Chapter 2 Motion in One Dimension

Transcription

1 Chaper Moion in One Dimension Concepual Problems 5 Sand in he cener of a large room. Call he direcion o your righ posiie, and he direcion o your lef negaie. Walk across he room along a sraigh line, using a consan acceleraion o quickly reach a seady speed along a sraigh line in he negaie direcion. Afer reaching his seady speed, keep your elociy negaie bu make your acceleraion posiie. (a) Describe how your speed aried as you walked. (b) Skech a graph of x ersus for your moion. Assume you sared a x. (c) Direcly under he graph of Par (b), skech a graph of x ersus. Deermine he Concep The imporan concep is ha when boh he acceleraion and he elociy are in he same direcion, he speed increases. On he oher hand, when he acceleraion and he elociy are in opposie direcions, he speed decreases. (a) Your speed increased from zero, sayed consan for a while, and hen decreased. (b) A graph of your posiion as a funcion of ime is shown o he righ. Noe ha he slope sars ou equal o zero, becomes more negaie as he speed increases, remains consan while your speed is consan, and becomes less negaie as your speed decreases. (c) The graph of () consiss of a sraigh line wih negaie slope (your acceleraion is consan and negaie) saring a (,), hen a fla line for a while (your acceleraion is zero), and finally an approximaely sraigh line wih a posiie slope heading o. x x Dr. Josiah S. Carberry sands a he op of he Sears Tower in Chicago. Waning o emulae Galileo, and ignoring he safey of he pedesrians below, he drops a bowling ball from he op of he ower. One second laer, he drops a second bowling ball. While he balls are in he air, does heir separaion (a) increase oer ime, (b) decrease, (c) say he same? Ignore any effecs due o air resisance. Deermine he Concep Neglecing air resisance, he balls are in free fall, each wih he same free-fall acceleraion, which is a consan. 7

2 8 Chaper A he ime he second ball is released, he firs ball is already moing. Thus, during any ime ineral heir elociies will increase by exacly he same amoun. Wha can be said abou he speeds of he wo balls? The firs ball will always be moing faser han he second ball. This being he case, wha happens o he separaion of he wo balls while hey are boh falling? Their separaion increases. (a) is correc. 3 Which of he elociy-ersus-ime cures in figure -9 bes describes he moion of an objec (a) wih consan posiie acceleraion, (b) wih posiie acceleraion ha is decreasing wih ime, (c) wih posiie acceleraion ha is increasing wih ime, and (d) wih no acceleraion? (There may be more han one correc answer for each par of he problem.) Deermine he Concep The slope of a () cure a any poin in ime represens he acceleraion a ha insan. (a) The correc answer is ( b). The slope of cure (b) is consan and posiie. Therefore he acceleraion is consan and posiie. (b) The correc answer is (c). The slope of cure (c) is posiie and decreasing wih ime. Therefore he acceleraion is posiie and decreasing wih ime. (c) The correc answer is (d). The slope of cure (d) is posiie and increasing wih ime. Therefore he acceleraion is posiie and increasing wih ime. (d) The correc answer is (e). The slope of cure (e) is zero. Therefore he elociy is consan and he acceleraion is zero. 5 An objec moes along a sraigh line. Is posiion ersus ime graph is shown in Figure -3. A which ime or imes is is (a) speed a a minimum, (b) acceleraion posiie, and (c) elociy negaie? Deermine he Concep Because his graph is of disance-ersus-ime we can use is insananeous slope o describe he objec s speed, elociy, and acceleraion. (a) The minimum speed is zero a B, D, and E. In he one-dimensional moion shown in he figure, he elociy is a minimum when he slope of a posiionersus-ime plo goes o zero (i.e., he cure becomes horizonal). A hese poins, he slope of he posiion-ersus-ime cure is zero; herefore, he speed is zero. (b) The acceleraion is posiie a poins A and D. Because he slope of he graph is increasing a hese poins, he elociy of he objec is increasing and is acceleraion is posiie.

3 Moion in One Dimension 9 (c) The elociy is negaie a poin C. Because he slope of he graph is negaie a poin C, he elociy of he objec is negaie. 9 A ball is hrown sraigh up. Neglec any effecs due o air resisance. (a) Wha is he elociy of he ball a he op of is fligh? (b) Wha is is acceleraion a ha poin? (c) Wha is differen abou he elociy and acceleraion a he op of he fligh if insead he ball impacs a horizonal ceiling ery hard and hen reurns. Deermine he Concep In he absence of air resisance, he ball will experience a consan acceleraion and he graph of is posiion as a funcion of ime will be parabolic. In he graphs o he righ, a coordinae sysem was chosen in which he origin is a he poin of release and he upward direcion is he +y direcion. The op graph shows he posiion of he ball as a funcion of ime and he boom graph shows he elociy of a ball as a funcion of ime. y (a) op of fligh (b) The acceleraion of he ball is he same a eery poin of is rajecory, including he poin a which (a he op of is fligh). Hence a g. fligh op of (c) If he ball impacs a horizonal ceiling ery hard and hen reurns, is elociy a he op of is fligh is sill zero and is acceleraion is sill downward bu greaer han g in magniude. 9 The posiions of wo cars in parallel lanes of a sraigh srech of highway are ploed as funcions of ime in he Figure -33.Take posiie alues of x as being o he righ of he origin. Qualiaiely answer he following: (a) Are he wo cars eer side by side? If so, indicae ha ime (hose imes) on he axis. (b) Are hey always raeling in he same direcion, or are hey moing in opposie direcions for some of he ime? If so, when? (c) Are hey eer raeling a he same elociy? If so, when? (d) When are he wo cars he farhes apar? (e) Skech (no numbers) he elociy ersus ime cure for each car. Deermine he Concep Gien he posiions of he wo cars as a funcion of ime, we can use he inersecions of he cures and heir slopes o answer hese quesions.

4 Car A Chaper (a) The posiions of cars A and B are he same a wo places where he graphs cross. x Car A Car B Cars are side by side 9 (s) (b) When he slopes of he cures hae opposie signs, he elociies of he cars are opposiely direced. Thus, afer approximaely 7 s, car A is moing lefward while car B is moing righward. Before 7 s, he wo cars are raeling in he same direcion. (c) The wo cars hae he same elociy when heir cures hae he same slopes. This occurs a abou 6 s. (d) The ime a which he wo cars are farhes apar is roughly 6 s as indicaed by he place a which, erically, he wo cures are farhes par. (e) Car B Esimaion and Approximaion 37 Occasionally, people can surie falling large disances if he surface hey land on is sof enough. During a raerse of he Eiger s infamous Nordand, mounaineer Carlos Ragone s rock anchor gae way and he plummeed 5 fee o land in snow. Amazingly, he suffered only a few bruises and a wrenched shoulder. Assuming ha his impac lef a hole in he snow 4. f deep, esimae his aerage acceleraion as he slowed o a sop (ha is, while he was impacing he snow).

5 Moion in One Dimension Picure he Problem In he absence of air resisance, Carlos acceleraion is consan. Because all he moion is downward, le s use a coordinae sysem in which downward is he posiie direcion and he origin is a he poin a which he fall began. Using a consan-acceleraion equaion, relae Carlos final elociy o his elociy jus before his impac, his sopping acceleraion a s upon impac, and his sopping disance Δy: + a s Δy a s Δy or, because, a s Δy () Using a consan-acceleraion equaion, relae Carlos speed jus before impac o his acceleraion during free-fall and he disance he fell h: Subsiuing for in equaion () yields: Subsiue numerical alues and ealuae a s : + afree-fallh or, because and a -fall g gh a s gh Δy a ( 9.8 m/s )( 5 f) ( 4. f) 3. m/s free, Remarks: The magniude of his acceleraion is abou 5g! Speed, Displacemen, and Velociy 43 A runner runs.5 km, in a sraigh line, in 9. min and hen akes 3 min o walk back o he saring poin. (a) Wha is he runner s aerage elociy for he firs 9. min? (b) Wha is he aerage elociy for he ime spen walking? (c) Wha is he aerage elociy for he whole rip? (d) Wha is he aerage speed for he whole rip? Picure he Problem In his problem he runner is raeling in a sraigh line bu no a consan speed - firs she runs, hen she walks. Le s choose a coordinae sysem in which her iniial direcion of moion is aken as he posiie x direcion. (a) Using he definiion of aerage elociy, calculae he aerage elociy for he firs 9 min:.5 km Δ 9. min a.8 km / min

6 Chaper (b) Using he definiion of aerage elociy, calculae her aerage elociy for he 3 min spen walking: a.5 km Δ 3 min 83 m / min (c) Express her aerage elociy for Δ round rip he whole rip: a x Δ Δ (d) Finally, express her aerage speed for he whole rip: disance raeled speed a elapsed ime (.5 km) 3 min + 9. min.3 km / min 47 Proxima Cenauri, he closes sar o us besides our own sun, is 4. 3 km from Earh. From Zorg, a plane orbiing his sar, a Zorgian places an order a Tony s Pizza in Hoboken, New Jersey, communicaing ia ligh signals. Tony s fases deliery craf raels a. 4 c (see Problem 46). (a) How long does i ake Gregor s order o reach Tony s Pizza? (b) How long does Gregor wai beween sending he signal and receiing he pizza? If Tony s has a "- years-or-i s-free" deliery policy, does Gregor hae o pay for he pizza? Picure he Problem In free space, ligh raels in a sraigh line a consan speed, c. We can use he definiion of aerage speed o find he elapsed imes called for in his problem. (a) Using he definiion of aerage speed (equal here o he assumed consan speed of ligh), sole for he ime Δ required o rael he disance o Proxima Cenauri: Subsiue numerical alues and ealuae Δ: Δ disance raeled speed of ligh 4. 6 m.37 s 3. m s 8 Δ 8 4.3y

7 Moion in One Dimension 3 (b) The deliery ime (Δ oal ) is he sum of he ime for he order o reach Hoboken and he rael ime for he deliery craf o rael o Proxima Cenauri: Δ Δ + Δ oal order o be sen o Hoboken 4.33 y + order o be deliered 4. km 4 8 (. )( 3. m s) y y 4.3 y 3 y s Because y >> y, Gregor does no hae o pay. 53 The cheeah can run as fas as 3 km/h, he falcon can fly as fas as 6 km/h, and he sailfish can swim as fas as 5 km/h. The hree of hem run a relay wih each coering a disance L a maximum speed. Wha is he aerage speed of his relay eam for he enire relay? Compare his aerage speed wih he numerical aerage of he hree indiidual speeds. Explain carefully why he aerage speed of he relay eam is no equal o he numerical aerage of he hree indiidual speeds. Picure he Problem We can find he aerage speed of he relay eam from he definiion of aerage speed. Using is definiion, relae he aerage speed o he oal disance raeled and he elapsed ime: a disance raeled elapsed ime Express he ime required for each animal o rael a disance L: L, cheeah and sailfish cheeah L sailfish falcon L falcon Express he oal ime Δ: Δ L + + cheeah falcon sailfish Use he oal disance raeled by he relay eam and he elapsed ime o calculae he aerage speed: 3L L km/h 6km/h 5 km/h a.3 km/h km/h

8 4 Chaper Calculaing he aerage of he hree speeds yields: Aerage hreespeeds 3 km/h + 6km/h + 5 km/h 6.33 km/h 6 km/h 3.4 a The aerage speed of he relay eam is no equal o he numerical aerage of he hree indiidual speeds because he runners did no run for he same ineral of ime. The aerage speed would be equal o one-hird he sum of he hree speeds if he hree speeds were each mainained for he same lengh of ime insead of for he same disance. 55 A car raeling a a consan speed of m/s passes an inersecion a ime. A second car raeling a a consan speed of 3 m/s in he same direcion passes he same inersecion 5. s laer. (a) Skech he posiion funcions x () and x () for he wo cars for he ineral s. (b) Deermine when he second car will oerake he firs. (c) How far from he inersecion will he wo cars be when hey pull een? (d) Where is he firs car when he second car passes he inersecion? Picure he Problem One way o sole his problem is by using a graphing calculaor o plo he posiions of each car as a funcion of ime. Ploing hese posiions as funcions of ime allows us o isualize he moion of he wo cars relaie o he (fixed) ground. More imporanly, i allows us o see he moion of he wo cars relaie o each oher. We can, for example, ell how far apar he cars are a any gien ime by deermining he lengh of a erical line segmen from one cure o he oher. (a) Leing he origin of our coordinae sysem be a he inersecion, he posiion of he slower car, x (), is gien by: Because he faser car is also moing a a consan speed, we know ha he posiion of his car is gien by a funcion of he form: We know ha when 5. s, his second car is a he inersecion (ha is, x (5. s) ). Using his informaion, you can conince yourself ha: x () where x is in meers if is in seconds. x () 3 + b b 5 m

9 Moion in One Dimension 5 Thus, he posiion of he faser car is gien by: ( ) x 3 5 One can use a graphing calculaor, graphing paper, or a spreadshee o obain he following graphs of x () (he solid line) and x () (he dashed line): x, m , s (b) Use he ime coordinae of he inersecion of he wo lines o deermine he ime a which he second car oerakes he firs: (c) Use he posiion coordinae of he inersecion of he wo lines o deermine he disance from he inersecion a which he second car caches up o he firs car: From he inersecion of he wo lines, one can see ha he second car will "oerake" (cach up o) he firs car a 5 s. From he inersecion of he wo lines, one can see ha he disance from he inersecion is 3 m. (d) Draw a erical line from 5 s o he solid line and hen read he posiion coordinae of he inersecion of he erical line and he solid line o deermine he posiion of he firs car when he second car wen hrough he inersecion. From he graph, when he second car passes he inersecion, he firs car was m ahead. Acceleraion 59 An objec is moing along he x axis. A 5. s, he objec is a x +3. m and has a elociy of +5. m/s. A 8. s, i is a x +9. m and is elociy is. m/s. Find is aerage acceleraion during he ime ineral 5. s 8. s.

10 6 Chaper Picure he Problem We can find he change in elociy and he elapsed ime from he gien informaion and hen use he definiion of aerage acceleraion. The aerage acceleraion is defined as he change in elociy diided by he change in ime: a a Δ Δ Subsiue numerical alues and ealuae a a : a a (. m/s) ( 5. m/s) ( 8. s) ( 5. s). m/s 6 [SSM] The posiion of a cerain paricle depends on ime according o he equaion x() 5. +., where x is in meers if is in seconds. (a) Find he displacemen and aerage elociy for he ineral 3. s 4. s. (b) Find he general formula for he displacemen for he ime ineral from o + Δ. (c) Use he limiing process o obain he insananeous elociy for any ime. Picure he Problem We can closely approximae he insananeous elociy by he aerage elociy in he limi as he ime ineral of he aerage becomes small. This is imporan because all we can eer obain from any measuremen is he aerage elociy, a, which we use o approximae he insananeous elociy. (a) The displacemen of he paricle during he ineral 3. s 4. s is gien by: The aerage elociy is gien by: Find x(4. s) and x(3. s): Subsiue numerical alues in equaion () and ealuae : ( 4. s) x( 3. s) x () a () Δ x(4. s) (4.) 5(4.) + 3. m and x(3. s) (3.) 5(3.) + 5. m ( 3. m) ( 5. m).m Δ x Subsiue numerical alues in equaion () and ealuae a :. m. s a. m/s (b) Find x( + Δ): x( + Δ) ( + Δ) 5( + Δ) + ( + Δ + (Δ) ) 5( + Δ) +

11 Express x( + Δ) x() : ( ) ( ) Δ x Moion in One Dimension 7 5 Δ + Δ where is in meers if is in seconds. (c) From (b) find /Δ as Δ : ( 5) Δ + ( Δ) Δ Δ 5 + Δ and ( / Δ) lim Δ 5 where is in m/s if is in seconds. Alernaiely, we can ake he deriaie of x() wih respec o ime o obain he insananeous elociy. () ( ) d ( a + + ) dx b d d a + b 5 Consan Acceleraion and Free-Fall 67 An objec raeling along he x axis a consan acceleraion has a elociy of + m/s when i is a x 6. m and of +5 m/s when i is a x m. Wha is is acceleraion? Picure he Problem Because he acceleraion of he objec is consan we can use consan-acceleraion equaions o describe is moion. Using a consan-acceleraion equaion, relae he elociy o he acceleraion and he displacemen: Subsiue numerical alues and ealuae a: a + a a ( 5 ) ( 6. m) m m s 6m s 7 A load of bricks is lifed by a crane a a seady elociy of 5. m/s when one brick falls off 6. m aboe he ground. (a) Skech he posiion of he brick y() ersus ime from he momen i leaes he palle unil i his he ground. (b) Wha is he greaes heigh he brick reaches aboe he ground? (c) How long does i ake o reach he ground? (d) Wha is is speed jus before i his he ground? Picure he Problem In he absence of air resisance, he brick experiences consan acceleraion and we can use consan-acceleraion equaions o describe is moion. Consan acceleraion implies a parabolic posiion-ersus-ime cure.

12 8 Chaper (a) Using a consan-acceleraion equaion, relae he posiion of he brick o is iniial posiion, iniial elociy, acceleraion, and ime ino is fall: y + y ( g) + Subsiue numerical alues o obain: ( 5.m s) ( 4.9m s ) y 6.m + () The following graph of ( ) ( ) spreadshee program: y 6.m + 5.m s 4.9m s was ploed using a y, m , s (b) Relae he greaes heigh reached by he brick o is heigh when i falls off he load and he addiional heigh i rises : Δy max Using a consan-acceleraion equaion, relae he heigh reached by he brick o is acceleraion and iniial elociy: Subsiue numerical alues and ealuae Δ y max : Subsiue numerical alues in equaion () and ealuae h: h y + Δy max () ( ) op + g Δymax or, because op, + ( g) Δ max y Δy Δy ( 5.m s) max ( ) 9.8m s max.3m g h 6. m +.3m 7.3m Noe ha he graph shown aboe confirms his resul.

13 Moion in One Dimension 9 (c) Seing y in equaion () yields: Use he quadraic equaion or your graphing calculaor o obain: (d) Using a consan-acceleraion equaion, relae he speed of he brick on impac o is acceleraion and displacemen: 6. m + ( 5. m s) ( 4.9m s ).7s and.7 s. The negaie alue for yields (from g ) m/s and so has no physical meaning. + gh or, because, gh gh Subsiue numerical alues and ealuae : ( )( 7.3 m ) m/s 9.8m/s 75 A sone is hrown erically downward from he op of a -m cliff. During he las half second of is fligh, he sone raels a disance of 45 m. Find he iniial speed of he sone. Picure he Problem In he absence of air resisance, he acceleraion of he sone is consan. Choose a coordinae sysem wih he origin a he boom of he rajecory and he upward direcion posiie. Le f - be he speed one-half second before impac and he speed a impac. f Using a consan-acceleraion equaion, express he final speed of he sone in erms of is iniial speed, acceleraion, and displacemen: Find he aerage speed in he las half second: Using a consan-acceleraion equaion, express he change in speed of he sone in he las half second in erms of he acceleraion and he elapsed ime and sole for he change in is speed: f a and + aδy + gδy () f - + f 9m s f las half second Δ ( 9m s) 8m s f - + f Δ gδ f f - ( 9.8m s )(.5s) 4.9m s 45m.5s

14 3 Chaper Add he equaions ha express he sum and difference of f ½ and f and sole for f : f 8m s + 4.9m s 9.5m s Subsiue numerical alues in equaion () and ealuae : ( 9.5 m s) + ( 9.8 m s )( m) 68 m s Remarks: The sone may be hrown eiher up or down from he cliff and he resuls afer i passes he cliff on he way down are he same. 8 In a classroom demonsraion, a glider moes along an inclined air rack wih consan acceleraion. I is projeced from he low end of he rack wih an iniial elociy. Afer 8. s hae elapsed, i is cm from he low end and is moing along he rack a a elociy of 5 cm/s. Find he iniial elociy and he acceleraion. Picure he Problem The acceleraion of he glider on he air rack is consan. Is aerage acceleraion is equal o he insananeous (consan) acceleraion. Choose a coordinae sysem in which he iniial direcion of he glider s moion is he posiie direcion. Using he definiion of acceleraion, express he aerage acceleraion of he glider in erms of he glider s elociy change and he elapsed ime: Using a consan-acceleraion equaion, express he aerage elociy of he glider in erms of he displacemen of he glider and he elapsed ime: a a a a Δ Δ + x Δ Δ Δ Subsiue numerical alues and ealuae : The aerage acceleraion of he glider is: ( ) cm 8.s ( 5 cm/s) 5 cm/s (4 cm/s) a 8. s 6.9 cm/s 4 cm/s 83 A ypical auomobile under hard braking loses speed a a rae of abou 7. m/s ; he ypical reacion ime o engage he brakes is.5 s. A local school board ses he speed limi in a school zone such ha all cars should be able o sop

15 Moion in One Dimension 3 in 4. m. (a) Wha maximum speed does his imply for an auomobile in his zone? (b) Wha fracion of he 4. m is due o he reacion ime? Picure he Problem Assume ha he acceleraion of he car is consan. The oal disance he car raels while sopping is he sum of he disances i raels during he drier s reacion ime and he ime i raels while braking. Choose a coordinae sysem in which he posiie direcion is he direcion of moion of he auomobile and apply a consan-acceleraion equaion o obain a quadraic equaion in he car s iniial speed. (a) Using a consan-acceleraion equaion, relae he elociy of he car o is iniial elociy, acceleraion, and displacemen during braking: + abrk or, because he final elociy is zero, + abrk Δ xbrk a Express he oal disance raeled by he car as he sum of he disance raeled during he reacion ime and he disance raeled while slowing down: o Δ reac reac + brk a Rearrange his quadraic equaion o obain: Subsiue numerical alues and simplify o obain: Use your graphing calculaor or he quadraic formula o sole he quadraic equaion for is posiie roo: Coner his speed o mi/h: (b) Find he reacion-ime disance: aδreac + ao + ( 7. m/s) 56m / s 4.76 m/s mi/h.447m/s ( 4.76 m/s) mi/h reac Δ reac (4.76 m/s)(.5s).38 m Express and ealuae he raio of he reacion disance o he oal disance: o.38 m 4. m reac.6 9 Consider measuring he free-fall moion of a paricle (neglec air resisance). Before he aden of compuer-drien daa-logging sofware, hese experimens ypically employed a wax-coaed ape placed erically nex o he

16 3 Chaper pah of a dropped elecrically conducie objec. A spark generaor would cause an arc o jump beween wo erical wires hrough he falling objec and hrough he ape, hereby marking he ape a fixed ime inerals Δ. Show ha he change in heigh during successie ime inerals for an objec falling from res follows Galileo s Rule of Odd Numbers: Δy 3Δy, Δy 3 5Δy,..., where Δy is he change in y during he firs ineral of duraion Δ, Δy is he change in y during he second ineral of duraion Δ, ec. Picure he Problem In he absence of air resisance, he paricle experiences consan acceleraion and we can use consan-acceleraion equaions o describe is posiion as a funcion of ime. Choose a coordinae sysem in which downward is posiie, he paricle sars from res ( o ), and he saring heigh is zero (y ). Using a consan-acceleraion equaion, relae he posiion of he falling paricle o he acceleraion and he ime. Ealuae he y-posiion a successie equal ime inerals Δ, Δ, 3Δ, ec: y g y g y3 y4 g ec. ( Δ) ( Δ) g( Δ) ( 3Δ) g( Δ) ( 4Δ) g( Δ) Ealuae he changes in hose posiions in each ime ineral: Δ y ( ) y g Δ Δy Δy 3 Δy 4 y y g 3 g 3 y y g Δy ( Δ) + g( Δ) ( Δ) 3 g( Δ) 9 g ( Δ) + g( Δ) 4 ( Δ) 5 g( Δ)

17 Δy ec. 43 y Moion in One Dimension g 7Δy y 3 6 g ( Δ) + g( Δ) 9 ( Δ) 7 g( Δ) 93 If i were possible for a spacecraf o mainain a consan acceleraion indefiniely, rips o he planes of he Solar Sysem could be underaken in days or weeks, while oyages o he nearer sars would only ake a few years. (a) Using daa from he ables a he back of he book, find he ime i would ake for a oneway rip from Earh o Mars (a Mars closes approach o Earh). Assume ha he spacecraf sars from res, raels along a sraigh line, acceleraes halfway a g, flips around, and deceleraes a g for he res of he rip. (b) Repea he calculaion for a km rip o Proxima Cenauri, our neares sellar neighbor ouside of he sun. (See Problem 47.) Picure he Problem Noe: No maerial body can rael a speeds faser han ligh. When one is dealing wih problems of his sor, he kinemaic formulae for displacemen, elociy and acceleraion are no longer alid, and one mus inoke he special heory of relaiiy o answer quesions such as hese. For now, ignore such subleies. Alhough he formulas you are using (i.e., he consanacceleraion equaions) are no quie correc, your answer o Par (b) will be wrong by abou %. (a) Le / represen he ime i akes o reach he halfway poin. Then he oal rip ime is: Use a consan- acceleraion equaion o relae he half-disance o Mars o he iniial speed, acceleraion, and half-rip ime / : / () Δ x + a Because and a g: / a Subsiue in equaion () o obain: The disance from Earh o Mars a closes approach is 7.8 m. Subsiue numerical alues and ealuae : () a.d ( m) 3.9 round rip 8 9.8m/s 4 s

18 34 Chaper (b) From Problem 47 we hae: d Proxima Cenauri 4. 3 km Subsiue numerical alues in equaion () o obain: round rip 5.8 y 3 ( 4. km) 9.8m/s 8 7 s Remarks: Our resul in Par (a) seems remarkably shor, considering how far Mars is and how low he acceleraion is. 99 A speeder raeling a a consan speed of 5 km/h races pas a billboard. A parol car pursues from res wih consan acceleraion of (8. km/h)/s unil i reaches is maximum speed of 9 km/h, which i mainains unil i caches up wih he speeder. (a) How long does i ake he parol car o cach he speeder if i sars moing jus as he speeder passes? (b) How far does each car rael? (c) Skech x() for each car. Picure he Problem This is a wo-sage consan-acceleraion problem. Choose a coordinae sysem in which he direcion of he moion of he cars is he posiie direcion. The picorial represenaion summarizes wha we know abou he moion of he speeder s car and he parol car. x S, xs, xs, S, 5 km/h S, 5 km/h S, 5 km/h a S, a S, a a P,. m/s P, x xp, xp, xp, P, P, 9 km/h km/h P, 9 Coner he speeds of he ehicles and he acceleraion of he police car ino SI unis: km km h m/s, h s h s 36s km km h m/s, h h 36s and km km h m/s h h 36s

19 Moion in One Dimension 35 (a) Express he condiion ha deermines when he police car caches he speeder; ha is, ha heir displacemens will be he same: Using a consan-acceleraion equaion, relae he displacemen of he parol car o is displacemen while acceleraing and is displacemen once i reaches is maximum elociy: Using a consan-acceleraion equaion, relae he displacemen of he speeder o is consan elociy and he ime i akes he parol car o cach i: Calculae he ime during which he police car is speeding up: Express he displacemen of he parol car: Equae he displacemens of he wo ehicles: Subsiue for P, o obain: P, S, Δ P, Δ xp, + P, P, + P, S, S,Δ ( 34.7 m/s) P, P, P, ( ) ΔP, P, P, ap, ap, 5.8m/s 4s.m/s Δ + a Δ P, + 63m P, P, + P, + 63m + P, P, (. m/s )( 4s) P, P,( ) ( 5.8m/s) ( 4s) (34.7 m/s) 63 m + (5.8 m/s)( 4 s) Soling for he ime o cach up yields: 35.9 s 35s (b) The disance raeled is he displacemen,,s, of he speeder during he cach: S, S,Δ.km ( 35m/s)( 35.9s)

20 36 Chaper (c) The graphs of x S and x P follow. The sraigh line (solid) represens x S () and he parabola (dashed) represens x P (). 4 Speeder Officer x, m , s Inegraion of he Equaions of Moion 3 The elociy of a paricle is gien by x () (6. m/s ) + (3. m/s). (a) Skech ersus and find he area under he cure for he ineral o 5. s. (b) Find he posiion funcion x(). Use i o calculae he displacemen during he ineral o 5. s. Picure he Problem The inegral of a funcion is equal o he "area" beween he cure for ha funcion and he independen-ariable axis. (a) The following graph was ploed using a spreadshee program:, m/s , s

21 Moion in One Dimension 37 The disance is found by deermining he area under he cure. There are approximaely 36 blocks each haing an area of (5. m/s)(.5 s).5 m. You can confirm his resul by using he formula for he area of a rapezoid: A under cure ( 36 blocks)(.5 m/block) 9 m 33m/s + 3m/s A 9m ( 5.s s) (b) To find he posiion funcion x(), we inegrae he elociy funcion () oer he ime ineral in x () () d quesion: ( 6. m/s ) + ( 3. m/s ) Now ealuae x() a s and 5. s respeciely and subrac o obain : [ ]d and x( ) ( 3.m/s ) + ( 3.m/s) x 5.s x s 9 m m ( ) ( ) 9 m 9 Figure -45 shows a plo of x ersus for a body moing along a sraigh line. For his moion, skech graphs (using he same axis) of (a) x as a funcion of, and (b) a x as a funcion of. (c) Use your skeches o compare he imes when he objec is a is larges disance from he origin o he imes when is speed is greaes. Explain why hey do no occur a he same ime. (d) Use your skeches o compare he ime(s) when he objec is moing fases when he ime(s) when is acceleraion is he larges. Explain why hey do no occur a he same ime. Commen [EPM]: See my commens on he soluion. Picure he Problem Because he posiion of he body is no described by a parabolic funcion, he acceleraion is no consan. (a) Selec a series of poins on he graph of x() (e.g., a he exreme alues and where he graph crosses he axis), draw angen lines a hose poins, and measure heir slopes. In doing his, you are ealuaing dx/d a hese poins. Plo hese slopes aboe he imes a which you measured he slopes. Your graph should closely resemble he following graph.

22 38 Chaper (b) Selec a series of poins on he graph of () (e.g., a he exreme alues and where he graph crosses he axis), draw angen lines a hose poins, and measure heir slopes. In doing his, you are ealuaing a d/d a hese poins. Plo hese slopes aboe he imes a which you measured he slopes. Your graph should closely resemble he following graph. 5 5 a (c) The poins a he greaes disances from he ime axis correspond o urnaround poins. The elociy of he body is zero a hese poins. (d) The body is moing fases as i goes hrough he origin. A hese imes he elociy is no changing and hence he acceleraion is zero. The maximum acceleraion occurs a he maximum disances where he elociy is zero bu changing direcion rapidly. In he ime ineral from. s o. s, he acceleraion of a paricle raeling in a sraigh line is gien by a x (. m/s 3 ). Le o he righ be he +x direcion. A paricle iniially has a elociy o he righ of 9.5 m/s and is locaed

23 Moion in One Dimension m o he lef of he origin. (a) Deermine he elociy as a funcion of ime during he ineral, (b) deermine he posiion as a funcion of ime during he ineral, (c) deermine he aerage elociy beween. s and. s, and compare i o he aerage of he insananeous elociies a he sar and ending imes. Are hese wo aerages equal? Explain. Picure he Problem The acceleraion is a funcion of ime; herefore i is no consan. The insananeous elociy can be deermined by inegraion of he acceleraion funcion and he aerage elociy from he general expression for he aerage alue of a non-linear funcion. (a) The insananeous elociy funcion () is he ime-inegral of he acceleraion funcion: () b a d b d + x C where b. m/s 3 The iniial condiions are: ) x ( ) 5. m and ) 9.5 m/s ( ) Use iniial condiion ) o obain: ( ) 9 Subsiuing in () for b and C yields: (b) The insananeous posiion funcion x() is he ime-inegral of he elociy funcion: Using iniial condiion ) yields: ( ) Subsiuing in x() for C and C yields: x 3 (. m/s ). 5 m/s C 3 (. m/s ) ( ) m/s () ( ) ( ) d ( c + C ) x c 3 + C + C 3 where c. m/s 3. x 5. m C 3 () + ( 9.5 m/s) 5. m/s 3 (c) The aerage alue of () oer he ineral Δ is gien by: () d Δ Subsiue for () and ealuae he inegral o obain: d

24 4 Chaper b b 3 b 3 b 3 + C d + C + C + C Δ Δ 6 Δ 6 6 Simplifying his expression yields: b 3 3 ( ) + C( ) Δ 6 Because : b Δ C Subsiue numerical alues and simplify o obain: 3. m/s 3. s 6 (. s) + ( 9.5 m/s)(. s) 3 m/s The aerage of he iniial insananeous and final insananeous elociies is gien by: Using equaion (), ealuae () and ( s): a ( ) ( ) + (. s) () 9.5 m/s and s. m/s 3 ( ) ( ) (. s) m/s 9.5 m/s Subsiue in equaion () o obain: a 9. 5 m/s m/s 5 m/s a is no he same as because he elociy does no change linearly wih ime. The elociy does no change linearly wih ime because he acceleraion is no consan. General Problems 5 Consider an objec ha is aached o a horizonally oscillaing pison. The objec moes wih a elociy gien by B sin(ω), where B and ω are consans and ω is in s. (a) Explain why B is equal o he maximum speed max. (b) Deermine he acceleraion of he objec as a funcion of ime. Is he acceleraion consan? (c) Wha is he maximum acceleraion (magniude) in erms of ω and max. (d) A, he objec's posiion is known o be x. Deermine he posiion as a funcion of ime in erms of in erms of, ω, x and max. Deermine he Concep Because he elociy aries nonlinearly wih ime, he acceleraion of he objec is no consan. We can find he acceleraion of he objec by differeniaing is elociy wih respec o ime and is posiion funcion

25 Moion in One Dimension 4 by inegraing he elociy funcion. (a) The maximum alue of he sine funcion (as in max sin(ω)) is. Hence he coefficien B represens he maximum possible speed max. (b) The acceleraion of he objec is he deriaie of is elociy wih respec o ime: a d d d d ω cos max [ sin( ω) ] max ( ω) Because a aries sinusoidally wih ime i is no consan. (c) Examinaion of he coefficien of he cosine funcion in he expression for a leads one o he conclusion ha a ω. max max (d) The posiion of he objec as a funcion of ime is he inegral of he elociy funcion: Inegraing he lef-hand side of he equaion and subsiuing for on he righ-hand side yields: ( ) dx d x sin ω max ( ) d + C Inegrae he righ-hand side o ω ω obain: x cos( ) + C max () Use he iniial condiion x() x o obain: Soling for C yields: x ω max max C x + ω + C Subsiue for C in equaion () o max x + x + ω ω max obain: cos( ω) Soling his equaion for x yields: x max x + ω [ cos( ω) ] 7 A rock falls hrough waer wih a coninuously decreasing acceleraion. Assume ha he rock s acceleraion as a funcion of elociy has he form ay g b y where b is a posiie consan. (The +y direcion is direcly downward.) (a) Wha are he SI unis of b? (b) Proe mahemaically ha if he rock eners he waer a ime, he acceleraion will depend exponenially on

26 4 Chaper ime according o a y b ( ) ge. (c) Wha is he erminal speed for he rock in erms of g and b? (See Problem 38 for an explanaion of he phenomenon of erminal speed.) Picure he Problem Because he acceleraion of he rock is a funcion of is elociy, i is no consan and we will hae o inegrae he acceleraion funcion in order o find he elociy funcion. Choose a coordinae sysem in which downward is posiie and he origin is a he poin of release of he rock. (a) All hree erms in ay g b y mus hae he same unis in order for he equaion o be alid. Hence he unis of b y mus be acceleraion unis. Because he SI unis of y are m/s, b mus hae unis of s. (b) Rewrie a y g b y explicily as a differenial equaion: Separae he ariables, y on he lef, on he righ: d y d d g b g b y y y d Inegrae he lef-hand side of his equaion from o y and he righhand side from o : y d y g b y d Inegraing his equaion yields: g b ln b g y Sole his expression for y o obain: ( e b ) g y () b Differeniae his expression wih respec o ime o obain an expression for he acceleraion and complee he proof: a y d y d g d d b b b ( e ) ge (c) Take he limi, as, of boh g b b lim ( ) sides of equaion (): y lim e and g b Noice ha his resul depends only on b (inersely so). Thus b mus include

27 Moion in One Dimension 43 geomeric facors like he shape and cross-secional area of he falling objec, as well as properies of he liquid such as densiy and emperaure.