Part 2 KINEMATICS Motion in One and Two Dimensions Projectile Motion Circular Motion Kinematics Problems

Transcription

1 Pa 2 KINEMATICS Min in One and Tw Dimensins Pjecile Min Cicula Min Kinemaics Pblems

2 KINEMATICS The Descipin f Min Physics is much me han jus he descipin f min. Bu being able descibe he min f an bjec mahemaically is an impan cmpnen he sudy f physics. Much f he geneal descipin will seem like i shuld be inuiive. Bu ha is lagely because we ae aleady familia wih he wds used f he descipin - ie, hey ae a pa f u evey day vcabulay. Bu "displacemen", "velciy", and "acceleain" have vey specific meanings - and he cnnecin he mahemaical descipin wih which yu will need be vey cnvesan is impan. Afe a bief discussin f he ideas invlved, he me fmal mahemaical descipin will be develped. Tha is, saing fm he fundamenal definiins f velciy and acceleain in ne-dimensin in ems f ime deivaives f he psiin cdinaes, epessins can be bained f he psiin and velciy as funcins f ime if hee is a knwn cnsan acceleain. Thse equains will be vey useful in sme special case pblems. We will hen eend he ideas pblems invlving min in w and hee dimensins. Bh pjecile min and cicula min ae impan eamples f w-dimensinal min. Upn cmpleing his bief sudy, yu shuld be cmfable wih he geneal descipin, he mahemaical descipin, and he gaphical descipin f min, and shuld be able g back and fh beween hse descipins. (And if yu cann, yu shuld wk caefully hugh he deailed descipins in he e.) ONE DIMENSIONAL MOTION When an bjec mves in ne dimensin - f eample a bead which is cnsained mve alng a seched wie - is min can be descibed in a numbe f diffeen ways. F eample, ne culd jus descibe in wds wha he bead des, ie, whee i sas, wha diecin i mves, hw quickly i mves, whehe i speeds up slws dwn changes diecin f cmes a sp. One culd even skech a seies f picues shwing he lcain f he bead a successive imes (hink f he individual fames f a mvie - f individual images f a "flip-bk"). Bu as we will see, a gaph f is psiin as a funcin f ime can cnvey a much me pecise descipin f he bjec's psiinal hisy han eihe he wds he individual "snap-shs" picues f is psiin a diffeen imes. And, when apppiae, a fmal mahemaical descipin can be me pecise ye - as a specific funcin which epesens he psiin as a funcin f ime can cnain all f he infmain ha can be knwn abu he bjec's min. Pa f he difficuly - if ha is he igh wd - f kinemaics invlves he pecise use f he wds and ideas, which ae als a pa f u cllquial language. F eample "speed" and "velciy" ae n inechangeable ems f descibing hw quickly smehing mves - paly because "velciy" als cnains infmain abu he diecin f avel. In addiin, i is fen necessay disinguish beween aveage and insananeus values f hse quaniies. (T say ha a ace ca a Indianaplis aveages 225 mph f fu laps duing qualifying des n mean is speed was cnsan a 225 mph. I's insananeus speed vaies - slwe in he uns and fase n he saighs. Is velciy is cninuusly changing as is diecin als changes. And is aveage velciy is ze f he fu laps - because is ne displacemen is ze if i finishes eacly whee i saed, a he Sa-Finish line!) In any descipin f min, i is necessay esablish a cdinae sysem - say he -ais, wih a well defined igin ha cdinae sysem. The bjec's displacemen is hen jus is psiin wih espec he igin, and can be epesened by he cdinae () epesening is disance alng he ais

3 =0 >0 () fm he igin as a funcin f ime. Ofen, i is useful descibe he bjec's iniial psiin as, whee he subscip simply implies ha be he value f he displacemen fm he igin f cdicaes a he ime =0, ie, when he pblem descipin sas. The funcin hen simply saes whee he bjec is a any he ime. When descibed in his way, i is cmmn ha psiive displacemens will be he igh and negaive displacemens will be he lef f he igin. Changes in displacemen ae defined as = 2 1, and can be eihe psiive negaive depending n whehe 1 2 is he lage value. The aveage velciy will always be he change in psiin divided by he ime ineval duing which he change ccued, ha is Aveage velciy: v avg = and he aveage velciy ve he ime ineval = 2 1 can be eihe psiive negaive depending n whehe he change in psiin is psiive negaive. This shuld n be a all cnfusing, hweve, because i simply says ha an bjec mving he igh - ie, in he psiive -diecin has a psiive velciy and he velciy is negaive if i mves he lef. An insananeus velciy can hen be defined in ems f he aveage velciy given abve f an infiniesimally small ime ineval - ie, as he limi f ges ze. Bu ha simply definess he deivaive f () wih espec ime. = Δ Δ Insananeus velciy: Δ v = lim Δ 0 Δ = d d Acceleains ae similaly defined, ha is Aveage acceleain: a avg = v 2 v = Δv Δ Insananeusly acceleain: Δv a = lim Δ 0 Δ = dv d = d 2 d 2 The acceleain can als be eihe psiive negaive, depending n whehe he velciy is inceasing deceasing. (Bu nice ha simply saying an bjec is slwing des n mean is acceleain is negaive. F eample, he velciy f an bjec ha is mving he lef bu slwing is becming less negaive - hence is acceleain is psiive.) The cncep f acceleain is ypically much less inuiive han ha f velciy. Pa f he easn f ha is ha u sense f min is vey visual - and we can see an bjec mving and can deec whehe i is mving quickly slwly. Tha is, u visual sense f an bjec's min is vey sensiive is velciy - bu i is much me difficul sense acceleain - a quaniy ha is defined in ems f he ae f change f velciy. I is f ha easn ha we need be caeful in dealing wih he cncep f acceleain. And i is acceleain ha is ulimaely cnneced Newn's laws ha elae fces min.

4 The Mahemaics f One-Dimensinal Min If () is a funcin ha epesens psiin as a funcin f ime, hen he velciy and acceleain ae given by he deivaives v = d d and a = dv d = d 2 d 2 by definiin. Ne ha a psiive velciy means he diecin f min is in he psiive diecin. If he acceleain is psiive, i means ha he change in velciy is psiive - ie, eihe a psiive velciy is inceasing a velciy is becming less negaive. The invese elains can als be wien. Tha is, given he acceleain as a funcin f ime, he velciy and he psiin as funcins f ime can be deived: v() v = a() d and () = v() d = v + a() d d The inegals ae paiculaly easy d when he acceleain is a cnsan - ie, when a() = a and hey lead he kinemaics equains ha ae cmmnly used in inducy pblems descibing min in ne w dimensins. The kinemaics equains f cnsan acceleain: If he acceleain is cnsan, he ime inegals simplify since a cmes uside he inegals: v() v = a and () = ( v + a) d = v + 1 a 2 2 Tha is, if he acceleain is a cnsan and equal he value f a, hen he velciy and psiin funcins ae given by v() = v + a and () = + v a 2 A hid useful equain elaes displacemen, velciy, and he cnsan acceleain by eliminaing he ime vaiable fm he abve equains. This hid kinemaics equain is bained by slving he v() equain f, subsiuing ha epessin in he psiin equain (), and hen eaanging ems. I is useful, bu is n independen f he he w. v 2 v 2 = 2a( ) Many pblems in bh ne and w dimensinal min will uilize hese equains. They will be he ms impan equains yu have descibe he min f an bjec. Bu yu shuld als ealize ha hey ae special case - ha is, hey nly apply when he acceleain is a cnsan.

5 Eamples: A ca acceleaes fm es wih an acceleain a. Deemine he disance avelled and he speed afe a ime. [Nice ha v =0 and he disance avelled is jus -. S he () equain cnains jus he igh infmain and yu can slve f he disance.] T find he disance he ca equies sp fm a given speed, i is he final speed ha is ze and he acceleain wuld be negaive. Nice ha if he ime sp is n needed, jus using he equain v 2 v 2 = 2a( ) is all ha is needed. An bjec is hwn saigh up wih an iniial speed v having been eleased a a heigh H abve he gund. Deemine he maimum heigh he bjec achieves, he al ime i is in he ai, and he speed wih which i his he gund, assuming ha he nly acceleain afe he bjec is eleased is due gaviy. [Rewie he kinemaics equains in ems f y(). Le y =H and he acceleain a=-g (since he acceleain is in he negaive y-diecin).] T slve f he maimum heigh, yu need knw hw lng i akes each ha heigh. Se vy=0 and slve f he ime. Subsiue ha value f ime in he y-equain find y ma. (Ie, yu mus maimize he funcin y().) T slve f he al ime i is in he ai, yu find he ime ha i his he gund - ie, se y() =0 and slve f. Finding he speed wih which i his he gund jus equies subsiuing ha value f in he velciy equain. Gaphing psiin, velciy, and acceleain. The kinemaics - min - f bjecs can be descibed by he equains ha lcae an bjec as a funcin f ime - by simply gaphing he psiin as a funcin f ime. Seeing he gaph f an bjecs psiin can be vey helpful in deducing wha he bjec is acually ding, ie, hw i is mving jus as he equains can be helpful in pecisely deemining he lcain f an bjec a any paicula mmen in ime. Cnsan Velciy: () = + v When an bjec is mving wih a cnsan velciy in ne dimensin, i s psiin changes equal amuns in equal incemens f ime. Hence, a gaph f i s psiin () will be a linea funcin f ime, ie, a saigh line wih he slpe f he gaph equalling he bjec s cnsan velciy. If he slpe is psiive, he bjec is mving in he psiive -diecin and if negaive, hen in he negaive -diecin. () v>0 Cnsan Acceleain: () = + v a2 When an bjec undeges a cnsan acceleain in ne dimensin, is velciy changes an equal amun in equal incemens f ime. Tha is, i changes a a cnsan ae. Hence, gaphing he velciy v() as a funcin f ime wuld esul in a linea gaph whse slpe (eihe psiive negaive) epesens he value f he acceleain (ie, a=dv/d). Similaly, a cnsan acceleain wuld mean ha he slpe f he psiin vs ime gaph wuld cnsanly be changing - esuling in a paablic funcin f vs.

6 v() v a>0 v() = v + a I is impan be able gaph psiin, velciy, and acceleain as funcins f ime - eihe saing fm he psiin funcin saing fm a knwn acceleain. If yu can pl he psiin f an bjec as a funcin f ime, hen yu can als pl is velciy as a funcin f ime because v() is jus he slpe f he vs gaph a each pin. Wheneve he slpe f vs is ze, v is ze. When he slpe f vs is cnsan, he velciy cuve is fla since ha means he velciy is n changing. If he slpe f vs is deceasing, hen he velciy is deceasing - eihe becming less psiive me negaive. A negaive velciy jus means ha is deceasing (ie, he bjec is mving he lef), ec. By he same ken, if he velciy is changing, he acceleain is nn-ze. If v vs is inceasing, he acceleain is psiive, if i is deceasing, hen a is negaive. () v () = + v + 1 a a 2 2 F eample, if an bjec saed wih ze velciy a sme psiin elaive he igin f cdinaes, acceleaed up sme cnsan speed mving he igh, hen slwed as sp and evesed diecin all in ne smh ansiin, acceleaed up a cnsan speed back wad he igin, wen pas he igin and slwed a sp, he psiin, velciy, and acceleain gaphs wuld lk like: Psiin v Velciy a Acceleain One shuld als be able sa wih he gaph f he acceleain vesus ime and geneae he velciy and psiin cuves, since a cnsan acceleain means a velciy ha changes linealy (eihe inceasing deceasing, depending n he sign f a), ec. Ne ha if a is psiive, v vs is linealy inceasing and vs cuves up (as a paabla). When he acceleain is negaive, vs cuves dwn. Nice ha in a pblem invlving w bjecs - like a ca and a uck - gaphing bh as funcins f ime n he same aes can fen give a clea indicain f hw a specific pblem shuld be slved. F eample: Suppse a uck is aveling a cnsan speed. And jus as i passes a ca which has spped, he ca acceleaes a a cnsan ae. Hw lng will i ake befe he ca passes he uck - and hw fa will hey have aveled duing ha ime? Gaphing bh uck and ca shws hw slve he pblem. The ca veakes he uck when he w funcins ha descibe hei min inesec. S he pblem can be slved by seing he equains equal and finding bh when he ca passes and S, hw fa hey've aveled. S S Tuck Ca T

7 MOTION IN TWO AND THREE DIMENSIONS The cmplee descipin f min in w and hee dimensins equies being able lcae bjecs in hee-dimensinal space (ahe han jus alng ne ais). Rahe han lcaing an bjec by a single cdinae, ne mus use vecs keep ack f w hee cdinaes simulaneusly and be able descibe hw hse cdinaes change in ime. Wha fllws is a bief eview f he essenial ppeies f vecs. VECTORS - Hw descibe quaniies which have diecin Any quaniy ha equies bh a magniude and a diecin f is cmplee descipin is said be a vec quaniy. Eamples include he displacemen f an bjec fm a pin f efeence (ie, he igin f is cdinae sysem), he velciy f an bjec (wih is speed epesening he magniude f he vec and sme angle giving is diecin), a fce n an bjec (whee he diecin f he fce is as impan as he "sengh" f he fce). The cmplee descipin f he vec A is given eihe by is cmpnens ( A, A y ) by is magniude and diecin (A, θ) whee he angle θ is measued wih espec he -ais. If he cmpnens ae knwn hen bh he magniude and angle can be deemined. Ay A A = A 2 + A y 2 and θ = acan A y A θ A If he magniude and diecin ae knwn, hen he cmpnens ae: A = A cs θ and A y = A sin θ A paiculaly useful nain descibe a vec quaniy is he ijk nain whee î is a "uni vec" in he diecin f he -ais, ˆ j is a uni vec in he y-diecin, and k ˆ is a uni vec in he z- diecin. Tha is, he abve vec A wuld be given by: A = A ˆ i + A yˆ j = ( Acsθ)ˆ i + ( Asinθ)ˆ j A Ne n Vec Nain: I is useful ne ha when efeing a vec quaniy whaeve i is he quaniy is idenified as a vec by making symbl in bldface ype and including a small aw ve he symbl. Scala quaniies including he cmpnens f vecs ae indicaed by ialicized and nmal ype. In yu wn wk, yu shuld always shw vec signs ve quaniies which have bh magniude and diecin. Tha is, he vec saemen R = A + B is n he same as R=A+B, since he vecs A and B ae n necessaily cllinea, s hei magniudes d n necessaily add.

8 Adding and Subacing Vecs Many physics pblems equie adding vecs. F eample, cnside he al displacemen f an bjec ha fis mves an amun descibed by he displacemen vec A fllwed by a secnd displacemen given by he vec B. The al displacemen - "ne" displacemen - culd be given by he vec R (f "esulan vec") which is wien: R = A + B = ( A + B ) ˆ i + ( A y + B y ) ˆ j Tha is, he and y cmpnens f he al displacemen ae jus he sum f he cmpnens and f he y cmpnens f he vecs ha wee added. B y Ay A A B R B Thee ae w ways display he vec sum gaphically. Eihe shw bh vecs dawn in he de f he summain (he s-called "ip--ail" mehd) shw bh vecs dawn fm he igin and fm he paallelgam defined by he w vecs. The w mehds ae necessaily equivalen. The paallelgam mehd may seem me apppiae when w fces ae acing n an bjec simulaneusly - and yu wan cnside he al fce ne fce ha is acing. The magniude f he esulan vec is given by: Ay A R R = R 2 + R 2 y = ( A + B ) 2 + ( A y + B y ) 2 B y A B B whee he cmpnens ae jus he sums f he cmpnens f A and B. And he diecin f he vec R wih espec he -ais is jus θ = acan R y R ( ) Subacing vecs - jus like subacing numbes - is jus he addiin f w bjecs, ne f hem negaive! Tha is, defining he vec C = A B is he same as jus adding he vecs A and B. In vec nain: C = ( A B ) ˆ i + A y B y j ( ) ˆ Ay By A B C Nice ha he diffeence f w vecs is epesened gaphically as he vec fm he "ip" f he secnd vec he "ip" f he fis - ie, fm B A. Als nice ha he vec is jus he "he" diagnal f he paallelgam defined by he vecs A and B. A B Adding and subacing vecs will be vey impan in dealing wih mins in w and hee dimensins, wih he ne fce n an bjec subjec seveal fces he al mmenum f a sysem f paicles. Be sue yu knw hw cmbine he cmpnens f vecs.

9 Muliplying Vecs Thee ae w ways muliply vecs. Neihe mehd will be impan unil lae in he cuse. Bu when hey mehds ae impan, yu will need be able disiguish beween hem and inepe he esuls f muliplying vecs. Scala Pduc The scala "d" pduc f w vecs yields a scala quaniy ahe han anhe vec. The inepeain f he scala pduc is ha i epesens he pduc f he magniude f ne f he vecs and he cmpnen f he he vec paallel he fis. Tha is: C = A B = A Bcsθ ( ) = B Acsθ ( ) θ A A cs θ B The scala pduc is impan in he definiin f wk and penial enegy and in he develpmen f he wk-enegy heem and ulimaely he idea f enegy cnsevain. Nice ha he scala pduc depends n he cmpnens f he vecs paallel each he. I is a maimum when he w vecs ae paallel and ze when A and B ae pependicula. Vec Pduc The vec "css" pduc f w vecs yields anhe vec quaniy and is defined in a paiculaly way. Tha is: C = A B whee he magniude f C is given by C=A(B sinθ ) B(A sinθ ) A θ B The magniude f he css pduc f w vecs is jus he aea f he ecangle fmed defined by he w vecs being muliplied. The diecin f he css pduc vec is pependicula he paallelgam. Nice ha he vec pduc is geaes when he w vecs being muliplied ae pependicula and is ze when hey ae paallel. The vec pduc is impan in he definiin f que and angula mmenum. TWO DIMENSIONAL MOTION PROBLEMS USING VECTORS Descibing min in w and hee dimensins jus equies cmbining he ideas f min in ne dimensin and he descipins and ppeies f vecs. The essenial "pinciple" is his: The, y, and z diecins can be eaed independenly. Tha is, he equains ha descibe he min in each diecin can be wien as if hey wee a ne-dimensinal pblem. (Thee ae ecepins his "ule" - alhugh hey ae ae. If sme fce acs in he y-diecin, bu depends n he value f, f eample, he and y equains cann be eaed independenly.) We will als assume ha alhugh a pblem migh be hee-dimensinal in naue, he min ha ccus can be cmpleely descibed in w

10 dimensins (f eample, when yu sh a baskeball, is min nce i leaves yu hand is all in a plane egadless f whee yu wee n he cu when yu k he sh). Thee ae eally nly w ypes f pblems in his chape. Pjecile min pblems ae idenical he fee-fall pblems in he chape n ne dimensinal min wih he added cmplicain ha hee is a hiznal cmpnen he min. Bu he acceleain is nly due he gaviainal fce, s nly appeas in he y-equains. In all such pblems, always begin wih he kinemaics equains f each cdinae. Assuming ha he acceleain is cnsan, wih cmpnens a and ay., he and y kinemaics equains becme: v y( ) = y + v y + 1 a 2 2 y ( ) = v + a v y ( ) = v y + a y () = + v a 2 Slving pblems hen becmes an eecise in using he abve kinemaics equains by making hem specific he pblem given (f eample by incpaing given values f he iniial psiin, velciy, and he acceleain cmpnens in he equains). Cicula min pblems invlve an idea ha is difficul see - bu nce seen, i will always be ue f any bjec which mves in a cicle (hence is wh leaning!). Pjecile Min Descibing he min f a pjecile is a gd eample f w dimensinal min. We will assume ha he nly fce ha acs n he pjecile afe i is launched (wih sme iniial velciy v) is he gaviainal fce - which implies ha he nly acceleain is g in he negaive y-diecin. The veical and hiznal mins ae hen independen f each he - and can be slved sepaaely. Suppse a pjecile is launched fm an iniial heigh H, wih an iniial velciy v which is a an angleθ wih espec he hiznal. The kinemaics equains hen becme: y =H θ =0 a = 0 a y = g v = v csθ v y = v sin θ y( )= y + v y a y 2 y( ) = H + v sinθ v y ( )= v y + a y v y ( ) = v sinθ g ( ) = + v ( ) = v csθ v ( ) = v = v csθ ( ) 1 2 g2 ( ) Thee ae nw a numbe f quaniies ha can be slved f: T find he maimum heigh f he ajecy, yu need find he ime a which i eaches ha pin f is pah. Bu ha is when he y- velciy ges ze. Then subsiue in he y() equain. T find he hw lng he pjecile is in he ai, find he ime ha i his he gund - ie, find when y=0. The ange al hiznal disance i avels while i is in he ai is jus he () evaluaed a he ime a which i his he gund. The velciy i has a ha mmen is hen jus fund fm he w velciy cmpnens evaluaed a ha ime - he vec epessin being wien in vec nain as v ( ) = v ( )ˆ whee ˆ i and ˆ j ae uni vecs in he and y diecins, especively. i + v y ( )ˆ j

11 The ajecy f he pjecile (ie, he equain f he pah i akes) can be bained by hinking f he and y equains as paameic equains. Slving () f and subsiuing in he y() equain yields an equain f y in ems f. Gaphing ha equain wuld yield a cuve which epesens he pah ha he pjecile wuld fllw. Thee ae many vaians n he abve pjecile min pblem, f cuse, bu hey all have he same essenial elemens. Cicula Min and Min in Cuved Pahs The he class f pblem ha will appea hughu he cuse ae hse in which he bjec yu ae ying descibe avels in a cicle ( jus a pin f a cicle). Think f all bi pblems, a ball n a sing which yu whil aund like a slingsh, a pendulum, a ca ging hugh a cnsan adius un, a lle case ide, a es-ube in a cenifuge. All f hese pblems invlve sme bjec subjec vaius fces mving in an ac cicle. The pinciples discussed hee will apply all hse pblems and me. The essenial pin make in his discussin is ha even if he speed f he bjec is cnsan, he acceleain is n. The easn, f cuse, is ha acceleain is a vec. And as he bjec mves hugh is pah, he diecin f is velciy vec keeps changing. And a changing velciy implies a nn-ze acceleain - even if i is jus he diecin f he velciy ha is changing. This pin will becme me clea when we eamine he fces ha cause he bjec being descibed mve in a cicle. y v θ R Cnside an bjec which mves in a cicle f adius R a cnsan speed. I csses he ais a ime =0 and he angle θ inceases a a cnsan ae - ha is he angle can be epessed θ=ω, whee ω is a cnsan. [The quaniy ω is called he angula velciy and is jus he ae a which he angle changes - ie, ω=dθ/d.] When he bjec is a he angle θ wih espec he ais, is cdinaes ae: = R csθ = Rcs(ω) and y = Rsinθ = Rsin(ω) The vec () is hen wien in vec nain as: Velciy: By definiin: () = ( )ˆ i + y( )ˆ j = R cs(ω ) ˆ i + R sin(ω ) ˆ j v() = d d = d d ˆ i + d y ˆ j d The velciy vec is hen bained by aking he deivaive f he psiin vec wih espec ime. The deivaives f he cmpnens f jus invlve aking he deivaives f he sine and csine funcins, since he adius f he cicula pah R des n change. By chain ule, he deivaives ae: d d cs(ω) = ωsin(ω) and d sin(ω) =ω cs(ω) d S he velciy f he bjec in unifm cicula min can be epessed in vec fm as

12 v() = ωrsin(ω) ˆ i + ωrcs(ω) ˆ j Alhugh i may n be bvius fm he mahemaical descipin, he velciy vec is angenial he cicle. Nice ha he magniude f he velciy vec is v = v 2 + v 2 y = ω 2 R 2 ( sin 2 (ω)+ cs 2 (ω)) Bu since sin 2 θ + cs 2 θ =1, he angenial velciy v can be elaed diecly he angula velciy ω. Tha is, V = ωr ω = v/r. This esul will be vey useful lae in he cuse. Als nice ha v=ωr=2πr/t whee T is he ime cmplee ne ain (called he peid f he ain) - ha is, he angenial speed is jus he cicumfeence f he cicle divided by he ime cmplee ne ain, which makes sense. Acceleain: By definiin: a() = d v d = d v d ˆ i + d v y d ˆ j Since we have he cmpnens f he velciy vec, he deivaives can be aken jus as was dne wih he () vec bain v(). Tha is, v = ωrsin(ω ) and v y = ωr cs(ω ) S he acceleain vec is: a = ω 2 R cs(ω ) ˆ i ω 2 Rsin(ω ) ˆ j = ω ( 2 Rcs(ω ) ˆ i + Rsin(ω ) ˆ j ) Nice ha he epessin in paenheses is jus he iginal () vec. Tha is, a = ω 2 = ( ω 2 R) ˆ = v 2 ˆ R whee ˆ is he uni vec pining fm he igin wad he bjec, s he () = R( ) ˆ. The significance f his esul is ha he acceleain assciaed wih an bjec in unifm cicula min is ppsie in diecin he vec ha lcaes he paicle (ie, a is in he ˆ diecin). Tha is, he acceleain is wad he cene f he cicle. Such an acceleain is called he cenipeal acceleain (meaning "cene seeking"). Cenipeal Acceleain Since ω can be elaed he angenial velciy by ω=v/r, hen he cenipeal acceleain assciaed wih an bjec mving a speed v in a cicula pah f adius R is given by: a c = v 2 ˆ R v 2 -vec is wien v 1 v 1 Δv v 2 v a c

13 Tha is, he acceleain f an bjec mving in a cicula pah is wad he cene and has magniude If he speed is n cnsan: If he angenial speed f he bjec is n cnsan, hen in addiin he adial cmpnen f he acceleain, hee will be a angenial cmpnen equal he ae a which he angenial speed is changing - ie, a an.= dv/d. Tha is a = a c + a an = v 2 R ˆ + dv θ d ˆ v 2 v 1 whee ˆ and diecins. ˆ θ ae uni vecs alng he adial and angenial Nice ha he descipin f he acceleain f an bjec aveling in an ac is based n he deivaives f he psiin vec - hence i is by definiin an insananeus acceleain. I des n even depend n he min being unifm, even being in a cicula pah. Tha is, if an bjec avels in any ac cuved pah, he acceleain a a any pin n he pah has, in geneal, a angenial and a adial ( cenipeal) cmpnen and is epessed, in geneal, as a vec sum f he w cmpnens. The angenial cmpnen f he acceleain is he ae a which he speed is changing and he cenipeal acceleain (which is always pependicula he pah) is deemined by he insananeus speed and he adius f cuvaue f he pah ajecy a ha pin. The esuling acceleain vec has w cmpnens - he cenipeal acceleain ha is pependicula he diecin f avel and pins wad he cene f he adius f cuvaue, and he angenial acceleain ha is assciaed wih any change in speed f he bjec as i avels aund is cuved pah. a a = a c + a an whee a c = v2 a c = v 2 R and a an = dv d The magniude and diecin f he acceleain can hen be fund by he usual ules f any vec quaniy: a = a 2 2 c + a an Sme impan pblems invlving hese ideas include cas avelling in cuved pahs and cicula bi pblems. When he eah bis he sun he mn bis he eah, he acceleain f he biing bjec is wad he cene f he cicula pah. Undesanding ha helped Isaac Newn deemine he law f he gaviainal fce which led he eplanain f he ellipical planeay bis - and ulimaely he eplanain f Keple's laws f planeay min. v 3 a a a c a c a c v a an a an

14 KINEMATICS QUESTIONS AND PROBLEMS 1. Cnside a bead which can slide alng a lng saigh wie (he -ais). Suppse ha is psiin as a funcin f ime is shwn in he gaph f vs. Descibe in wds he min f he bead. On he secnd gaph, skech he velciy as a funcin f ime. () v() Idenify n he psiin and velciy cuves when: a. The bead is mving he igh a cnsan speed. c. The bead is acceleaing he igh. b. The bead is mving he lef a cnsan speed. d. The bead is acceleaing he lef. 2. =0 2v Suppse a mable wih iniial speed v lls dwn a amp, as shwn, and dubles is speed. Skech gaphs f is displacemen in he -diecin and f is speed as funcins f ime. v 3. Suppse yu dp a glf ball fm a heigh f ne mee and i ebunds a heigh f 0.8 mees. Wha is he change in velciy ha ccus when i sikes he gund? If i was in cnac wih he gund f 200 msec, wha was is acceleain while i was uching he gund? Wha was he al ime fm when i was dpped when i was caugh a he 0.8 mee heigh? Skech he psiin vs ime, velciy vs ime, and acceleain vs ime gaphs. 4. Suppse yu dp a wae balln fm he p f a building. The wae balln akes fu secnds hi he gund. Deemine he heigh f he building. [Se up he equains caefully and slve.] Deemine he speed f he balln jus befe i his he gund. Deemine he balln s acceleain as i is being spped by he gund if i cmes a sp in a disance f en cenimees.

15 5. Suppse yu hw a ball veically upwad wih a speed f 10 m/s as i leaves yu hand 2 mees abve he gund. Deemine is maimum heigh, when i ges he p f is ajecy, and when i his he gund. Caefully skech a gaph f is heigh as a funcin f ime fm he mmen yu elease i. 6. A mden dag ace can acceleae fm es cve a quae mile in five secnds. Assume he acceleain is cnsan. (Meic unis ae easie, assume ha 1/4 mi=400 m.) a. Deemine a value f he acceleain. [Epess yu answe as a cnsan imes "g": a= ( )g] b. Assuming cnsan acceleain, deemne he final speed. [Ne: 1 m/s 2 mph] c. If he ca can sp wih an acceleain equal -g, deemine he disance and he ime sp fm ha speed. TRAJECTORY PROBLEMS 7. Cnside ha yu give yu physics bk a shve n a able (a heigh h abve he fl). The bk has a speed v a disance S fm he edge f he able Biefly jusify yu easning in each pa belw. (Yu shuld assume yu knw he values f H, S, g, and v ) Se up and slve he pblems algebaically. H v S v F numeical values, use he values: S = 1 m v = 4 m/s H = 1 m g = 10 m/s2 a. Obain an epessin f he ime he bk is in he ai afe i leaves he able. b. Obain an epessin f he hiznal disance he bk avels afe i leaves he able. c. Obain an epessin f he speed wih which i his he gund. 8. Cnside ha a mable is sh hiznally fm a able wih iniial speed v. The ange f he ball is 2 mees if he heigh f he able is ne mee. F his pblem, yu can assume g = 10 m/s 2 Saing wih he () and y() equains, deemine he iniial speed f he mable. Deemine he speed f he mable when i his he fl.

16 9. Cnside ha a pjecile is fied fm he p f a building f heigh H wih an iniial speed f v a an angle θ wih espec he hiznal. Begin wih he geneal epessin f y() and deive he equain f he ajecy, ie he y() ha descibes he pah he pjecile akes. ange = Ouline he seps yu wuld ake slve f he ange f ma he pjecile. [NOTE: Yu d n need d he algeba - jus eplain wha yu wuld d. Be eplici enugh ha fllwing he seps shuld lead he cec ange.] 10. Shw ha he hiznal ange f a pjecile is he same if he pjecile is launched a angle θ a angle 90-θ. heigh. Shw a wha angle θ he hiznal ange will be geaes. And shw ha a ha angle, he hiznal ange will be fu imes he maimum 11. Suppse Bill hws a wae balln fm he p f a building a an angle f 30 wih espec he hiznal. The building is 20 m high and he iniial speed f he wae balln is 10 m/s. [Le g=10 m/s-.] Deemine he ime Bill's bes fiend, a physicis, has calculae whehe he is ging ge hi by he balln. If he is 25 m fm he building, shuld he mve? (Deemine bh he ime f fligh and he disance he pin f impac.) 12. Cnside ha a pjecile is sh wih an iniial speed f v a an angle θ wih espec he hiznal. Begin wih he geneal epessin f y() and deive: (a) The maimum heigh (in ems f v, g, and θ). (b) The hiznal ange f he pjecile (in ems f v, g, and θ). (c) An equain f he ajecy, ie he y() ha descibes he pah he pjecile akes. (d) Using yu epessin f he hiznal ange, deemine he angle a which he ange is maimized. (e) Shw ha he hiznal ange is he same f angles θ and (90 - θ).

17 13. Suppse Tige Wds can launch a glf ball wih an iniial ball speed f 180 mph a a launch angle f 14. Saing fm he kinemaics equains, deemine hw fa he culd hi a glf ball. Sae he assumpins yu ae making in seing up his pblem. [Yu pbably wan cnve SI unis i.e., since ne mee is abu 1.1 yads and hee ae 1780 yads pe mile and 3600 sec. pe hu, 1 mph cnves abu 0.45 m/s.] Discuss hw he assumpins affec he sluin - i.e., wuld yu epec he ball g fahe less fa han yu calculains suggess. If Wds dives can cay 300 yads, can yu hink f he effecs f he deails ha we have lef u f his analysis and hw hey migh affec he esul? 14. v Anyne wh plays glf has had a pu, mving quickly, "skip" acss he cup wihu falling in. I happens when less ha half he ball is belw he fa edge f he cup, s i jus bunces up and cninues fwad. Se up he pblem and slve f he maimum speed which a ball can have as i eaches he hle and sill dp in d he cup. Yu can d he pblem eihe algebaically in ems f D and d d i numeically. D Assume he fllwing dimensins (app.): Diamee f cup: D=12 cm Diamee f ball: d=4 cm CIRCULAR MOTION v Suppse yu can hw a baseball wih a speed f 30 m/s. The ball acually avels in an ac. A he "p" f is ac (whee is min is hiznal), deemine he adius f cuvaue f is pah. Eplain yu easning caefully. Suppse a bulle is fied fm M. Evees hiznally wih he speed necessay bi he eah (assume n ai esisance). If he adius f he eah (and he bi) is m, deemine he speed f he bulle. Deemine he ime f ne cmplee bi. 17. An Indianaplis ace ca can ciculae he 2.5 mile val a an aveage speed f 225 mph. Assuming he ack is cicula (ahe han an val) wih a cicumfeence f 2.5 miles and ha he speed is cnsan a 225 mph, deemine he aveage acceleain f he ace ca. [Epess he acceleain in he fm a=(cns)g - whee he cnsan in paenheses is jus he ai a/g. F eample, a=(.5)g means he acceleain is half ha f feefall acceleain.] NOTE: 1 mi = 1.6 km

18 18. Wih gea ies, yu Psche can geneae abu (0.9) g f laeal acceleain in a un. Deemine he speed wih which he ca culd g hugh an unbanked un wih a adius f 40 m. R Deemine whehe he ime cmplee a lap aund a cicula pah wuld incease, decease, say he same as he adius f he cicle inceases. (F eample, culd yu cmplee a lap aund a 100 m adius cicle a 200 m adius cicle in less ime if he cenipeal acceleain wee 0.9 g?) 19. Suppse an bjec sas a es and acceleaes unifmly while aveling in a cicle f adius R. Assume is angenial acceleain is given by a. Shw ha he bjec s cenipeal acceleain will equal he angenial acceleain afe i has cmpleed 2π cmplee bis subjec ha cnsan acceleain. [HINT: The bjec s angenial speed v depends n is acceleain and hw lng i has been acceleaing. I s cenipeal acceleain depends n is angenial speed and he adius. Bu i s angenial speed als depends n he adius and he peid f he bi a ha mmen even hugh ha peid is cninuusly changing.] 20. Cnside ha a ball n a sing is in cicula min in a veical cicle f adius R. v L a. In de say in a cicle a he p f he pah, he cenipeal acceleain mus be a leas as gea as he gaviainal acceleain. Using ha infmain, find an epessin f he minimum speed a he p. Suppse he sing is ne mee in lengh, calculae he value f he speed a he p jus say in he cicula pah. b. As he ball falls cmplee he cicle, gaviy acs speed he ball up cnsideably. On he figue shwing he ball a seveal psiins, shw he acceleain vec a each f hse psiins. Yu shuld shw bh he hiznal and veical cmpnens f he acceleain as well as he vec sum (ie, he acceleain vec). Jusify yu hinking. Wie an epessin f he magniude f he acceleain a ne f hse w "side" psiins - assuming yu knw he speed v a ha pin. A he vey bm, cmpae he magniudes f he cenipeal acceleain and he gaviainal acceleain. Biefly jusify. a c < g a c = g a c > g

19 21. If he gaviainal acceleain diminishes by he squae f he adius fm he cene f he eah, deemine he elainship beween he peid f a cicula bi and is adius. Tha is, bain an epessin f he peid f an bi f adius (whee is geae han he eah s adius R). Yu final epessin f he peid T shuld be in ems f R, g, and. R [HINT: Assume ha a g =g when =R, whee R is eah's adius, s ha when >R, he acceleain is given by he epessin a g = (R/) 2 g..] The weny-fu biing GPS saellies ha make up he newk f glbal psiining each bi he eah wice pe day. Deemine he adii f hei bis.