One-Dimensional Kinematics

Transcription

1 One-Dimensional Kinemaics Michael Fowler Physics 14E Lec Jan 19, 009 Reference Frame Mechanics sars wih kinemaics, which is jus a quaniaive descripion of moion. Then i goes on o dynamics, which aemps o accoun for he observed moion in erms of forces, or some equivalen heory. We begin wih kinemaics, and he simples case: moion in one dimension. Kinemaics is abou moving (i s he same word as cinema, meaning movies) so we hink of some small objec, like a lile ball, which sars somewhere and moves o somewhere else. This is all along a line, and we neglec for now oher ineresing moions, such as he ball spinning. To sae wha is posiion is, we need somehing o refer o, his is called a reference frame, and is jus a se of coordinae axes: for moion along a line, we only need one axis, of course, we call i he x-axis. x-axis x 1 x Δx The simples frame of reference: jus one dimension If our lile ball moves from x 1 o x, we say he displacemen Δx = x x 1. This can of course be a posiive or a negaive disance. The oal displacemen for a sequence of moves is given by adding hem wih he righ sign: if you drive o Richmond, hen drive back, your oal displacemen is zero. Inermediae posiions are irrelevan he displacemen is only he same as he oal disance raveled if all moves are in he same direcion! Speed and Velociy If you drive o Richmond and back a 65 mph, obviously your average speed is 65 mph: speed = disance driven/ime. Bu average is defined by: Average = oal displacemen/ime elapsed Therefore, for your round rip o Richmond, your average is zero!

2 As we shall see, raher han speed occupies a cenral place in dynamics, even if i doesn relae o gas consumpion. speed ime ime Insananeous Velociy This means a some given ime: wha your speedomeer is reading a ha momen. Of course, a a given insan you re a a definie place, so o make sense of insananeous we have o ake a shor inerval of ime, find he displacemen in ha shor ime inerval, and from ha figure he average. Then we ake a really shor ime inerval, shor enough ha he hasn changes significanly, and define he insananeous as he average aking shorer and shorer inervals. Obviously, he appropriae ime inerval for a car isn he same as ha for finding he insananeous of he ip of a mosquio s wing. Consider a car acceleraing from res: displacemen x Δ Δx ime 1

3 3 To find is insananeous a ime, choose imes 1, jus before and afer as shown. The average in he ime inerval 1, is v x x x vavge 1 The a an insan v() is he limi of his as he ime inerval beween 1 and is made smaller and smaller, v lim 10 since his is he definiion of he derivaive of a funcion. x x 1 dx d Acceleraion Jus as is rae of change of posiion, acceleraion is rae of change of. The average acceleraion beween imes 1 and is The acceleraion a ime a v a v v v dv lim d Suppose a car acceleraes from res, he increasing a a seady rae so ha afer one second i s going m per sec, afer secs 4 m per sec, and so on for a few seconds: m per sec 4 Time in seconds

4 4 During his period, he car has a consan acceleraion of + meers per sec per sec. The acceleraion is he derivaive of he, ha is, he acceleraion is he slope of he curve in he graph of as a funcion of ime. For he case of consan acceleraion a, and his is easily inegraed o find dv a d v v0 a. (We ve aken he iniial zero in he graph above.) In fac, we can ake he nex sep for he case of consan acceleraion: inegraes o give dx This formula is very useful for moion wih graviy. v v a 0 d x x v a 0 0. Essenial calculus! if v 0, a are consans, where c, c' are consans of inegraion, fixed by he given daa. v0d v0 c, ad a d a c Disance Traveled is Area Under Curve in Velociy/Time Graph Look firs a he consan acceleraion graph above: he increases from zero a a seady rae, reaching 4 meers per second afer seconds. How far has he car raveled in hose four seconds? The average is clearly half he final (since i sared from res) so disance raveled x is given by x = ½ x4x = 4. Bu his is jus he area of he riangle lying under he sraigh line plo of! I urns ou ha his is always rue: for a car moving wih any variable,

5 5 v() m per sec Time in seconds Δ during he shor ime inerval Δ beween he wo verical dashed lines, he car ravels a disance x ime = v()δ. Bu his is almos exacly he area under he curve beween he wo doed lines: ha is, corresponding o he ime inerval Δ. This becomes exac as we ake smaller and smaller Δ, and we see ha he oal disance raveled is equal o he oal area when he graph is divided ino narrow verical srips, wih he srip widhs finally going o zero. v() m per sec Time in seconds The area under he curve equals he oal disance x() raveled in ime This is jus he definiion of he inegral: so oal disance x raveled from ime zero o ime is given by: 0 x v d Exercise: prove from his formula ha dx()/d = v().

6 6 Equaions for Consan Acceleraion From above, and v v0 a, x x v a 0 0. I s also useful o have a formula for he as a funcion of disance raveled in consan acceleraion: o find his we jus eliminae beween he wo equaions above: so from which vv0 / a, 1 x x v vv / a a v v / a v v a x x. 0 0 I s also worh bearing in mind ha in consan acceleraion, he average over a period of ime v, is jus he average of he iniial v 0 and he final v 1 : v v0 v 1. Esimaing Acceleraion As we ll discuss in he nex lecure, a falling ball has consan downward acceleraion of 9.8 meers per sec per sec. This is denoed by he leer g: g 9.8 m. sec I s ofen convenien o give acceleraions in unis of g: as we shall see shorly, his gives a comparison of he magniude of he acceleraing force on he objec wih he objec s weigh. Thus an asronau in a ship acceleraing a 6g feels a force equal o six imes his or her own weigh. This migh lead o problems wih, for example, breahing. To find ou wha happens o a es pilo who experimens on himself wih huge acceleraions check on John Sapp! Exercises

7 7 1. If I drop a ball on he floor, wha (very approximaely) is is acceleraion during he bounce off he floor while i s in conac wih he floor? (Answer: use v ah : if you drop i from h, say one meer. I hen comes o a hal afer one cm or so. Of course, is upward acceleraion during ha period won be uniform, bu le s say i is, he same formula will give a = 100g! And nonuniformiy means i s even more a he peak.) Noe: his example is explained more fully in he nex lecure.. This shows ime for a quarer mile, and speed afer quarer mile (Indianopolis, 004). (Answer: Tha s 31 mph, say 500 kph, afer 4.5 seconds, abou 10 kph per sec, or 10,000/3600 m sec per sec, say 30 m sec -, or 3g, very roughly.) According o Wikipedia: A Top Fuel dragser acceleraes from 0 o 100 mph (160 km/h) in as lile as 0.8 seconds. Wha is he g-force his driver experiences? (I s 5.7g.) A producion Porsche 911 akes 8 seconds o reach 100 mph.