle moton Assumptons of Projectle Moton The free-fall acceleraton s constant over the range of moton It s drected downward Ths s the same as assumng a flat Earth over the range of the moton It s reasonable as long as the range s small compared to the radus of the Earth The effect of ar frcton s neglgble Wth these assumptons, an object n projectle moton wll follow a parabolc path Ths path s called the trajectory Projectle Moton Dagram Clcker Queston If a baseball player throws a ball wth a fxed ntal speed, but wth varable angles, the ball wll move furthest f the angle from horzontal drecton s: A: 0 degrees B: 30 degrees C: 45 degrees D: 60 degrees E: 90 degrees Analyzng Projectle Moton Consder the moton as the superposton of the motons n the x- and y-drectons The actual poston at any tme s gven by: r r v t 1 g t f The ntal velocty can be expressed n terms of ts components v x = v cos and v y = v sn The x-drecton has constant velocty a x = 0 The y-drecton s free fall a y = -g Effects of Changng Intal Condtons The velocty vector components depend on the value of the ntal velocty Change the angle and note the effect Change the magntude and note the effect 4
Analyss Model The analyss model s the superposton of two motons Moton of a partcle under constant velocty n the horzontal drecton Moton of a partcle under constant acceleraton n the vertcal drecton Specfcally, free fall Projectle Moton Vectors r r v t 1 g t f The fnal poston s the vector sum of the ntal poston, the poston resultng from the ntal velocty and the poston resultng from the acceleraton Projectle Moton Implcatons The y-component of the velocty s zero at the maxmum heght of the trajectory The acceleraton stays the same throughout the trajectory Range and Maxmum Heght of a Projectle When analyzng projectle moton, two characterstcs are of specal nterest The range, R, s the horzontal dstance of the projectle The maxmum heght the projectle reaches s h Heght of a Projectle, equaton The maxmum heght of the projectle can be found n terms of the ntal velocty vector: v sn h g Ths equaton s vald only for symmetrc moton Range of a Projectle, equaton The range of a projectle can be expressed n terms of the ntal velocty vector: v sn R g Ths s vald only for symmetrc trajectory 5
More About the Range of a Projectle Range of a Projectle, fnal The maxmum range occurs at = 45 o Complementary angles wll produce the same range The maxmum heght wll be dfferent for the two angles The tmes of the flght wll be dfferent for the two angles Non-Symmetrc Projectle Moton Follow the general rules for projectle moton Break the y-drecton nto parts up and down or symmetrcal back to ntal heght and then the rest of the heght Apply the problem solvng process to determne and solve the necessary equatons May be non-symmetrc n other ways Unform Crcular Moton Unform crcular moton occurs when an object moves n a crcular path wth a constant speed The assocated analyss moton s a partcle n unform crcular moton An acceleraton exsts snce the drecton of the moton s changng Ths change n velocty s related to an acceleraton The velocty vector s always tangent to the path of the object Clcker Queston A partcle s undergong constant-speed crcular moton, whch of the followng statements s correct? A. The moton velocty s a constant. B. The velocty s perpendcular to acceleraton. C. The velocty s parallel to the dsplacement. D. The acceleraton s perpendcular to dsplacement. E. The acceleraton s perpendcular to the plane of moton. Changng Velocty n Unform Crcular Moton The change n the velocty vector s due to the change n drecton The vector dagram shows v f v v 6
Centrpetal Acceleraton The acceleraton s always perpendcular to the path of the moton The acceleraton always ponts toward the center of the crcle of moton Ths acceleraton s called the centrpetal acceleraton Centrpetal Acceleraton, cont The magntude of the centrpetal acceleraton vector s gven by v ac r The drecton of the centrpetal acceleraton vector s always changng, to stay drected toward the center of the crcle of moton Perod The perod, T, s the tme requred for one complete revoluton The speed of the partcle would be the crcumference of the crcle of moton dvded by the perod Therefore, the perod s defned as r T v Tangental Acceleraton The magntude of the velocty could also be changng In ths case, there would be a tangental acceleraton The moton would be under the nfluence of both tangental and centrpetal acceleratons Note the changng acceleraton vectors Total Acceleraton The tangental acceleraton causes the change n the speed of the partcle The radal acceleraton comes from a change n the drecton of the velocty vector Total Acceleraton, equatons dv The tangental acceleraton: at dt v The radal acceleraton: ar ac r The total acceleraton: Magntude a a a r t Drecton Same as velocty vector f v s ncreasng, opposte f v s decreasng 7
Relatve Velocty Two observers movng relatve to each other generally do not agree on the outcome of an experment However, the observatons seen by each are related to one another A frame of reference can descrbed by a Cartesan coordnate system for whch an observer s at rest wth respect to the orgn Dfferent Measurements, example Observer A measures pont P at +5 m from the orgn Observer B measures pont P at +10 m from the orgn The dfference s due to the dfferent frames of reference beng used Dfferent Measurements, another example The man s walkng on the movng beltway The woman on the beltway sees the man walkng at hs normal walkng speed The statonary woman sees the man walkng at a much hgher speed The combnaton of the speed of the beltway and the walkng The dfference s due to the relatve velocty of ther frames of reference Relatve Velocty, generalzed Reference frame S A s statonary Reference frame S B s movng to the rght relatve to S A at v AB Ths also means that S A moves at v relatve to BA S B Defne tme t = 0 as that tme when the orgns concde Relatve Velocty, equatons The postons as seen from the two reference frames are related through the velocty r r v t PA PB BA The dervatve of the poston equaton wll gve the velocty equaton u PA u u PB v BA s the velocty of the partcle P measured by observer A PA u s the velocty of the partcle P measured by observer B PB These are called the Gallean transformaton equatons Acceleraton n Dfferent Frames of Reference The dervatve of the velocty equaton wll gve the acceleraton equaton The acceleraton of the partcle measured by an observer n one frame of reference s the same as that measured by any other observer movng at a constant velocty relatve to the frst frame. 8