2.7 Freely Falling Objects

Transcription

1 4 Chaper Moion in One Dimension.8 coninued Rearrange o give a quadraic equaion: 1 a x v x car x 5 Solve he quadraic equaion for he ime a which he rooper caches he car (for help in solving quadraic equaions, see Appendix B..): Evaluae he soluion, choosing he posiive roo because ha is he only choice consisen wih a ime. : 5 v x car 6 "v x car 1 a x x a x (1) 5 v x car a x 6 Å v x car a x 1 x a x 45. m/s 5 3. m/s m/s 145. m 1 Å 13. m/s 3. m/s s Why didn we choose 5 as he ime a which he car passes he rooper? If we did so, we would no be able o use he paricle under consan acceleraion model for he rooper. Her acceleraion would be zero for he firs second and hen 3. m/s for he remaining ime. By defining he ime 5 as when he rooper begins moving, we can use he paricle under consan acceleraion model for her movemen for all posiive imes. WHAT IF? Wha if he rooper had a more powerful moorcycle wih a larger acceleraion? How would ha change he ime a which he rooper caches he car? Answer If he moorcycle has a larger acceleraion, he rooper should cach up o he car sooner, so he answer for he ime should be less han 31 s. Because all erms on he righ side of Equaion (1) have he acceleraion a x in he denominaor, we see symbolically ha increasing he acceleraion will decrease he ime a which he rooper caches he car..7 Freely Falling Objecs Galileo Galilei Ialian physicis and asronomer ( ) Galileo formulaed he laws ha govern he moion of objecs in free fall and made many oher significan discoveries in physics and asronomy. Galileo publicly defended Nicolaus Copernicus s asserion ha he Sun is a he cener of he Universe (he heliocenric sysem). He published Dialogue Concerning Two New World Sysems o suppor he Copernican model, a view ha he Caholic Church declared o be hereical. Georgios Kollidas/Shuersock.com I is well known ha, in he absence of air resisance, all objecs dropped near he Earh s surface fall oward he Earh wih he same consan acceleraion under he influence of he Earh s graviy. I was no unil abou 16 ha his conclusion was acceped. Before ha ime, he eachings of he Greek philosopher Arisole (384 3 BC) had held ha heavier objecs fall faser han ligher ones. The Ialian Galileo Galilei ( ) originaed our presen-day ideas concerning falling objecs. There is a legend ha he demonsraed he behavior of falling objecs by observing ha wo differen weighs dropped simulaneously from he Leaning Tower of Pisa hi he ground a approximaely he same ime. Alhough here is some doub ha he carried ou his paricular experimen, i is well esablished ha Galileo performed many experimens on objecs moving on inclined planes. In his experimens, he rolled balls down a sligh incline and measured he disances hey covered in successive ime inervals. The purpose of he incline was o reduce he acceleraion, which made i possible for him o make accurae measuremens of he ime inervals. By gradually increasing he slope of he incline, he was finally able o draw conclusions abou freely falling objecs because a freely falling ball is equivalen o a ball moving down a verical incline. You migh wan o ry he following experimen. Simulaneously drop a coin and a crumpled-up piece of paper from he same heigh. If he effecs of air resisance are negligible, boh will have he same moion and will hi he floor a he same ime. In he idealized case, in which air resisance is absen, such moion is referred

2 .7 Freely Falling Objecs 41 o as free-fall moion. If his same experimen could be conduced in a vacuum, in which air resisance is ruly negligible, he paper and he coin would fall wih he same acceleraion even when he paper is no crumpled. On Augus, 1971, asronau David Sco conduced such a demonsraion on he Moon. He simulaneously released a hammer and a feaher, and he wo objecs fell ogeher o he lunar surface. This simple demonsraion surely would have pleased Galileo! When we use he expression freely falling objec, we do no necessarily refer o an objec dropped from res. A freely falling objec is any objec moving freely under he influence of graviy alone, regardless of is iniial moion. Objecs hrown upward or downward and hose released from res are all falling freely once hey are released. Any freely falling objec experiences an acceleraion direced downward, regardless of is iniial moion. We shall denoe he magniude of he free-fall acceleraion, also called he acceleraion due o graviy, by he symbol g. The value of g decreases wih increasing aliude above he Earh s surface. Furhermore, sligh variaions in g occur wih changes in laiude. A he Earh s surface, he value of g is approximaely 9.8 m/s. Unless saed oherwise, we shall use his value for g when performing calculaions. For making quick esimaes, use g 5 1 m/s. If we neglec air resisance and assume he free-fall acceleraion does no vary wih aliude over shor verical disances, he moion of a freely falling objec moving verically is equivalen o he moion of a paricle under consan acceleraion in one dimension. Therefore, he equaions developed in Secion.6 for he paricle under consan acceleraion model can be applied. The only modificaion for freely falling objecs ha we need o make in hese equaions is o noe ha he moion is in he verical direcion (he y direcion) raher han in he horizonal direcion (x) and ha he acceleraion is downward and has a magniude of 9.8 m/s. Therefore, we choose a y 5 g m/s, where he negaive sign means ha he acceleraion of a freely falling objec is downward. In Chaper 13, we shall sudy how o deal wih variaions in g wih aliude. Pifall Prevenion.6 g and g Be sure no o confuse he ialic symbol g for free-fall acceleraion wih he nonialic symbol g used as he abbreviaion for he uni gram. Pifall Prevenion.7 The Sign of g Keep in mind ha g is a posiive number. I is emping o subsiue 9.8 m/s for g, bu resis he empaion. Downward graviaional acceleraion is indicaed explicily by saing he acceleraion as a y 5 g. Pifall Prevenion.8 Acceleraion a he Top of he Moion A common misconcepion is ha he acceleraion of a projecile a he op of is rajecory is zero. Alhough he velociy a he op of he moion of an objec hrown upward momenarily goes o zero, he acceleraion is sill ha due o graviy a his poin. If he velociy and acceleraion were boh zero, he projecile would say a he op. Q uick Quiz.7 Consider he following choices: (a) increases, (b) decreases, (c) increases and hen decreases, (d) decreases and hen increases, (e) remains he same. From hese choices, selec wha happens o (i) he acceleraion and (ii) he speed of a ball afer i is hrown upward ino he air. Concepual Example.9 The Daring Skydivers A skydiver jumps ou of a hovering helicoper. A few seconds laer, anoher skydiver jumps ou, and hey boh fall along he same verical line. Ignore air resisance so ha boh skydivers fall wih he same acceleraion. Does he difference in heir speeds say he same hroughou he fall? Does he verical disance beween hem say he same hroughou he fall? A any given insan, he speeds of he skydivers are differen because one had a head sar. In any ime inerval D afer his insan, however, he wo skydivers increase heir speeds by he same amoun because hey have he same acceleraion. Therefore, he difference in heir speeds remains he same hroughou he fall. The firs jumper always has a greaer speed han he second. Therefore, in a given ime inerval, he firs skydiver covers a greaer disance han he second. Consequenly, he separaion disance beween hem increases.

3 4 Chaper Moion in One Dimension Example.1 No a Bad Throw for a Rookie! AM A sone hrown from he op of a building is given an iniial velociy of. m/s sraigh upward. The sone is launched 5. m above he ground, and he sone jus misses he edge of he roof on is way down as shown in Figure.14. (A) Using 5 as he ime he sone leaves he hrower s hand a posiion, deermine he ime a which he sone reaches is maximum heigh. You mos likely have experience wih dropping objecs or hrowing hem upward and waching hem fall, so his problem should describe a familiar experience. To simulae his siuaion, oss a small objec upward and noice he ime inerval required for i o fall o he floor. Now imagine Figure.14 (Example.1) Posiion, velociy, and acceleraion values a various imes for a freely falling sone hrown iniially upward wih a velociy v yi 5. m/s. Many of he quaniies in he labels for poins in he moion of he sone are calculaed in he example. Can you verify he oher values ha are no? hrowing ha objec upward from he roof of a building. Because he sone is in free fall, i is modeled as a paricle under consan acceleraion due o graviy. Recognize ha he iniial velociy is posiive because he sone is launched upward. The velociy will change sign afer he sone reaches is highes poin, bu he acceleraion of he sone will always be downward so ha i will always have a negaive value. Choose an iniial poin jus afer he sone leaves he person s hand and a final poin a he op of is fligh. y v y. m/s a y 9.8 m/s 5. m.4 s y.4 m v y a y 9.8 m/s 4.8 s y v y. m/s a y 9.8 m/s 5. s y.5 m v y 9. m/s a y 9.8 m/s 5.83 s y 5. m v y 37.1 m/s a y 9.8 m/s Use Equaion.13 o calculae he ime a which he sone reaches is maximum heigh: v yf 5 v yi 1 a y S 5 v yf v yi a y Subsiue numerical values: 5 5 (B) Find he maximum heigh of he sone.. m/s 9.8 m/s 5.4 s As in par (A), choose he iniial and final poins a he beginning and he end of he upward fligh. Se y 5 and subsiue he ime from par (A) ino Equaion.16 o find he maximum heigh: y max 5 y 5 y 1 v x 1 1 a y y m/s1.4 s m/s 1.4 s 5.4 m (C) Deermine he velociy of he sone when i reurns o he heigh from which i was hrown. Choose he iniial poin where he sone is launched and he final poin when i passes his posiion coming down. Subsiue known values ino Equaion.17: v y 5 v y 1 a y (y y ) v y 5 (. m/s) 1 (9.8 m/s )( ) 5 4 m /s v y 5. m/s

4 .8 Kinemaic Equaions Derived from Calculus 43.1 coninued When aking he square roo, we could choose eiher a posiive or a negaive roo. We choose he negaive roo because we know ha he sone is moving downward a poin. The velociy of he sone when i arrives back a is original heigh is equal in magniude o is iniial velociy bu is opposie in direcion. (D) Find he velociy and posiion of he sone a 5 5. s. Choose he iniial poin jus afer he hrow and he final poin 5. s laer. Calculae he velociy a from Equaion.13: Use Equaion.16 o find he posiion of he sone a 5 5. s: v y 5 v y 1 a y 5. m/s 1 (9.8 m/s )(5. s) 5 9. m/s y 5 y 1 v y 1 1 a y 5 1 (. m/s)(5. s) 1 1 (9.8 m/s )(5. s) 5.5 m The choice of he ime defined as 5 is arbirary and up o you o selec as he problem solver. As an example of his arbirariness, choose 5 as he ime a which he sone is a he highes poin in is moion. Then solve pars (C) and (D) again using his new iniial insan and noice ha your answers are he same as hose above. WHAT IF? Wha if he hrow were from 3. m above he ground insead of 5. m? Which answers in pars (A) o (D) would change? Answer None of he answers would change. All he moion akes place in he air during he firs 5. s. (Noice ha even for a hrow from 3. m, he sone is above he ground a 5 5. s.) Therefore, he heigh of he hrow is no an issue. Mahemaically, if we look back over our calculaions, we see ha we never enered he heigh of he hrow ino any equaion..8 Kinemaic Equaions Derived from Calculus This secion assumes he reader is familiar wih he echniques of inegral calculus. If you have no ye sudied inegraion in your calculus course, you should skip his secion or cover i afer you become familiar wih inegraion. The velociy of a paricle moving in a sraigh line can be obained if is posiion as a funcion of ime is known. Mahemaically, he velociy equals he derivaive of he posiion wih respec o ime. I is also possible o find he posiion of a paricle if is velociy is known as a funcion of ime. In calculus, he procedure used o perform his ask is referred o eiher as inegraion or as finding he aniderivaive. Graphically, i is equivalen o finding he area under a curve. Suppose he v x graph for a paricle moving along he x axis is as shown in Figure.15 on page 44. Le us divide he ime inerval f i ino many small inervals, each of duraion D n. From he definiion of average velociy, we see ha he displacemen of he paricle during any small inerval, such as he one shaded in Figure.15, is given by Dx n 5 v xn,avg D n, where v xn,avg is he average velociy in ha inerval. Therefore, he displacemen during his small inerval is simply he area of he shaded recangle in Figure.15. The oal displacemen for he inerval f i is he sum of he areas of all he recangles from i o f : Dx 5 an v xn,avg D n where he symbol o (uppercase Greek sigma) signifies a sum over all erms, ha is, over all values of n. Now, as he inervals are made smaller and smaller, he number of erms in he sum increases and he sum approaches a value equal o he area

5 44 Chaper Moion in One Dimension Figure.15 Velociy versus ime for a paricle moving along he x axis. The oal area under he curve is he oal displacemen of he paricle. v xn,avg v x The area of he shaded recangle is equal o he displacemen in he ime inerval n. i f n under he curve in he velociy ime graph. Therefore, in he limi n S `, or D n S, he displacemen is Dx 5 lim Dn S a n v xn,avg D n (.18) If we know he v x graph for moion along a sraigh line, we can obain he displacemen during any ime inerval by measuring he area under he curve corresponding o ha ime inerval. The limi of he sum shown in Equaion.18 is called a definie inegral and is wrien v xi v x i Definie inegral v x v xi consan v xi f lim D n S a n v xn,avg D n 5 3 f i v x 1 d (.19) where v x () denoes he velociy a any ime. If he explici funcional form of v x () is known and he limis are given, he inegral can be evaluaed. Someimes he v x graph for a moving paricle has a shape much simpler han ha shown in Figure.15. For example, suppose an objec is described wih he paricle under consan velociy model. In his case, he v x graph is a horizonal line as in Figure.16 and he displacemen of he paricle during he ime inerval D is simply he area of he shaded recangle: Dx 5 v xi D (when v x 5 v xi 5 consan) Figure.16 The velociy ime curve for a paricle moving wih consan velociy v xi. The displacemen of he paricle during he ime inerval f i is equal o he area of he shaded recangle. Kinemaic Equaions We now use he defining equaions for acceleraion and velociy o derive wo of our kinemaic equaions, Equaions.13 and.16. The defining equaion for acceleraion (Eq..1), a x 5 dv x d may be wrien as dv x 5 a x d or, in erms of an inegral (or aniderivaive), as v xf v xi 5 3 a x d For he special case in which he acceleraion is consan, a x can be removed from he inegral o give v xf v xi 5 a x 3 d 5 a x 1 5 a x (.) which is Equaion.13 in he paricle under consan acceleraion model. Now le us consider he defining equaion for velociy (Eq..5): v x 5 dx d

6 General Problem-Solving Sraegy 45 We can wrie his equaion as dx 5 v x d or in inegral form as x f x i 5 3 v x d Because v x 5 v xf 5 v xi 1 a x, his expression becomes x f x i 5 3 1v xi 1 a x d 5 3 v xi d 1 a x 3 d 5 v xi 1 1 a x a x f x i 5 v xi 1 1 a x which is Equaion.16 in he paricle under consan acceleraion model. b Besides wha you migh expec o learn abou physics conceps, a very valuable skill you should hope o ake away from your physics course is he abiliy o solve complicaed problems. The way physiciss approach complex siuaions and break hem ino manageable pieces is exremely useful. The following is a general problem-solving sraegy o guide you hrough he seps. To help you remember he seps of he sraegy, hey are Concepualize, Caegorize, Analyze, and Finalize. GENERAL PROBLEM-SOLVING STRATEGY Concepualize are o hink abou and undersand he siuaion. Sudy carefully any represenaions of he informaion (for example, diagrams, graphs, ables, or phoographs) ha accompany he problem. Imagine a movie, running in your mind, of wha happens in he problem. should almos always make a quick drawing of he siuaion. Indicae any known values, perhaps in a able or direcly on your skech. ion is given in he problem. Carefully read he problem saemen, looking for key phrases such as sars from res (v i 5 ), sops (v f 5 ), or falls freely (a y 5 g m/s ). lem. Exacly wha is he quesion asking? Will he final resul be numerical or algebraic? Do you know wha unis o expec? own experiences and common sense. Wha should a reasonable answer look like? For example, you wouldn expec o calculae he speed of an auomobile o be m/s. Caegorize abou, you need o simplify he problem. Remove he deails ha are no imporan o he soluion. For example, model a moving objec as a paricle. If appropriae, ignore air resisance or fricion beween a sliding objec and a surface. caegorize he problem. Is i a simple subsiuion problem such ha numbers can be subsiued ino a simple equaion or a definiion? If so, he problem is likely o be finished when his subsiuion is done. If no, you face wha we call an analysis problem: he siuaion mus be analyzed more deeply o generae an appropriae equaion and reach a soluion. furher. Have you seen his ype of problem before? Does i fall ino he growing lis of ypes of problems ha you have solved previously? If so, idenify any analysis model(s) appropriae for he problem o prepare for he Analyze sep below. We saw he firs hree analysis models in his chaper: he paricle under consan velociy, he paricle under consan speed, and he paricle under consan acceleraion. Being able o classify a problem wih an analysis model can make i much easier o lay ou a plan o solve i. For example, if your simplificaion shows ha he problem can be reaed as a paricle under consan acceleraion and you have already solved such a problem (such as he examples in Secion.6), he soluion o he presen problem follows a similar paern. coninued

7 46 Chaper Moion in One Dimension Analyze mahemaical soluion. Because you have already caegorized he problem and idenified an analysis model, i should no be oo difficul o selec relevan equaions ha apply o he ype of siuaion in he problem. For example, if he problem involves a paricle under consan acceleraion, Equaions.13 o.17 are relevan. bolically for he unknown variable in erms of wha is given. Finally, subsiue in he appropriae numbers, calculae he resul, and round i o he proper number of significan figures. Finalize correc unis? Does i mee your expecaions from your concepualizaion of he problem? Wha abou he algebraic form of he resul? Does i make sense? Examine he variables in he problem o see wheher he answer would change in a physically meaningful way if he variables were drasically increased or decreased or even became zero. Looking a limiing cases o see wheher hey yield expeced values is a very useful way o make sure ha you are obaining reasonable resuls. you have solved. How was i similar? In wha criical ways did i differ? Why was his problem assigned? Can you figure ou wha you have learned by doing i? If i is a new caegory of problem, be sure you undersand i so ha you can use i as a model for solving similar problems in he fuure. When solving complex problems, you may need o idenify a series of subproblems and apply he problemsolving sraegy o each. For simple problems, you probably don need his sraegy. When you are rying o solve a problem and you don know wha o do nex, however, remember he seps in he sraegy and use hem as a guide. For pracice, i would be useful for you o revisi he worked examples in his chaper and idenify he Concepualize, Caegorize, Analyze, and Finalize seps. In he res of his book, we will label hese seps explicily in he worked examples. Many chapers in his book include a secion labeled Problem-Solving Sraegy ha should help you hrough he rough spos. These secions are organized according o he General Problem- Solving Sraegy oulined above and are ailored o he specific ypes of problems addressed in ha chaper. To clarify how his Sraegy works, we repea Example.7 below wih he paricular seps of he Sraegy idenified. When you Concepualize a problem, ry o undersand he siuaion ha is presened in he problem saemen. Sudy carefully any represenaions of he informaion (for example, diagrams, graphs, ables, or phoographs) ha accompany he problem. Imagine a movie, running in your mind, of wha happens in he problem. Simplify he problem. Remove he deails ha are no imporan o he soluion. Then Caegorize he problem. Is i a simple subsiuion problem such ha numbers can be subsiued ino a simple equaion or a definiion? If no, you face an analysis problem. In his case, idenify he appropriae analysis model. Example.7 Carrier Landing AM A je lands on an aircraf carrier a a speed of 14 mi/h (< 63 m/s). (A) Wha is is acceleraion (assumed consan) if i sops in. s due o an arresing cable ha snags he je and brings i o a sop? Concepualize You migh have seen movies or elevision shows in which a je lands on an aircraf carrier and is brough o res surprisingly fas by an arresing cable. A careful reading of he problem reveals ha in addiion o being given he iniial speed of 63 m/s, we also know ha he final speed is zero. Caegorize Because he acceleraion of he je is assumed consan, we model i as a paricle under consan acceleraion.

8 Summary 47.7 coninued Analyze We define our x axis as he direcion of moion of he je. Noice ha we have no informaion abou he change in posiion of he je while i is slowing down. Equaion.13 is he only equaion in he paricle under consan acceleraion model ha does no involve posiion, so we use i o find he acceleraion of he je, modeled as a paricle: a x 5 v xf v xi (B) If he je ouches down a posiion x i 5, wha is is final posiion? m/s 63 m/s. s Use Equaion.15 o solve for he final posiion: x f 5 x i 1 1 1v xi 1 v xf m/s 1 1. s 5 63 m Finalize Given he size of aircraf carriers, a lengh of 63 m seems reasonable for sopping he je. The idea of using arresing cables o slow down landing aircraf and enable hem o land safely on ships originaed a abou he ime of World War I. The cables are sill a vial par of he operaion of modern aircraf carriers. WHAT IF? Suppose he je lands on he deck of he aircraf carrier wih a speed higher han 63 m/s bu has he same acceleraion due o he cable as ha calculaed in par (A). How will ha change he answer o par (B)? Answer If he je is raveling faser a he beginning, i will sop farher away from is saring poin, so he answer o par (B) should be larger. Mahemaically, we see in Equaion.15 ha if v xi is larger, x f will be larger. Now Analyze he problem. Selec relevan equaions from he analysis model. Solve symbolically for he unknown variable in erms of wha is given. Subsiue in he appropriae numbers, calculae he resul, and round i o he proper number of significan figures. Finalize he problem. Examine he numerical answer. Does i have he correc unis? Does i mee your expecaions from your concepualizaion of he problem? Does he answer make sense? Wha abou he algebraic form of he resul? Examine he variables in he problem o see wheher he answer would change in a physically meaningful way if he variables were drasically increased or decreased or even became zero. Wha If? quesions will appear in many examples in he ex, and offer a variaion on he siuaion jus explored. This feaure encourages you o hink abou he resuls of he example and assiss in concepual undersanding of he principles. Summary Definiions When a paricle moves along he x axis from some iniial posiion x i o some final posiion x f, is displacemen is Dx ; x f x i (.1) The average velociy of a paricle during some ime inerval is he displacemen Dx divided by he ime inerval D during which ha displacemen occurs: v x,avg ; Dx (.) D The average speed of a paricle is equal o he raio of he oal disance i ravels o he oal ime inerval during which i ravels ha disance: v avg ; d D (.3) coninued

9 48 Chaper Moion in One Dimension The insananeous velociy of a paricle is defined as he limi of he raio Dx/D as D approaches zero. By definiion, his limi equals he derivaive of x wih respec o, or he ime rae of change of he posiion: Dx v x ; lim D S D 5 dx d (.5) The insananeous speed of a paricle is equal o he magniude of is insananeous velociy. The average acceleraion of a paricle is defined as he raio of he change in is velociy Dv x divided by he ime inerval D during which ha change occurs: a x,avg ; Dv x D 5 v xf v xi f i (.9) The insananeous acceleraion is equal o he limi of he raio Dv x /D as D approaches. By definiion, his limi equals he derivaive of v x wih respec o, or he ime rae of change of he velociy: Dv x a x ; lim D S 5 dv x D d (.1) Conceps and Principles When an objec s velociy and acceleraion are in he same direcion, he objec is speeding up. On he oher hand, when he objec s velociy and acceleraion are in opposie direcions, he objec is slowing down. Remembering ha F x ~ a x is a useful way o idenify he direcion of he acceleraion by associaing i wih a force. An objec falling freely in he presence of he Earh s graviy experiences free-fall acceleraion direced oward he cener of he Earh. If air resisance is negleced, if he moion occurs near he surface of he Earh, and if he range of he moion is small compared wih he Earh s radius, he free-fall acceleraion a y 5 g is consan over he range of moion, where g is equal o 9.8 m/s. Complicaed problems are bes approached in an organized manner. Recall and apply he Concepualize, Caegorize, Analyze, and Finalize seps of he General Problem- Solving Sraegy when you need hem. An imporan aid o problem solving is he use of analysis models. Analysis models are siuaions ha we have seen in previous problems. Each analysis model has one or more equaions associaed wih i. When solving a new problem, idenify he analysis model ha corresponds o he problem. The model will ell you which equaions o use. The firs hree analysis models inroduced in his chaper are summarized below. Analysis Models for Problem-Solving Paricle Under Consan Velociy. If a paricle moves in a sraigh line wih a consan speed v x, is consan velociy is given by v x 5 Dx D and is posiion is given by (.6) x f 5 x i 1 v x (.7) Paricle Under Consan Speed. If a paricle moves a disance d along a curved or sraigh pah wih a consan speed, is consan speed is given by v v 5 d D (.8) v Paricle Under Consan Acceleraion. If a paricle moves in a sraigh line wih a consan acceleraion a x, is moion is described by he kinemaic equaions: v xf 5 v xi 1 a x (.13) v x,avg 5 v xi 1 v xf (.14) v a x f 5 x i 1 1 1v xi 1 v xf (.15) x f 5 x i 1 v xi 1 1 a x (.16) v xf 5 v xi 1 a x (x f x i ) (.17)

10 Objecive Quesions 49 Objecive Quesions 1. denoes answer available in Suden Soluions Manual/Sudy Guide 1. One drop of oil falls sraigh down ono he road from he engine of a moving car every 5 s. Figure OQ.1 shows he paern of he drops lef behind on he pavemen. Wha is he average speed of he car over his secion of is moion? (a) m/s (b) 4 m/s (c) 3 m/s (d) 1 m/s (e) 1 m/s 6 m Figure OQ.1. A racing car sars from res a 5 and reaches a final speed v a ime. If he acceleraion of he car is consan during his ime, which of he following saemens are rue? (a) The car ravels a disance v. (b) The average speed of he car is v/. (c) The magniude of he acceleraion of he car is v/. (d) The velociy of he car remains consan. (e) None of saemens (a) hrough (d) is rue. 3. A juggler hrows a bowling pin sraigh up in he air. Afer he pin leaves his hand and while i is in he air, which saemen is rue? (a) The velociy of he pin is always in he same direcion as is acceleraion. (b) The velociy of he pin is never in he same direcion as is acceleraion. (c) The acceleraion of he pin is zero. (d) The velociy of he pin is opposie is acceleraion on he way up. (e) The velociy of he pin is in he same direcion as is acceleraion on he way up. 4. When applying he equaions of kinemaics for an objec moving in one dimension, which of he following saemens mus be rue? (a) The velociy of he objec mus remain consan. (b) The acceleraion of he objec mus remain consan. (c) The velociy of he objec mus increase wih ime. (d) The posiion of he objec mus increase wih ime. (e) The velociy of he objec mus always be in he same direcion as is acceleraion. 5. A cannon shell is fired sraigh up from he ground a an iniial speed of 5 m/s. Afer how much ime is he shell a a heigh of m above he ground and moving downward? (a).96 s (b) 17.3 s (c) 5.4 s (d) 33.6 s (e) 43. s 6. An arrow is sho sraigh up in he air a an iniial speed of 15. m/s. Afer how much ime is he arrow moving downward a a speed of 8. m/s? (a).714 s (b) 1.4 s (c) 1.87 s (d).35 s (e) 3. s 7. When he pilo reverses he propeller in a boa moving norh, he boa moves wih an acceleraion direced souh. Assume he acceleraion of he boa remains consan in magniude and direcion. Wha happens o he boa? (a) I evenually sops and remains sopped. (b) I evenually sops and hen speeds up in he forward direcion. (c) I evenually sops and hen speeds up in he reverse direcion. (d) I never sops bu loses speed more and more slowly forever. (e) I never sops bu coninues o speed up in he forward direcion. 8. A rock is hrown downward from he op of a 4.-m-all ower wih an iniial speed of 1 m/s. Assuming negligible air resisance, wha is he speed of he rock jus before hiing he ground? (a) 8 m/s (b) 3 m/s (c) 56 m/s (d) 784 m/s (e) More informaion is needed. 9. A skaeboarder sars from res and moves down a hill wih consan acceleraion in a sraigh line, raveling for 6 s. In a second rial, he sars from res and moves along he same sraigh line wih he same acceleraion for only s. How does his displacemen from his saring poin in his second rial compare wih ha from he firs rial? (a) one-hird as large (b) hree imes larger (c) one-ninh as large (d) nine imes larger (e) 1/!3 imes as large 1. On anoher plane, a marble is released from res a he op of a high cliff. I falls 4. m in he firs 1 s of is moion. Through wha addiional disance does i fall in he nex 1 s? (a) 4. m (b) 8. m (c) 1. m (d) 16. m (e). m 11. As an objec moves along he x axis, many measuremens are made of is posiion, enough o generae a smooh, accurae graph of x versus. Which of he following quaniies for he objec canno be obained from his graph alone? (a) he velociy a any insan (b) he acceleraion a any insan (c) he displacemen during some ime inerval (d) he average velociy during some ime inerval (e) he speed a any insan 1. A pebble is dropped from res from he op of a all cliff and falls 4.9 m afer 1. s has elapsed. How much farher does i drop in he nex. s? (a) 9.8 m (b) 19.6 m (c) 39 m (d) 44 m (e) none of he above 13. A suden a he op of a building of heigh h hrows one ball upward wih a speed of v i and hen hrows a second ball downward wih he same iniial speed v i. Jus before i reaches he ground, is he final speed of he ball hrown upward (a) larger, (b) smaller, or (c) he same in magniude, compared wih he final speed of he ball hrown downward? 14. You drop a ball from a window locaed on an upper floor of a building. I srikes he ground wih speed v. You now repea he drop, bu your friend down on he ground hrows anoher ball upward a he same speed v, releasing her ball a he same momen ha you drop yours from he window. A some locaion, he balls pass each oher. Is his locaion (a) a he halfway poin beween window and ground, (b) above his poin, or (c) below his poin? 15. A pebble is released from res a a cerain heigh and falls freely, reaching an impac speed of 4 m/s a he floor. Nex, he pebble is hrown down wih an iniial speed of 3 m/s from he same heigh. Wha is is speed a he floor? (a) 4 m/s (b) 5 m/s (c) 6 m/s (d) 7 m/s (e) 8 m/s

11 5 Chaper Moion in One Dimension 16. A ball is hrown sraigh up in he air. For which siuaion are boh he insananeous velociy and he acceleraion zero? (a) on he way up (b) a he op of is fligh pah (c) on he way down (d) halfway up and halfway down (e) none of he above 17. A hard rubber ball, no affeced by air resisance in is moion, is ossed upward from shoulder heigh, falls o he sidewalk, rebounds o a smaller maximum heigh, and is caugh on is way down again. This moion is represened in Figure OQ.17 Figure OQ.17, where he successive posiions of he ball hrough are no equally spaced in ime. A poin he cener of he ball is a is lowes poin in he moion. The moion of he ball is along a sraigh, verical line, bu he diagram shows successive posiions offse o he righ o avoid overlapping. Choose he posiive y direcion o be upward. (a) Rank he siuaions hrough according o he speed of he ball uv y u a each poin, wih he larges speed firs. (b) Rank he same siuaions according o he acceleraion a y of he ball a each poin. (In boh rankings, remember ha zero is greaer han a negaive value. If wo values are equal, show ha hey are equal in your ranking.) 18. Each of he srobe phoographs (a), (b), and (c) in Figure OQ.18 was aken of a single disk moving oward he righ, which we ake as he posiive direcion. Wihin each phoograph, he ime inerval beween images is consan. (i) Which phoograph shows moion wih zero acceleraion? (ii) Which phoograph shows moion wih posiive acceleraion? (iii) Which phoograph shows moion wih negaive acceleraion? a b c Figure OQ.18 Objecive Quesion 18 and Problem 3. Cengage Learning/Charles D. Winers Concepual Quesions 1. denoes answer available in Suden Soluions Manual/Sudy Guide 1. If he average velociy of an objec is zero in some ime inerval, wha can you say abou he displacemen of he objec for ha inerval?. Try he following experimen away from raffic where you can do i safely. Wih he car you are driving moving slowly on a sraigh, level road, shif he ransmission ino neural and le he car coas. A he momen he car comes o a complee sop, sep hard on he brake and noice wha you feel. Now repea he same experimen on a fairly genle, uphill slope. Explain he difference in wha a person riding in he car feels in he wo cases. (Brian Popp suggesed he idea for his quesion.) 3. If a car is raveling easward, can is acceleraion be wesward? Explain. 4. If he velociy of a paricle is zero, can he paricle s acceleraion be zero? Explain. 5. If he velociy of a paricle is nonzero, can he paricle s acceleraion be zero? Explain. 6. You hrow a ball verically upward so ha i leaves he ground wih velociy 15. m/s. (a) Wha is is velociy when i reaches is maximum aliude? (b) Wha is is acceleraion a his poin? (c) Wha is he velociy wih which i reurns o ground level? (d) Wha is is acceleraion a his poin? 7. (a) Can he equaions of kinemaics (Eqs ) be used in a siuaion in which he acceleraion varies in ime? (b) Can hey be used when he acceleraion is zero? 8. (a) Can he velociy of an objec a an insan of ime be greaer in magniude han he average velociy over a ime inerval conaining he insan? (b) Can i be less? 9. Two cars are moving in he same direcion in parallel lanes along a highway. A some insan, he velociy of car A exceeds he velociy of car B. Does ha mean ha he acceleraion of car A is greaer han ha of car B? Explain.