Chapter 2: One-Dimensional Kinematics

Transcription

1 Chaper : One-Dimensional Kinemaics Answers o Een-Numbered Concepual Quesions. An odomeer measures he disance raeled by a car. You can ell his by he fac ha an odomeer has a nonzero reading afer a round rip. 4. No. Their elociies are differen because hey rael in differen direcions. 6. Since he car circles he rack is direcion of moion mus be changing. Therefore, is elociy changes as well. Is speed, howeer, can be consan. 8. (a) The ime required o sop is doubled. (b) The disance required o sop increases by a facor of four.. Yes, if i moes wih consan elociy.. (a) No. If air resisance can be ignored, he acceleraion of he ball is he same a each poin on is fligh. (b) Same answer as par (a). 4. (a) No. Displacemen is he change in posiion, and herefore i is independen of he locaion chosen for he origin. (b) Yes. In order o know wheher an objec s displacemen is posiie or negaie, we need o know which direcion has been chosen o be posiie. Soluions o Problems and Concepual Exercises. Picure he Problem: You walk in boh he posiie and negaie direcions along a sraigh line. Sraegy: The disance is he oal lengh of rael, and he displacemen is he ne change in posiion. We place he origin a he locaion labeled Your house. Soluion:. (a) Add he lenghs:.75.6 mi.6 mi.95 mi. (b) Subrac x i from x f o find he displacemen. x xf xi.75. mi.75 mi Insigh: The disance raeled is always posiie, bu he displacemen can be negaie.. Picure he Problem: You walk in boh he posiie and negaie direcions along a sraigh line. Sraegy: The disance is he oal lengh of rael, and he displacemen is he ne change in posiion. We place he origin a he locaion labeled Your house. Soluion:. (a) Add he lenghs:.6.35 mi mi.65 mi. (b) Subrac x i from x f o find he displacemen. x xf xi..75 mi.75 mi Insigh: The disance raeled is always posiie, bu he displacemen can be negaie. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No

2 Chaper : One-Dimensional Kinemaics 3. Picure he Problem: Player A walks in he posiie direcion and player B walks in he negaie direcion. Sraegy: In each case he disance is he oal lengh of rael, and he displacemen is he ne change in posiion. Soluion:. (a) Noe he disance raeled by player A: 5 m. The displacemen of player A is posiie: x xf xi 5 m m 5 m 3. (b) Noe he disance raeled by player B: m 4. The displacemen of player B is negaie. Le he origin be a he iniial posiion of player A. Insigh: The disance raeled is always posiie, bu he displacemen can be negaie. x xf xi 5 m 7 m m 4. Picure he Problem: The ball is pued in he posiie direcion and hen he negaie direcion. Sraegy: The disance is he oal lengh of rael, and he displacemen is he ne change in posiion. Soluion:. (a) Add he lenghs:.5 m.5 m 5 m. (b) Subrac x i from x f o find he displacemen. x xf xi m m Insigh: The disance raeled is always posiie, bu he displacemen can be negaie. 5. Picure he Problem: The runner moes along he oal rack. Sraegy: The disance is he oal lengh of rael, and he displacemen is he ne change in posiion. Soluion:. (a) Add he lenghs: 5 m m 5 m 3 m. Subrac x i from x f o find he displacemen. x xf xi m m 3. (b) Add he lenghs: m 6 m 4. Subrac x i from x f o find he displacemen. x xf xi m m Insigh: The disance raeled is always posiie, bu he displacemen can be negaie. The displacemen is always zero for a complee circui, as in his case. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No

3 Chaper : One-Dimensional Kinemaics 6. Picure he Problem: The pony walks around he circular rack. Sraegy: The disance is he oal lengh of rael, and he displacemen is he ne change in posiion. A 5.5 m B Soluion: (a). The disance raeled is half he circumference: d r r 5.5 m 6.5 m. The displacemen is he disance from A o B: x x x r f i 5.5 m.5 m 3. (b) The disance raeled will increase when he child complees one circui, because he pony will hae aken more seps. 4. (c) The displacemen will decrease when he child complees one circui, because he displacemen is maximum when he child has gone halfway around, and is zero when he child reurns o he saring posiion. 5. (d) The disance raeled equals he circumference: d r 6. The displacemen is zero because he child has reurned o her saring posiion. 5.5 m 33. m Insigh: The disance raeled is always posiie, bu he displacemen can be negaie. The displacemen is always zero for a complee circui, as in his case. 7. Picure he Problem: You drie your car in a sraigh line a wo differen speeds. Sraegy: We could calculae he aerage speed wih he gien informaion by deermining he oal disance raeled and diiding by he elapsed ime. Howeer, we can arrie a a concepual undersanding of he answer by remembering ha aerage speed is an aerage oer ime, no an aerage oer he disance raeled. Soluion:. (a) The aerage speed will be less han m/s because you will spend a longer ime driing a he lower speed. You will coer he second km disance in less ime a he higher speed han you did a he lower speed.. (b) The bes answer is I. More ime is spen a 5 m/s han a 5 m/s because he disances raeled a each speed are he same, so ha i will ake a longer ime a he slower speed o coer he same disance. Saemen II is rue bu irrelean and saemen III is false. Insigh: The ime elapsed a he lower speed is, m 5 m/s 667 s and he ime elapsed a he higher speed is, m 5 m/s 4 s, hence he aerage speed is, m 67 s 8.7 m/s. 8. Picure he Problem: You drie your car in a sraigh line a wo differen speeds. Sraegy: We could calculae he aerage speed wih he gien informaion by deermining he oal disance raeled and diiding by he elapsed ime. Howeer, we can arrie a a concepual undersanding of he answer by remembering ha aerage speed is an aerage oer ime, no an aerage oer he disance raeled. Soluion:. (a) The aerage speed will be equal o m/s because you will spend an equal amoun of ime driing a he lower speed as a he higher speed. The aerage speed is herefore he mean alue of he wo speeds.. (b) The bes answer is III. Equal ime is spen a 5 m/s and 5 m/s because ha fac is saed in he quesion. Saemens I and II are boh false. Insigh: The disance raeled a he lower speed would be 5 m/s6 s 9 m and he disance raeled a he higher speed would be 5 m/s6 s 5, m so he aerage speed is 4, m s. m/s. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 3

4 Chaper : One-Dimensional Kinemaics 9. Picure he Problem: A runner sprins in he forward direcion. Sraegy: The aerage speed is he disance diided by elapsed ime. Soluion: Diide he disance by he ime: disance. m mi 36 s s.4 m/s 3.3 mi/h ime 9.9 s 69 m h Insigh: The displacemen would be complicaed in his case because he -m dash usually akes place on a cured rack. Forunaely, he aerage speed depends upon disance raeled, no displacemen.. Picure he Problem: A kangaroo hops in he forward direcion. Sraegy: The disance is he aerage speed muliplied by he ime elapsed. The ime elapsed is he disance diided by he aerage speed. Soluion:. (a) Muliply he aerage speed by he ime elapsed:. (b) Diide he disance by he aerage speed: km h d s min 3.5 km h 6 min d.5 km 6 min 4 s s 65 km/h h Insigh: The insananeous speed migh ary from 65 km/h, bu he ime elapsed and he disance raeled depend only upon he aerage speed during he ineral in quesion.. Picure he Problem: Rubber ducks drif along he ocean surface. Sraegy: The aerage speed is he disance diided by elapsed ime. Soluion:. (a) Diide he disance by he ime: d 6 mi 69 m mo d s 4 mo mi 3.5 d 8.64 s.98 m/s. (b) Diide he disance by he ime: d 6 mi mo d s. mi/h mo 3.5 d 4 h Insigh: The insananeous speed migh ary from.98 m/s, bu we can calculae only aerage speed from he oal disance raeled and ime elapsed.. Picure he Problem: Radio waes propagae in a sraigh line. Sraegy: The ime elapsed is he disance diided by he aerage speed. The disance o he Moon is.39 5 mi. We mus double his disance because he signal raels here and back again. Soluion: Diide he disance by he aerage speed: 5.39 mi d.57 s s 5.86 mi/s Insigh: The ime is slighly shorer han his because he gien disance is from he cener of he Earh o he cener of he Moon, bu presumably any radio communicaions would occur beween he surfaces of he Earh and Moon. When he radii of he wo spheres is aken ino accoun, he ime decreases o.5 s. 3. Picure he Problem: Sound waes propagae in a sraigh line from a hunderbol o your ears. Sraegy: The disance is he aerage speed muliplied by he ime elapsed. We will neglec he ime i akes for he ligh wae o arrie a your eyes because i is asly smaller han he ime i akes he sound wae o rael. Soluion: Muliply he aerage speed by he ime elapsed: d s 34 m/s 6.5 s m. km Insigh: The speed of sound, 34 m/s, works ou o approximaely one mile eery fie seconds, a useful rule of humb for esimaing he disance o an approaching hundersorm! Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 4

5 Chaper : One-Dimensional Kinemaics 4. Picure he Problem: Human nere impulses propagae a a fixed speed. Sraegy: The ime elapsed is he disance diided by he aerage speed. The disance from your oes o your brain is on he order of wo meers. d m Soluion: Diide he disance by he aerage speed:. s s m/s Insigh: This nere impulse rael ime is no he limiing facor for human reacion ime, which is abou. s. 5. Picure he Problem: A finch raels a shor disance on he back of he oroise and a longer disance hrough he air, wih boh displacemens along he same direcion. Sraegy: Firs find he oal disance raeled by he finch and hen deermine he aerage speed by diiding by he oal ime elapsed. Soluion:. Deermine he oal disance raeled:. Diide he disance by he ime elapsed: d s s d.6 m/s.5 min m/s.5 min 6 s/min d 995 m d 995 m s 5.5 m/s 3. min 6 s/min Insigh: Mos of he disance raeled by he finch occurred by air. In fac, if we neglec he 5.4 m he finch raeled while on he oroise s back, we sill ge an aerage speed of 5.5 m/s oer he 3. min ime ineral! The bird migh as well hae been a res during he ime i perched on he oroise s back. 6. Picure he Problem: You jog for 5. km and hen rael an addiional 3 km by car, wih boh displacemens along he same direcion. Sraegy: Firs find he oal ime elapsed by diiding he disance raeled by he aerage speed. Find he ime elapsed while jogging, and subrac i from he oal ime o find he ime elapsed while in he car. Finally use he rael-by-car disance and ime informaion o find he aerage speed wih which you mus drie he car. Soluion:. Use he definiion of aerage speed o deermine he oal ime elapsed.. Find he ime elapsed while jogging: d 5. 3 km.7 h s 5 km/h a d 5. km.55 h 9. km/h 3. Find he ime elapsed while in he car:.7 h.55 h.7 h 4. Find he speed of he car: s d 3 km 76 km/h.7 h Insigh: Noice ha he aerage speed is no he aerage of 9. km/h and 76 km/h (which would be 43 km/h) because you spend a much longer ime jogging a low speed han you spend driing a high speed. 7. Picure he Problem: A dog coninuously runs back and forh as he owners close he disance beween each oher. Sraegy: Firs find he ime ha will elapse before he owners mee each oher. Then deermine he disance he dog will coer if i coninues running a consan speed oer ha ime ineral. Soluion:. Find he ime i akes each owner o walk half he disance (4. m) before meeing each oher: d 4. m 3.5 s s.3 m/s. Find he disance he dog runs: d s Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 5 a.7 m/s.7 m/s 3.5 s 8.5 m Insigh: The dog will acually run a shorer disance han his, because i is impossible for i o mainain he same.7 m/s as i urns around o run o he oher owner. I mus firs slow down o zero speed and hen accelerae again. 8. m

6 Chaper : One-Dimensional Kinemaics 8. Picure he Problem: Blood flows a wo differen speeds hrough areries during a specified ime ineral. Sraegy: Deermine he aerage speed by firs calculaing he oal disance raeled and hen diiding i by he oal ime elapsed. Soluion:. (a) Because he ime inerals are he same, he blood spends equal imes a. m/s and.6 m/s, and is aerage speed will be equal o.8 m/s.. (b) Diide he oal disance by he ime elapsed: s s s. m/s.5 s.6 m/s.5 s.8 m.5.5 s. s a.8 m/s Insigh: Aerage speed is a weighed aerage according o how much ime he blood spends raeling a each speed. 9. Picure he Problem: Blood flows a wo differen speeds hrough areries oer a specified disance. Sraegy: Deermine he aerage speed by firs calculaing he oal disance raeled and hen diiding i by he oal ime elapsed. Soluion:. (a) The disance inerals are he same bu he ime inerals are differen. The blood will spend more ime a he lower speed han a he higher speed. Because he aerage speed is a ime-weighed aerage, i will be less han.8 m/s.. (b) Diide he oal disance by he ime elapsed: s a d d s a d d d d. m.5 m.5 m s s. m/s.6 m/s.75 m/s Insigh: The blood spends.5 s flowing a. m/s and.83 s flowing a.6 m/s.. Picure he Problem: You rael in a sraigh line a wo differen speeds during he specified ime ineral. Sraegy: Deermine he disance raeled during each leg of he rip in order o plo he graph. Soluion:. (a) Calculae he disance raeled in he firs leg: d s m/s.5 min 6 s/min 8 m. Calculae he disance raeled in he second leg: d s m/s 3.5 min m 3. Calculae he disance raeled in he hird leg: d s 4. Calculae he oal disance raeled: d d d d3 333 m m/s.5 min 6 s/min 5 m 5. Draw he graph: 6. (b) Diide he oal disance by he ime elapsed: s d d d 333 m 3 a min 6 s/min 7.4 m/s Insigh: The aerage speed is a weighed aerage according o how much ime you spend raeling a each speed. Here you spend he mos amoun of ime a res, so he aerage speed is less han eiher m/s or 5 m/s. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 6

7 Chaper : One-Dimensional Kinemaics. Picure he Problem: As specified in he posiion-ersus-ime graph, he faher walks forward, sops, walks forward again, and hen walks backward. Sraegy: Deermine he direcion of he elociy from he slope of he graph along each segmen. Then deermine he magniude of he elociy by calculaing he slope of he graph a each specified poin. Soluion:. (a) The slope a A is posiie so he elociy is posiie. (b) The elociy a B is zero. (c) The elociy a C is posiie. (d) The elociy a D is negaie. x. m. (e) Find he slope of he graph a A: a. m/s. s x. m 3. (f) Find he slope of he graph a B: a. m/s. s x. m 4. (g) Find he slope of he graph a C: a. m/s. s x 3. m 5. (h) Find he slope of he graph a D: a.5 m/s. s Insigh: The signs of each answer in (e) hrough (h) mach hose prediced in pars (a) hrough (d). Wih pracice you can form boh a qualiaie and quaniaie moie of he moion in your head simply by examining he posiionersus-ime graph.. Picure he Problem: The gien posiion funcion indicaes he paricle begins raeling in he negaie direcion bu is acceleraing in he posiie direcion. Sraegy: Creae he x-ersus- plo using a spreadshee, or calculae indiidual alues by hand and skech he cure using graph paper. Use he known x and informaion o deermine he aerage elociy. To find he aerage speed, we mus find he oal disance ha he paricle raels beween and. s, and hen diide by. s. Soluion:. (a) Use a spreadshee or similar program o creae he plo shown a righ. Noice ha he aerage elociy oer he firs second of ime is equal o he slope of a sraigh line drawn from he origin o he alue of he cure a =. s. A ha ime he posiion is. m.. (b) Find he aerage elociy from = o =. s: 3. (c) To find he aerage speed, we mus deermine he disance raeled. Firs calculae he ime a which x = : 4. The ime a which he paricle urns around is half he ime found in sep 3. Find x a he urnaround ime: 5. A = s, he paricle is a x = m, so i has raeled an addiional.83 m afer urning around. Find he aerage speed: a 5 m/s. s 3 m/s. s. m x. m/s. s 5 m/s 3 m/s Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 7 5 m/s 3 m/s 5 3 s.67 s x 5 m/s5 6 s 3 m/s5 6 s.83 m s a m. m/s. s Insigh: The insananeous speed is always he magniude of he insananeous elociy, bu he aerage speed is no always he magniude of he aerage elociy. For insance, in his problem he paricle reurns o x = afer.67 s, a which ime is aerage speed is sa 4.7 m.67 s.5 m/s, bu is aerage elociy is zero because x =.

8 Chaper : One-Dimensional Kinemaics 3. Picure he Problem: The gien posiion funcion indicaes he paricle begins raeling in he posiie direcion bu is acceleraing in he negaie direcion. Sraegy: Creae he x-ersus- plo using a spreadshee, or calculae indiidual alues by hand and skech he cure using graph paper. Use he known x and informaion o deermine he aerage speed and elociy. Soluion:. (a) Use a spreadshee o creae he plo shown a righ:. (b) Find he aerage elociy from = o =. s: a a x 6 m/s. s m/s. s. m 4. m/s. s 3. (c) The aerage speed is he magniude of he aerage elociy: sa a 4. m/s Insigh: Noice ha he aerage elociy oer he firs second of ime is equal o he slope of a sraigh line drawn from he origin o he cure a =. s. A ha ime he posiion is 4. m. 4. Picure he Problem: Following he moion specified in he posiionersus-ime graph, he ennis player moes lef, hen righ, hen lef again, if we ake lef o be in he negaie direcion. Sraegy: Deermine he direcion of he elociy from he slope of he graph. The speed will be greaes for he segmen of he cure ha has he larges slope magniude. Soluion:. (a) The magniude of he slope a B is larger han A or C so we conclude he speed is greaes a B. x. m. (b) Find he slope of he graph a A: sa. m/s. s x. m 3. (c) Find he slope of he graph a B: sa. m/s. s x. m 4. (d) Find he slope of he graph a C: sa.5 m/s. s Insigh: The speed during segmen B is larger han he speed during segmens A and C, as prediced. Speeds are always posiie because hey do no inole direcion, bu elociies can be negaie o indicae heir direcion. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 8

9 Chaper : One-Dimensional Kinemaics 5. Picure he Problem: You rael in he forward direcion along he roads leading o he wedding ceremony, bu your aerage speed is differen during he firs and second porions of he rip. Sraegy: Firs find he disance raeled during he firs 5 minues in order o calculae he disance ye o rael. Then deermine he speed you need during he second 5 minues of rael. Soluion:. Use he definiion of aerage mi h d speed o deermine he disance raeled: s min.5 mi h 6 min. Find he remaining disance o rael: d doal d..5 mi 8.8 mi 3. Find he required speed for he second par of he rip: s d 8.8 mi.5 h 35 mi/h Insigh: The car needs an aerage speed of mi/.5 h = mi/h for he enire rip. Howeer, in order o make i on ime i mus go seen imes faser in he second half (ime-wise) of he rip han i did in he firs half of he rip. 6. Picure he Problem: The graph in he problem saemen depics he posiion of a boa as a funcion of ime. Sraegy: The elociy of he boa is equal o he slope of is posiion-ersus-ime graph. Soluion: By examining he graph we can see ha he seepes slope in he negaie direcion (down and o he righ) is a poin C. Therefore, he boa had is mos negaie elociy a ha ime. Poins A, B, D, and F all correspond o imes of zero elociy because he slope of he graph is zero a hose poins. Poin E has a large posiie slope and we conclude he boa had is mos posiie elociy a ha ime. Therefore, he ranking is: C < A = B = D = F < E. Insigh: The porion of he graph o he lef of poin B also corresponds o a ime of high posiie elociy. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 9

10 Chaper : One-Dimensional Kinemaics 7. Picure he Problem: The gien posiion funcion indicaes he paricle begins raeling in he posiie direcion bu is acceleraing in he negaie direcion. Sraegy: Creae he x-ersus- plo using a spreadshee, or calculae indiidual alues by hand and skech he cure using graph paper. Use he known x and informaion o deermine he aerage speed and elociy. Soluion:. (a) Use a spreadshee o creae he plo:. (b) Find he aerage elociy from =.35 o =.45 s: 3. (c) Find he aerage elociy from =.39 o =.4 s: a a 3 3 m/s.45 s 3 m/s.45 s m/s.35 s 3 m/s.35 s 3 3 x. s.55 m/s 3 3 m/s.4 s 3 m/s.4 s m/s.39 s 3 m/s.39 s 3 3 x.4.39 s.56 m/s 4. (d) The insananeous speed a =.4 s will be closer o.56 m/s. As he ime ineral becomes smaller he aerage elociy is approaching.56 m/s, so we conclude he aerage speed oer an infiniesimally small ime ineral will be ery close o ha alue. Insigh: Noice ha he insananeous elociy a.4 s is equal o he slope of a sraigh line drawn angen o he cure a ha poin. Because i is difficul o accuraely draw a angen line, we ofen resor o mahemaical mehods like hose illusraed aboe o deermine he insananeous elociy. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No

11 Chaper : One-Dimensional Kinemaics 8. Picure he Problem: The gien posiion funcion indicaes he paricle begins raeling in he negaie direcion bu is acceleraing in he posiie direcion. Sraegy: Creae he x-ersus- plo using a spreadshee, or calculae indiidual alues by hand and skech he cure using graph paper. Use he known x and informaion o deermine he aerage speed and elociy. Soluion:. (a) Use a spreadshee o creae he plo:. (b) Find he aerage elociy from =.5 o =.5 s: a 3 m/s.5 s 3 m/s.5 s x m/s.5 s 3 m/s.5 s.63 m/s.5.5 s 3. (c) Find he aerage elociy from =.9 o =. s: a 3 m/s. s 3 m/s. s x m/s.9 s 3 m/s.9 s.64 m/s..9 s 4. (d) The insananeous speed a =. s will be closer o.64 m/s. As he ime ineral becomes smaller he aerage elociy approaches.64 m/s, and we conclude he aerage speed oer an infiniesimally small ime ineral will be ery close o ha alue. Insigh: Noice ha he insananeous elociy a. s is equal o he slope of a sraigh line drawn angen o he cure a ha poin. Because i is difficul o accuraely draw a angen line, we ofen resor o mahemaical mehods like hose illusraed aboe o deermine he insananeous elociy. 9. Picure he Problem: You accelerae your car from res along wo on-ramps of differen lenghs. Sraegy: Use he definiions of aerage speed and acceleraion o compare your moion along he wo on-ramps. Soluion:. (a) We can reason ha because you accelerae beween he same iniial and final elociies, you mus hae he same aerage speed along boh on-ramps. If you hae he same aerage speed, hen you will accelerae for a shorer period of ime along he shorer on-ramp A. Your acceleraion mus be greaer o achiee he same final elociy in a shorer ime. We conclude ha your acceleraion along on-ramp A is greaer han your acceleraion along on-ramp B.. (b) As discussed aboe, he bes explanaion is I. The shorer acceleraion disance along ramp A requires a greaer acceleraion. Saemen II is rue bu is no a complee explanaion, and saemen III is false. Insigh: We could also se in he equaion, ax and sole for a: a x From his expression we can see ha for he same final elociy, you will hae a smaller acceleraion when you accelerae oer he greaer disance x. 3. Picure he Problem: An airplane acceleraes uniformly along a sraigh runway. Sraegy: The aerage acceleraion is he change of he elociy diided by he elapsed ime. 56 mi/h.447 m/s Soluion: Diide he change in elociy by he ime: aa.98 m/s 35. s mi/h Insigh: The insananeous acceleraion migh ary from.98 m/s, bu we can calculae only aerage acceleraion from he ne change in elociy and ime elapsed. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No

12 Chaper : One-Dimensional Kinemaics 3. Picure he Problem: A runner acceleraes uniformly along a sraigh rack. Sraegy: The change in elociy is he aerage acceleraion muliplied by he elapsed ime. Soluion:. (a) Muliply he acceleraion by he ime: a. (b) The runner s speed will be he same a he end of he race as i is a = 5. s: m/s.9 m/s. s 3.8 m/s a m/s.9 m/s 5. s 9.9 m/s Insigh: World class spriners hae op speeds of oer m/s and can ge up o speed in much less han 5. s. 3. Picure he Problem: An airplane slows down uniformly along a sraigh runway as i raels oward he eas. Sraegy: The aerage acceleraion is he change of he elociy diided by he elapsed ime. Assume ha eas is in he posiie direcion. Soluion:. Diide he change in elociy by he ime: a 7.6 m/s 3. s f i a 5.43 m/s. We noe from he preious sep ha he acceleraion is negaie. Because eas is he posiie direcion, negaie acceleraion mus be oward he wes. Thus he je has an acceleraion of 5.43 m/s oward he wes. Insigh: In physics we almos neer alk abou deceleraion. Insead, we call i negaie acceleraion. 33. Picure he Problem: A car raels in a sraigh line due norh, eiher speeding up or slowing down, depending upon he direcion of he acceleraion. Sraegy: Use he definiion of acceleraion o deermine he final elociy oer he specified ime ineral. Le norh be he posiie direcion. Soluion:. (a) Calculae he elociy: a 3.6 m/s.3 m/s 7. s 3.8 m/s norh. (b) Calculae he elociy: a 3.6 m/s.5 m/s 7. s 5.4 m/s norh Insigh: In physics we almos neer alk abou deceleraion. Insead, we call i negaie acceleraion. In his problem souh is considered he negaie direcion, and in par (b) he car is slowing down or undergoing negaie acceleraion. 34. Picure he Problem: Following he moion specified in he elociyersus-ime graph, he moorcycle is speeding up, hen moing a consan speed, hen slowing down. Sraegy: Deermine he acceleraion from he slope of he graph. Soluion:. (a) Find he slope a A: a a. m/s m/s 5. s. (b) Find he slope of he graph a B: a a m/s. s. m/s 3. (c) Find he slope of he graph a C: a a 5. m/s.5 m/s. s Insigh: The acceleraion during segmen A is larger han he acceleraion during segmens B and C because he slope here has he greaes magniude. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No

13 Chaper : One-Dimensional Kinemaics 35. Picure he Problem: Following he moion specified in he elociyersus-ime graph, he person on horseback is speeding up, hen acceleraing a an een greaer rae, hen slowing down. Sraegy: We could deermine he acceleraion from he slope of he graph, and hen use he acceleraion and iniial elociy o deermine he displacemen. Alernaiely, we could use he iniial and final elociies in each segmen o deermine he aerage elociy and he ime elapsed o find he displacemen during each ineral. Soluion:. (a) Use he aerage elociy during ineral A o calculae he displacemen: x. m/s s m. (b) Calculae he displacemen during segmen B: x. 6. m/s 5. s m 3. (c) Calculae he displacemen during segmen C: x 6.. m/s s 4 m Insigh: There are ofen seeral ways o sole moion problems inoling consan acceleraion, some easier han ohers. 36. Picure he Problem: A horse raels in a sraigh line in he posiie direcion while acceleraing in he negaie direcion (slowing down). Sraegy: Use he definiion of acceleraion o deermine he ime elapsed for he specified change in elociy m/s Soluion: Calculae he ime ineral:. s a.8 m/s Insigh: An acceleraion of greaer magniude would decrease he horse s elociy in a shorer period of ime. 37. Picure he Problem: Your car raels in a sraigh line in he posiie direcion while acceleraing in he negaie direcion (slowing down). Sraegy: Use he consan acceleraion equaion of moion o deermine he ime elapsed for he specified change in elociy. Soluion:. (a) The ime required o come o a sop is he change in elociy diided by he acceleraion. In boh cases he final elociy is zero, so he change in elociy doubles when you double he iniial elociy. Therefore, he sopping ime will increase by a facor of wo when you double your driing speed.. (b) Calculae he sopping ime: 8 m/s 4. m/s a 4.3 s 3. (c) Calculae he sopping ime: 36 m/s 8.6 s a 4. m/s Insigh: Noice ha he deceleraion is reaed as a negaie acceleraion in his problem and elsewhere in he ex. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 3

14 Chaper : One-Dimensional Kinemaics 38. Picure he Problem: A rain raels in a sraigh line in he posiie direcion while acceleraing in he posiie direcion (speeding up). Sraegy: Firs find he acceleraion and hen deermine he final elociy. 4.7 m/s Soluion:. Use he definiion of acceleraion: a.94 m/s 5. s. Calculae he final speed of he second segmen, using he final speed from he firs segmen as he iniial speed: a 4.7 m/s.94 m/s 5. s 9.4 m/s Insigh: Anoher way o ackle his problem is o se up similar riangles on a elociy-ersus-ime graph. The answer would hen be calculaed as = (4.7 m/s) s / 5 s = 9.4 m/s. Try i! 39. Picure he Problem: A paricle raels in a sraigh line in he posiie direcion while acceleraing in he posiie direcion (speeding up). Sraegy: Use he consan acceleraion equaion of moion o find he iniial elociy. Soluion: Calculae : a 9.3 m/s 6.4 m/s.45 s 6.5 m/s Insigh: As expeced, he iniial elociy is less han he final elociy because he paricle is speeding up. 4. Picure he Problem: A je raels in a sraigh line oward he souh while acceleraing in he norherly direcion (slowing down). Sraegy: Use he relaionship beween acceleraion, elociy, and displacemen (Equaion -). The acceleraion should be negaie if we ake he direcion of he je s moion (o he souh) o be posiie. Soluion: Sole for he acceleraion: 7.4 m/s a x 949 m.69 m/s.69 m/s o he norh Insigh: The negaie acceleraion indicaes he je is slowing down during ha ime ineral. Noice ha Equaion - is a good choice for problems in which no ime informaion is gien or requesed. 4. Picure he Problem: Your car raels in a sraigh line oward he wes while acceleraing in he easerly direcion (slowing down). Sraegy: The aerage elociy is simply half he sum of he iniial and final elociies because he acceleraion is uniform. Soluion: Calculae half he sum of he elociies: a 8 m/s 9. m/s o he wes Insigh: The aerage elociy of any objec ha slows down and comes o a sop is jus half he iniial elociy. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 4

15 Chaper : One-Dimensional Kinemaics 4. Picure he Problem: A ball rolls down an inclined plane wih consan acceleraion. Sraegy: The ball sars a a posiie alue of is posiion x and mus herefore rael in he negaie direcion in order o reach he locaion x =. Soluion:. (a) No maer how fas he ball migh iniially moe in he posiie direcion, away from x =, a consan negaie acceleraion will eenually slow i down, bring i briefly o res, and speed i up back oward x =. Therefore, in cases 3 and 4, where a <, he ball will cerainly pass x =.. (b) I is possible for he iniial elociy o be so large in he negaie direcion ha a posiie acceleraion canno bring i o res before i passes x =. Therefore, in case where and a, i is possible ha he ball will pass x =, bu we need more informaion abou he relaie magniudes of and a in order o be cerain. 3. (c) Wheneer he iniial elociy is opposie in sign o he acceleraion, he ball will eenually come o res briefly and hen speed up in he direcion of he acceleraion. Therefore, in cases and 3 we know ha he ball will momenarily come o res. Insigh: If we suppose ha a = +4. m/s and ha x. m, we can deermine ha an iniial elociy of ax 4. m/s. m 8.83 m/s is he hreshold iniial elociy for he ball o reach he x = posiion. Wih ha iniial elociy he ball will come o res momenarily a x = before speeding up in he posiie direcion again. 43. Picure he Problem: A boa raels in a sraigh line wih consan posiie acceleraion. Sraegy: The aerage speed is simply half he sum of he iniial and final elociies because he acceleraion is uniform. Soluion:. (a) Calculae half he sum of he elociies:. (b) The disance raeled is he aerage 4.8 m/s.4 m/s a elociy muliplied by he ime elapsed: d a.4 m/s 4.77 s.5 m Insigh: The aerage elociy of any objec ha speeds up from res is jus half he final elociy. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 5

16 Chaper : One-Dimensional Kinemaics 44. Picure he Problem: The gien posiion funcion indicaes he car begins a a posiie posiion, bu is raeling in he negaie direcion and acceleraing in he negaie direcion. Sraegy: Compare he gien posiion as a funcion of ime wih he symbolic expression o deermine he iniial posiion, iniial elociy, and acceleraion of he car. Creae he x-ersus- plo using a spreadshee, or calculae indiidual alues by hand and skech he cure using graph paper. Finally, use he known x and informaion o deermine he disance raeled and he aerage elociy. Soluion:. (a) Compare he symbolic formula wih he gien equaion o find he iniial posiion:. Compare he symbolic formula wih he gien equaion o find he iniial elociy: 3. Compare he symbolic formula wih he gien equaion o find acceleraion: 4. (b) Use a spreadshee or similar program o creae he blue plo shown a righ. The aerage elociy of he car beween. and. s is equal o he slope of a sraigh line drawn from is posiion a =. s and ha a =. s as shown. 5 m 5. m/s m/s x x a x 5 m 5 m 5. m/s m/s x x a 5. m/s a m/s a m/s 5 m 5. m/s m/s x x a 5. (c) Because he car raels in a sraigh line and does no reerse direcion, he disance raeled equals he magniude of he displacemen: i f x 5 m 5. m/s s m/s s 5 m x 5 m 5. m/s. s m/s. s 35 m x x x 35 5 m 5 m disance x 5 m f i 6. (d) Find he aerage elociy from =. s o =. s: i f x 5 m 5. m/s. s m/s. s 35 m x 5 m 5. m/s. s m/s. s m x x x 35 m f i a f i.. s 35 m/s Insigh: Aerage speed and aerage elociy are always he same as long as he objec coninuously raels in he same direcion. If i reerses course or raels in wo (or hree) dimensions, he relaionship beween he wo is more complex, bu he disance raeled will always be greaer han or equal o he displacemen, so he aerage speed will always be greaer han or equal o he aerage elociy. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 6

17 Chaper : One-Dimensional Kinemaics 45. Picure he Problem: The gien posiion funcion indicaes he ball begins raeling in he posiie direcion bu is acceleraing in he negaie direcion. Sraegy: Compare he gien posiion as a funcion of ime wih he symbolic expression o deermine he iniial posiion, iniial elociy, and acceleraion of he ball. Creae he x-ersus- plo using a spreadshee, or calculae indiidual alues by hand and skech he cure using graph paper. Finally, use he known x and informaion o deermine he aerage elociy and he aerage speed. Soluion:. (a) Compare he symbolic formula wih he gien equaion o find he iniial posiion:. Compare he symbolic formula wih he gien equaion o find he iniial elociy: 3. Compare he symbolic formula wih he gien equaion o find acceleraion: 4. (b) Use a spreadshee or similar program o creae he blue plo shown a righ. The aerage speed of he ball beween. and. s is equal o he slope of a sraigh line drawn from is posiion a =. s and ha a =. s as shown. m 5. m/s m/s x x a x m m 5. m/s m/s x x a 5. m/s a m/s a m/s m 5. m/s m/s x x a 5. (c) Because he ball reerses direcion, he aerage elociy from = o =. s should be calculaed wih careful aenion o he signs: 6. (d) Because he ball does no reerse direcion beween =. s o =. s, he aerage speed is he magniude of he aerage elociy: i f a x m 5. m/s s m/s s m x m 5. m/s. s m/s. s 5. m x x x 5. m 5 m f i x 5. m. s 5. m/s i f x m 5. m/s. s m/s. s 5. m x x x 3 5. m x m 5. m/s. s m/s. s 3 m f i a a f i.. s 5 m/s 5 m/s Insigh: The insananeous speed is always he magniude of he insananeous elociy, bu he aerage speed is no always he magniude of he aerage elociy. For insance, in his problem he ball raels o +.65 m a =.5 s and hen o 5. m a =. s, a oal disance of 6.5 m, while is displacemen is 5. m. Hence is aerage speed is 6.5 m/s while is aerage elociy is 5. m/s oer he ime ineral beween = and =. s Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 7

18 Chaper : One-Dimensional Kinemaics 46. Picure he Problem: A cheeah runs in a sraigh line wih consan posiie acceleraion. Sraegy: The aerage elociy is simply half he sum of he iniial and final elociies because he acceleraion is uniform. The disance raeled is he aerage elociy muliplied by he ime elapsed. Soluion:. (a) Calculae half he sum of he elociies: 5. m/s.5 m/s a. Use he aerage elociy o find he disance: d a.5 m/s 6. s 77.8 m 3. (b) For a consan acceleraion he elociy aries linearly wih ime. Therefore we expec he elociy o be equal o.5 m/s afer half he ime (3. s) has elapsed. 4. (c) Calculae half he sum of he elociies:.5 m/s 6.5 m/s a, 5. Calculae half he sum of he elociies:.5 5. m/s 8.8 m/s a, 6. (d) Use he aerage elociy o find he disance: d a, 6.5 m/s 3. s 9.4 m 7. Use he aerage elociy o find he disance: d a, 8.8 m/s 3. s 58.5 m Insigh: The disance raeled is always he aerage elociy muliplied by he ime. This is a consequence of he definiion of aerage elociy. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 8

19 Chaper : One-Dimensional Kinemaics 47. Picure he Problem: Measuremens aken from a ideo of a sled raeling down an icy slope can be used o deermine he aerage speed and he acceleraion of he sled. Sraegy: Creae an x-ersus- plo using a spreadshee, or plo he indiidual alues by hand using graph paper. Use he known x and informaion o deermine he aerage elociy oer he specified ime ineral. Use eiher he spreadshee feaures or he equaion x x a o deermine he aerage acceleraion of he sled. Soluion:. (a) Use a spreadshee or similar program o creae he plo shown a righ. The aerage elociy of he sled beween.5 and.3 s is equal o he slope of a sraigh line drawn from is posiion a =.5 s and ha a =.3 s as shown.. (b) Draw a smooh cure o represen he sled daa, and use he smooh cure o deermine he approximae aerage speed: 3. Check your answer using he leas-squares regression from he spreadshee: 4. (c) Calculae he aerage acceleraion of he sled from he equaion, x x a : From he plo, x.5 m and x.5 m a x x i f i x.5.5 m. m/s.3.5 s f.93 m.4 m/s.5 s.443 m/s.5 s.74 m.93 m.4 m/s.3 s.443 m/s.3 s.37 m x x x m.9 m/s confirmed f i a f i.3.5 s x x a a x 8.8 m a.8 m/s.5 s 5. Check your answer using he leassquares regression from he spreadshee: a.443 m/s a.8486 m/s confirmed Insigh: Spreadshee sofware usually includes powerful ools like regression o analyze daa like hese. 48. Picure he Problem: A child slides down he hill in a sraigh line wih consan posiie acceleraion. Sraegy: Use he known acceleraion and imes o deermine he posiions of he child. In each case x and are zero. Soluion:. (a) Calculae her posiion: x x a.6 m/s. s.8 m. (b) Calculae her posiion: x x a.6 m/s. s 3. m 3. (c) Calculae her posiion: x x a.6 m/s 3. s 7. m Insigh: Her posiion aries wih he square of he ime because of her consan acceleraion. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 9

20 Chaper : One-Dimensional Kinemaics 49. Picure he Problem: Passengers on he Deonaor ride accelerae sraigh downward. Sraegy: Use he known iniial and final elociies and he elapsed ime o find he acceleraion. f i 45 mi/h.447 m/s Soluion: Calculae he acceleraion: a 9. m/s. s mi/h Insigh: The passenger s acceleraion is jus less han ha for a free-falling objec. Wha a hrill! 5. Picure he Problem: Jules Verne s Columbiad spaceship acceleraes from res down he barrel of he cannon. Sraegy: Employ he relaionship beween acceleraion, displacemen, and elociy (Equaion -) o find he acceleraion. Soluion: Calculae he acceleraion: yd/s 3 f/yd.35 m/f a.8 m/s x 7 f.35 m/f Insigh: An acceleraion his grea would ear he occupans of he spacecraf apar! Noice ha he equaion a x is a good choice for problems in which no ime informaion is gien or requesed Picure he Problem: An Escherichia coli bacerium acceleraes from res in he forward direcion. Sraegy: Employ he definiion of acceleraion o find he ime elapsed, and he relaionship beween acceleraion, displacemen, and elociy (Equaion -) o find he disance raeled. Soluion:. (a) Calculae he ime o accelerae:. (b) Calculae he displacemen: m/s 56 m/s a.77 s m/s x.46 m a 56 m/s Insigh: The acceleraions are iny bu so are he baceria! The aerage speed here is abou 3 body lenghs per second if each bacerium were µm long. If his were a human ha would be 6 m/s or 3 mi/h, much faser han we can swim! 5. Picure he Problem: Two cars are raeling in opposie direcions. Sraegy: Wrie he equaions of moion based upon Equaion -, and se hem equal o each oher o find he ime a which he wo cars pass each oher. Soluion:. (a) Wrie an equaion for he posiion of car, which is raeling eas and speeding up. Le eas be he posiie direcion:. m/s.5 m/s x x a x,,. Wrie an equaion for he posiion of car, which is raeling wes bu slowing down, which means i is acceleraing oward he eas: 3. (b) Se x x x x a x m 3. m/s 3. m/s,, and sole for :. m/s.5 m/s m 3. m/s.6 m/s , 4 s. s.7 Insigh: We chose smaller of he wo roos, which corresponds o he firs ime he cars pass each oher. The larger acceleraion of car means ha i ll come o res, speed up in he posiie direcion, and oerake car a = 4 s. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No

21 Chaper : One-Dimensional Kinemaics 53. Picure he Problem: A meeorie acceleraes from a high speed o res afer impacing he car. Sraegy: Employ he relaionship beween acceleraion, displacemen, and elociy (Equaion -) o find he acceleraion. Soluion: Calculae he acceleraion: 3 m/s 4 a 3.8 m/s x. m Insigh: The high siffness of seel is responsible for he remendous (negaie) acceleraion of he meeorie. 54. Picure he Problem: A rocke acceleraes sraigh upward. Sraegy: Employ he relaionship beween acceleraion, displacemen, and ime (Equaion -) o find he acceleraion. Because he rocke was a res before blas off, he iniial elociy is zero, and so is he iniial posiion x. Once he acceleraion is known, we can use he consan acceleraion equaion (Equaion -7) o find he speed. Soluion:. (a) Wrie ou he posiion s. ime equaion:. Le x and sole for acceleraion: x x a x 9 m a.8 s 3 m/s upward 3. (b) Calculae he final speed: a 3. m/s.8 s 65 m/s Insigh: The posiion s. ime equaion simplifies considerably if he iniial posiion and he iniial elociy are zero. 55. Picure he Problem: You drie in a sraigh line and hen slow down o a sop. Sraegy: Employ he relaionship beween acceleraion, displacemen, and elociy (Equaion -) o find he displacemen. Equaion - is a good choice for problems in which no ime informaion is gien or requesed. In his case he acceleraion is negaie because he car is slowing down. Soluion:. (a) Calculae he displacemen:. m/s x m a a a 3.5 m/s. (b) Because elociy is proporional o he square roo of displacemen, cuing he disance in half will reduce he elociy by, no. Therefore he speed will be greaer han 6. m/s afer raeling half he disance. 3. (c) Calculae he speed afer half he displacemen:. m/s a x a 8.49 m/s a Insigh: For consan acceleraion, he elociy changes linearly wih ime, bu nonlinearly wih disance. 56. Picure he Problem: You drie in a sraigh line and hen slow down o a sop. Sraegy: Use he consan acceleraion equaion of moion (Equaion -7) o find he ime. Once he ime is known, we can use he same equaion o find he speed. In his case, he acceleraion is negaie because he car is slowing down. 6 m/s Soluion:. (a) Calculae he sopping ime: 5. s a 3. m/s. (b) Because he elociy aries linearly wih ime for consan acceleraion, he elociy will be half he iniial elociy when you hae braked for half he ime. Therefore he speed afer braking.5 s will be equal o 8. m/s. 3. (c) Calculae he speed afer half he ime: a 6 m/s 3. m/s.5 s 8. m/s Insigh: For consan acceleraion, he elociy changes linearly wih ime, bu nonlinearly wih disance. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No

22 Chaper : One-Dimensional Kinemaics 57. Picure he Problem: A chameleon s ongue acceleraes in a sraigh line unil i is exended o is full lengh. Sraegy: Employ he relaionship beween acceleraion, displacemen, and ime (Equaion -) o find he acceleraion. Le he iniial elociy and he iniial posiion x of he ongue each be zero. Soluion:. (a) Le x and calculae he acceleraion: x.6 m a 3 m/s. s. (b) Because he displacemen aries wih he square of he ime for consan acceleraion, he displacemen will be less han half is final alue when half he ime has elapsed. Mos of he displacemen occurs when he ongue's speed is greaes, lae in he ime ineral. Therefore we expec he ongue o hae exended less han 8. cm afer.5 s. 3. (c) Calculae he posiion of he ongue afer half he ime: x a 3 m/s.5 s 4. cm Insigh: For consan acceleraion, he displacemen changes nonlinearly wih boh ime and elociy. Noice ha he acceleraion of he chameleon s ongue is oer hree imes he acceleraion of graiy! 58. Picure he Problem: Daid Purley raels in a sraigh line, slowing down a a uniform rae unil coming o res. Sraegy: Use he ime-free relaionship beween displacemen, elociy, and acceleraion (Equaion -) o find he acceleraion. Soluion: Calculae he acceleraion:.78 m/s 73 km/h km/h a x.66 m.g a 8 m/s 8g a 8g 9.8 m/s Insigh: Mr. Purley was lucky o escape deah when experiencing an acceleraion his large! We ll learn in Chaper 5 ha a large acceleraion implies a large force, which in his case mus hae been applied o his body in jus he righ way o produce a non-lehal injury. 59. Picure he Problem: A boa slows down a a uniform rae as i coass in a sraigh line. Sraegy: Because he iniial and final elociies are known, he ime can be deermined from he aerage elociy and he disance raeled. Then use he consan acceleraion equaion of moion (Equaion -7) o find he acceleraion and he ime-free equaion (Equaion -) o find he elociy afer he boa had coased half he disance. Soluion:. (a) Use he displacemen and he x m 5.7 s aerage elociy o find he ime elapsed:.6.6 m/s. (b) Apply he definiion of acceleraion: 3. (c) Calculae he elociy afer coasing 6. m using he ime-free equaion of moion:.6.6 m/s a 5.7 s sign means opposie he direcion of moion..75 m/s where he negaie a x.6 m/s.75 m/s 6. m. m/s Insigh: For consan acceleraion, he elociy changes linearly wih ime bu nonlinearly wih disance. Tha is why he.-m/s elociy afer coasing 6. m is greaer han he.-m/s aerage speed he boa has oer he enire -m disance i coased. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No

23 Chaper : One-Dimensional Kinemaics 6. Picure he Problem: A model rocke acceleraes sraigh upward a a consan rae. Sraegy: Because he iniial and final elociies are known, he ime can be deermined from he aerage elociy and he disance raeled. The consan acceleraion equaion of moion (Equaion -7) can hen be used o find he acceleraion. Once ha is known, he posiion of he rocke as a funcion of ime is gien by Equaion -, and he elociy as a funcion of ime is gien by Equaion -7. Soluion:. (a) Use he displacemen and he x 4. m.33 s.3 s aerage elociy o find he ime elapsed: 6. m/s. (b) Apply he definiion of acceleraion: 6. m/s a.33 s 8 m/s 3. (c) Find he rocke s heigh, assuming x 4. Find he elociy of he rocke, assuming : x a 8 m/s. s.4 m : a 8 m/s. s 8. m/s Insigh: Model rockes accelerae a ery large raes, bu only for a ery shor ime. Sill, een inexpensie sarer rockes can reach 5 f in aliude and can be grea fun o build and launch! 6. Picure he Problem: The infamous chicken dashes oward home plae while playing baseball, and hen slides along a sraigh line and comes o res. Sraegy: Because he iniial and final elociies and he ime elapsed are known, he acceleraion can be deermined from he consan acceleraion equaion of moion (Equaion -7). The disance raeled can be found from he aerage elociy and he ime elapsed (Equaion -). Soluion:. (a) Calculae he acceleraion:. (b) Use he aerage elociy and ime 5.7 m/s a. s means opposie he direcion of moion, or oward hird base. 4.8 m/s, where he negaie sign o find he disance he chicken slides: x 5.7 m/s. s 3.4 m Insigh: If he dir had acceleraed he chicken a a lesser rae, he chicken would hae had nonzero speed as i crossed home plae. A larger magniude acceleraion would sop he chicken before reaching he plae, and i would be ou! Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 3

24 Chaper : One-Dimensional Kinemaics 6. Picure he Problem: The disance-ersus-ime plo a righ shows how a bicyclis can oerake his friend by pedaling a consan acceleraion. Sraegy: To find he ime elapsed when he wo bicycliss mee, we mus se he consan elociy equaion of moion of he friend (Equaion -8) equal o he consan acceleraion equaion of moion (Equaion -) of he bicyclis. Once he ime is known, he displacemen and elociy of he bicyclis can be deermined from Eqs. - and -7, respeciely. Soluion:. (a) Se he wo equaions of moion equal o each oher. For he friend, use Equaion -8 wih x and for he bicyclis, use Equaion - wih x and :. Sole he wo equaions for by rearranging hem ino a quadraic expression: 3. Now use he quadraic formula: x friend friend x bicyclis a bicyclis friend a bicyclis 4 4 friend abicyclis.4 m/s m/s ,.64 s 4. We choose he larger roo because he ime mus be greaer han. s, he ime a which he bicyclis began pursuing his friend. The bicyclis will oerake his friend 6.3 s afer his friend passes him. 5. (b) Use he known ime o find he posiion: xfriend 3.5 m/s6.3 s m x a 6. (c) Use Equaion -7 o find bicyclis. Keep in mind ha and ha he bicyclis doesn begin acceleraing unil wo seconds hae elapsed: bicyclis bicyclis.4 m/s 4.3 s m a.4 m/s 6.3. s m/s Insigh: Een a smaller acceleraion would allow he bicyclis o cach up o he friend, because he speed is always increasing for any nonzero acceleraion. Hence he bicyclis s speed would eenually exceed he friend s speed and he wo would mee some ime afer ha. bicyclis Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 4

25 Chaper : One-Dimensional Kinemaics 63. Picure he Problem: The elociy-ersus-ime plo a righ indicaes he elociy of a car as i acceleraes in he forward direcion, mainains a consan speed, and hen rapidly slows down o a sop. Sraegy: The disance raeled by he car is equal o he area under he elociy-ersus-ime plo. Because he disance raeled is known o be m, we can use ha fac o deermine he unknown speed V. Once we know he elociy as a funcion of ime we can answer any oher quesion abou is moion during he ime ineral. Soluion:. (a) Deermine he area under he cure by adding he area of he riangle from o 4 s, he recangle from 4 o 6 s, and he riangle from 6 o 8 s. 4 s 6 4 s 8 6 s 5 s x V V V V. Se x equal o m and sole for V: x V V 3. Now find he area of he riangle from o 4 s: 5. s m / 5. m/s 4.4 m/s x 4 s 4.4 m/s 8.8 m 4. (b) Find he area of he riangle from 6 o 8 s: x 8 6 s 4.4 m/s 4.4 m 5. (c) We found he unknown speed in sep : V 4.4 m/s Insigh: The elociy-ersus-ime graph is a rich source of informaion. Besides elociy and ime informaion, you can deermine acceleraion from he slope of he graph and disance raeled from he area under he graph. 64. Picure he Problem: The elociy-ersus-ime plos of he car and he ruck are shown a righ. The car begins wih a posiie posiion and a negaie elociy, so i mus be represened by he lower line. The ruck begins wih a negaie posiion and a posiie elociy, so i is represened by he upper line. Sraegy: The disances raeled by he car and he ruck are equal o he areas under heir elociy-ersus-ime plos. We can deermine he disances raeled from he plos and use he known iniial posiions o find he final posiions and he final separaion. Soluion:. Find he final posiion of he ruck. The ruck s displacemen x ruck is he area under is s. graph:. Find he final posiion of he car. The car s displacemen xcar is he area under is s. graph: ruck,ruck ruck x x x 35 m.5 s m/s.5 m car,car car x x x 5 m 3.5 s 5 m/s.5 m 3. Now find he separaion: x x car ruck.5 m.5 m.3 m Insigh: The elociy-ersus-ime graph is a rich source of informaion. Besides elociy and ime informaion, you can deermine acceleraion from he slope of he graph, and disance raeled from he area under he graph. In his case, we can see he acceleraion of he car (4.9 m/s ) has a greaer magniude han he acceleraion of he ruck ( 4. m/s ). Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 5

26 Chaper : One-Dimensional Kinemaics 65. Picure he Problem: Penguins slide down hree differen fricionless ramps, A, B, and C. The disance along each ramp and he aerage sliding imes are recorded. Sraegy: Use he relaionship beween disance, acceleraion, and ime (Equaion -) o deermine he acceleraions of he penguins. Then use a gsin o deermine he angle of inclinaion for each ramp. Soluion:. (a) Wrie an expression for he acceleraion, assuming ha x :. Calculae he acceleraion of penguins ha slide along ramp A: 3. Calculae he acceleraion of penguins ha slide along ramp B: 4. Calculae he acceleraion of penguins ha slide along ramp C: 5. (b) Wrie an expression for he angle of incline : 6. Calculae he angle of incline for ramp A: 7. Calculae he angle of incline for ramp B: x a a x a a x 4.9 m.7 m/s A A A.9 s x.96 m 3.36 m/s B B B.8 s x.8 m a 4.9 m/s C C C.663 s a a gsin sin g.7 m/s A sin. 9.8 m/s 3.36 m/s B sin. 9.8 m/s 6. Calculae he angle of incline for ramp C: 4.9 m/s C sin m/s Insigh: Along a seeper ramp here is a greaer componen of graiaional force ha is parallel o he ramp, resuling in a larger acceleraion. 66. Picure he Problem: Two balls are each hrown wih speed from he same iniial heigh. Ball is hrown sraigh upward and ball is hrown sraigh downward. Sraegy: Use he known se of kinemaic equaions ha describe moion wih consan acceleraion o deermine he relaie speeds of balls and when hey hi he ground. Soluion:. Sole Equaion - for, assuming he ball is hrown upward wih elociy :. Sole Equaion - for, assuming he ball is hrown downward wih elociy : g x g x g x g x 3. By comparing he wo expressions for aboe we can conclude ha he bes answer is B. The speed of ball is equal o he speed of ball. Insigh: In a laer chaper we ll come o he same conclusion from an undersanding of he conseraion of mechanical energy. The balls hae he same speed jus before hey land because hey boh hae he same downward speed when hey are a he leel of he roof. Ball simply sars off wih he speed downward. Ball raels upward iniially, bu when i reurns o he leel of he roof i is moing downward wih he speed, jus like ball. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 6

27 Chaper : One-Dimensional Kinemaics 67. Picure he Problem: A cliff dier drops from res, picking up speed wih he acceleraion of graiy. Sraegy: Coner he speed in mi/h o m/s, and hen sole Equaion -7 for he ime required for he cliff dier o reach ha speed when she acceleraes a 9.8 m/s. Soluion:. Coner o unis of m/s : mi m/s m/s h.4 mi/h. Sole Equaion -7 for he ime: g g 6.8 m/s g 9.8 m/s.7 s Insigh: This is significanly less han he 3.5 s required for a powerful car o achiee 6. mi/h from res. 68. Picure he Problem: A juggler hrows a ball sraigh upward and laer caches i a he same heigh i was hrown. Sraegy: Use he known acceleraion of graiy (9.8 m/s ) and Equaion -7 o find he iniial speed of he ball, assuming by symmery ha he final speed is he same as he iniial speed, excep he final elociy is downward (negaie). Soluion: Sole Equaion -7 for he iniial elociy: g g g 9.8 m/s 3. s 6 m/s Insigh: This speed is equialen o 35 mi/h, a reasonably easy hrow for an accomplished juggler. 69. Picure he Problem: Snowboarder Shaun Whie soars sraigh upward a disance 6.4 m aboe he rim of a half-pipe. Sraegy: Because he heigh of he snowboard and rider is known, he ime-free equaion of moion (elociy in erms of displacemen, Equaion -) can be used o find he akeoff speed. Soluion: Sole he ime-free equaion of moion for g x : 9.8 m/s 6.4 m m/s Insigh: Tha speed is abou 5 mi/h sraigh upward! Olympic snowboarders mus be ery ahleic as well as acrobaic o perform he feas we winess during he Games. 7. Picure he Problem: A gull drops a clam shell, which falls from res sraigh down under he influence of graiy. Sraegy: Because he disance of he fall is known, use he ime-free equaion of moion (elociy in erms of displacemen, Equaion -) o find he landing speed. Soluion: Sole he ime-free equaion of moion for. Le and le downward be he posiie direcion. g x 9.8 m/s 7 m 8 m/s Insigh: Tha speed (abou 4 mi/h) is sufficien o shaer he shell and proide a asy meal! Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 7

28 Chaper : One-Dimensional Kinemaics 7. Picure he Problem: A olcano launches a laa bomb sraigh upward. I slows down under he influence of graiy, coming o res momenarily before falling downward. Sraegy: Because he acceleraion of graiy is known, he consan acceleraion equaion of moion (elociy as a funcion of ime, Equaion -7) can be used o find he speed and elociy as a funcion of ime. Le upward be he posiie direcion. Soluion:. (a) Apply Equaion -7 g direcly wih a = g: 8 m/s 9.8 m/s. s 8.4 m/s. (b) Apply Equaion -7 direcly wih a = g: g 8 m/s 9.8 m/s 3. s.4 m/s 3. The posiie sign for he elociy in par (a) indicaes ha he laa bomb is raeling upward, and he negaie sign for par (b) means i is raeling downward. Insigh: We can see he laa bomb mus hae reached is peak beween. and 3. seconds. In fac, i reached i a a 8 m/s 9.8 m/s.85 s. 7. Picure he Problem: Volcanic maerial on Io raels sraigh upward, slowing down under he influence of graiy unil i momenarily comes o res a is maximum aliude. Sraegy: Because he maximum aliude is known, use he ime-free equaion of moion (elociy in erms of displacemen, Equaion -) o find he iniial elociy. Le upward be he posiie direcion, so ha a =.8 m/s. Soluion: Sole he ime-free equaion of moion for : a x 5.8 m/s 3. m 4 m/s.4 km/s Insigh: On Earh ha speed would only hurl he maerial o an aliude of 55 km, as opposed o 3 km on Io. Sill, ha s a ery impressie iniial elociy! I is equialen o he muzzle elociy of a bulle, and is.5 imes he speed of sound on Earh. 73. Picure he Problem: A ruler falls sraigh down under he influence of graiy. Sraegy: Because he acceleraion and iniial elociy (zero) of he ruler are known, use he posiion as a funcion of ime equaion of moion (Equaion -) o find he ime. Soluion: Sole Equaion - for. Le and le downward be he posiie direcion. x.5 m. s g 9.8 m/s Insigh: This is a ery good reacion ime, abou half he aerage human reacion ime of. s. 74. Picure he Problem: A hammer drops sraigh downward and passes by wo windows of equal heigh. Sraegy: Use he definiion of acceleraion ogeher wih he knowledge ha a falling hammer undergoes consan acceleraion o answer he concepual quesion. Soluion:. (a) The acceleraion of he hammer is a consan hroughou is fligh (neglecing air fricion) so is speed increases by he same amoun for each equialen ime ineral. Howeer, i passes by he second window in a smaller amoun of ime han i ook o pass by he firs window because is speed has increased. We conclude ha increase in speed of he hammer as i drops pas window is greaer han he increase in speed as i drops pas window.. (b) The bes explanaion (see he discussion aboe) is III. The hammer spends more ime dropping pas window. Saemen is false because acceleraion is independen of speed, and saemen II is false because acceleraion is rae of change of speed per ime no disance. Insigh: If he hammer were hrown upward, is speed decrease as i passes window would be less han he decrease in is speed as i passes window, again because i is raeling slower as i passes window. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 8

29 Chaper : One-Dimensional Kinemaics 75. Picure he Problem: A hammer drops sraigh downward and passes by wo windows of equal heigh. Sraegy: The elociy-ersus-ime graph conains wo pieces of informaion: he slope of he graph is he acceleraion, and he area under he graph is he disance raeled. Use his knowledge o answer he concepual quesion. Soluion:. (a) The wo windows hae he same heigh, so he hammer raels he same disance as i passes each window. We conclude ha he area of he shaded region corresponding o window is equal o he area of he shaded region corresponding o window.. (b) The bes explanaion (see he discussion aboe) is II. The windows are equally all. Saemen I is rue, bu no relean, and saemen III is rue, bu no relean. Insigh: If he hammer were hrown upward, he elociy-ersus-ime graph would hae a negaie slope, bu he shaded areas corresponding o each window would sill be equal, wih he all and narrow shaded area for window on he lef (because he hammer passes i firs) and he shor and wide shaded area for window on he righ. 76. Picure he Problem: Two balls are hrown upward wih he same iniial speed bu a differen imes. The second ball is hrown a he insan he firs ball has reached he peak of is fligh. Sraegy: The aerage speed of he ball is smaller a aliudes aboe h, so ha i spends a greaer fracion of ime in ha region han i does a aliudes below h. Use his insigh o answer he concepual quesion. Soluion: The second ball will reach h on is way up sooner han he firs ball will reach h on is way down because he speed of each ball is greaer a low aliudes han a high aliudes. We conclude ha he wo balls pass a an aliude ha is aboe h. Insigh: A careful analysis reeals ha he wo balls will pass each oher a aliude of 3 h Picure he Problem: Seeral swimmers fall sraigh down from a bridge ino he Snohomish Rier. Sraegy: The iniial elociies of he swimmers are zero because hey sep off he bridge raher han jump up or die downward. Use he equaion of moion for posiion as a funcion of ime and acceleraion, realizing ha he acceleraion in each case is 9.8 m/s. Se x and le downward be he posiie direcion for simpliciy. The known acceleraion can be used o find elociy as a funcion of ime for par (b). Finally, he same equaion of moion for par (a) can be soled for ime in order o answer par (c). Soluion:. (a) Calculae he fall disance:. (b) Calculae he final speed if : 3. (c) Calculae he fall ime for wice he disance: x x a. m 9.8 m/s.5 s x m a 9.8 m/s.5 s 5 m/s x m. s a 9.8 m/s Insigh: The ime in par (c) doesn double because i depends upon he square roo of he disance he swimmer falls. If you wan o double he fall ime you mus quadruple he heigh of he bridge! Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 9

30 Chaper : One-Dimensional Kinemaics 78. Picure he Problem: Waer in he highes founain is projeced wih a large upward elociy, rises sraigh upward, and momenarily comes o res before falling sraigh back down again. Sraegy: By analyzing he ime-free equaion of moion (Equaion -) wih (because he waer briefly comes o res a he op of is rajecory), we can see ha he iniial elociy increases wih he square roo of he founain heigh. The known founain heigh and acceleraion of graiy can also be used o deermine he ime i akes for he waer o reach he peak using he posiion as a funcion of ime (Equaion -). Soluion:. (a) Calculae assuming he gx waer comes o res ( ) a he op: gx 9.8 m/s 56 f.35 m/f 58 m/s. (b) Calculae he ime required for he waer o reach he op of he founain: x 56 f.35 m/f 5.9 s a 9.8 m/s Insigh: The speed of 58 m/s corresponds o 3 mi/h. The founain is produced by a world-class waer pump! 79. Picure he Problem: A baskeball bounces sraigh up, momenarily comes o res, and hen falls sraigh back down. Sraegy: If air fricion is negleced, he ime i akes he ball o fall is he same as he ime i akes he ball o rise. Therefore, he maximum heigh of he ball is also he disance a ball will fall for.6 s. Use he equaion of moion for posiion as a funcion of ime and acceleraion, realizing ha he acceleraion in each case is 9.8 m/s. Se x and le downward be he posiie direcion for simpliciy. Soluion: Calculae he maximum heigh: x x a. m 9.8 m/s.6 s.6 m 3 m Insigh: The.6-m heigh corresponds o 4 f. The ball mus hae rebounded from he floor wih a speed of 5.7 m/s or 35 mi/h. The player was prey angry! 8. Picure he Problem: A baseball gloe rises sraigh up, momenarily comes o res, and hen falls sraigh back down. Sraegy: The gloe will land wih he same speed i was released, neglecing any air fricion, so he final elociy 6.5 m/s. We can use he equaion of moion for elociy as a funcion of ime (Eq. -7) o find he ime of fligh. Soluion:. (a) Calculae he oal ime of fligh. (b) Calculae he ime o reach maximum heigh: m/s a 9.8 m/s 6.5 m/s 9.8 m/s a.66 s Insigh: Throwing he gloe upward wih wice he speed will double he ime of fligh, bu he maximum heigh aained by he gloe (.5 m for a 6.5 m/s iniial speed) will increase by only a facor of..3 s Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 3

31 Chaper : One-Dimensional Kinemaics 8. Picure he Problem: Two balls fall sraigh down under he influence of graiy. The firs ball falls from res bu he second ball is gien an iniial downward elociy. Sraegy: Because he fall disance is known in each case, use he elociy in erms of displacemen equaion of moion (Equaion -) o predic he final elociy. Le downward be he posiie direcion for simpliciy. Soluion:. (a) The speed increases linearly wih ime bu nonlinearly wih disance. Because he firs ball has a lower iniial elociy and hence a lower aerage elociy, i spends more ime in he air. The firs (dropped) ball will herefore experience a larger increase in speed.. (b) Firs ball: Sole Eq. - for, seing : g x 9.8 m/s 3.5 m 4.5 m/s 3. Second ball: Sole Eq. - for : g x. m/s 9.8 m/s 3.5 m 6.9 m/s 4. Compare he alues: 4.5 m/s 4.5 m/s for he firs ball and 6.9. m/s 5.7 m/s for he second ball. Insigh: The second ball is cerainly going faser, bu is change in speed is less han he firs ball. 8. Picure he Problem: An arrow rises sraigh upward, slowing down due o he acceleraion of graiy. Sraegy: Because he posiion, ime, and acceleraion are all known, we can use he equaion of moion for posiion as a funcion of ime (Equaion -) o find he iniial elociy. The same equaion could be used o find he ime required o rise o a heigh of 5. m aboe is launch poin. Le he launch posiion be x and le upward be he posiie direcion. Soluion:. (a) Calculae from a rearrangemen of Equaion -: x 3. m a 9.8 m/s. s. s. (b) Sole Equaion - wih x = 5. m: 5. m 4.8 m/s 9.8 m/s 4.95 m/s 4.8 m/s 5. m 4.8 m/s 3. Now use he quadraic formula: b b 4ac a s, 4.36 s Insigh: The second roo of he soluion o par (b) corresponds o he ime when he arrow, afer rising o is maximum heigh, falls back o a posiion 5. m aboe he launch poin. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 3

32 Chaper : One-Dimensional Kinemaics 83. Picure he Problem: A book acceleraes sraigh downward and his he floor of an eleaor ha is descending a consan speed. Sraegy: The consan speed moion of he eleaor does no affec he acceleraion of he book. From he perspecie of an obserer ouside he eleaor, boh he book and he floor hae an iniial downward elociy of 3. m/s. Therefore, from your perspecie he moion of he book is no differen han if he eleaor were a res. Sole he posiion as a funcion of ime and acceleraion equaion (Equaion -) for, seing and leing downward be he posiie direcion. Then use elociy as a funcion of ime (Equaion -7) o find he speed of he book when i lands. Soluion:. (a) Sole Equaion - for, seing x : x. m.49 s g 9.8 m/s. (b) Apply Equaion -7 o find : g 9.8 m/s.49 s 4.8 m/s Insigh: The speed in par (b) is relaie o you. Relaie o he ground he elociy of he book is = 7.8 m/s in he downward direcion. 84. Picure he Problem: A camera has an iniial downward elociy of.3 m/s when i is dropped from a ho-air balloon. The camera acceleraes sraigh downward before sriking he ground. Sraegy: One way o sole his problem is o use he quadraic formula o find from he posiion as a funcion of ime and acceleraion equaion (Equaion -). Then he definiion of acceleraion can be used o find he final elociy. Here s anoher way: Find he final elociy from he ime-free equaion of moion (Equaion -) and use he relaionship beween aerage elociy, posiion, and ime (Equaion -) o find he ime. We ll herefore be soling his problem backwards, finding he answer o (b) firs and hen (a). Le upward be he posiie direcion, so ha.3 m/s and x x x 4 m 4 m. Soluion:. (a) Sole Equaion - for : g x.3 m/s 9.8 m/s 4 m 8 m/s. Sole Equaion - for : x 4 m 8.3 m/s.7 s 3. (b) We found in sep : 8 m/s Insigh: There is ofen more han one way o approach consan acceleraion problems, some easier han ohers. In his case our sraegy allowed us o aoid using he quadraic formula o find. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 3

33 Chaper : One-Dimensional Kinemaics 85. Picure he Problem: A model rocke rises sraigh upward, acceleraing oer a disance of 9 m and hen slowing down and coming o res a some aliude higher han 9 m. Sraegy: Use he gien acceleraion and disance and he ime-free equaion of moion (Equaion -) o find he elociy of he rocke a he end of is acceleraion phase, when is aliude is 9 m. Use ha as he iniial elociy of he free-fall sage in order o find he maximum aliude (Equaion - again). Then apply Equaion - a hird ime o find he elociy of he rocke when i reurns o he ground. The gien and calculaed posiions a arious sages of he fligh can hen be used o find he elapsed ime in each sage and he oal ime of fligh. Soluion:. (a) Find he elociy a he end g x of he boos phase using Equaion -:. Find he heigh change during he boos phase using Equaion - and a final speed of zero: boos m/s 9 m 6.4 m/s boos boos gxboos xboos g 3. Now find he oerall maximum heigh: hmax h xboos 9 m g boos 6.4 m/s 9 m 9 36 m 65 m 9.8 m/s 4. (b) Apply Equaion - once again beween he end of he boos phase and gx boos he poin where i his he ground: gx 6.4 m/s 9.8 m/s 9 m boos 36 m/s 5. (c) Firs find he duraion of he boos phase. Use he known posiions and Equaion -: 6. Now find he ime for he rocke o reach is maximum aliude from he end of he boos phase: x 9 m. s boos boos boos 6.4 m/s x 36 m.7 s boos up boos op 6.4 m/s 7. Now find he ime for he rocke o down down fall back o he ground: 36 m/s x op ground 65 m 8. Sum he imes o find he ime of fligh: oal boos up down s 8.5 s Insigh: Noice how knowledge of he iniial and final elociies in each sage, and he disance raeled in each sage, allowed he calculaion of he elapsed imes using he relaiely simple Equaion -, as opposed o he quadraic Equaion -. Learning o recognize he easies roue o he answer is an imporan skill o obain. 3.6 s Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 33

34 Chaper : One-Dimensional Kinemaics 86. Picure he Problem: The erical posiion-ersus-ime plo of a flying squirrel is shown a righ. The squirrel sars from res and drops a disance of x = 4. m in =. s. Sraegy: Because he squirrel sars from res and lands a x =, he equaion of moion for posiion as a funcion of ime (Equaion -) can be soled o find he acceleraion a. We expec he acceleraion o be negaie because he squirrel begins from a posiie heigh and ends a a zero heigh. The negaie slope of he plo also indicaes he elociy of he squirrel is downward and increasing in magniude. Soluion:. Sole he posiion as a funcion of ime equaion for he acceleraion a:. Calculae he squirrel s acceleraion: x x a x x a a x 4. m a 6.6 m/s. s Insigh: If he squirrel did no hae a paagium o slow is descen, is acceleraion would be close o 9.8 m/s. 87. Picure he Problem: The heigh-ersus-ime plo of a high sriker plug is shown a righ. The plug sars wih a high elociy and begins o slow down when i his he bell afer.6 s. Sraegy: The aerage elociy is he disance raeled by he plug diided by he ime (Equaion -). Assuming here is no fricion, he ime and free fall acceleraion ( 9.8 m/s ) can be used o find he change in elociy (Equaion -7). The iniial elociy can hen be deermined from he change in elociy and aerage elociies by combining Equaions -7 and -9. Soluion:. (a) Find he aerage elociy using Equaion -: x x 4. m 6.7 m/s.6 s a. (b) Find he change in elociy using Eq. -7: a 3. (c) Sole Equaion -7 for : 9.8 m/s.6 s 5.9 m/s a a 4. Sole Equaion -9 for : 5. Subsiue he expression for ino he equaion for : a 6. Now sole ha expression for : a a a a 6.7 m/s 9.8 m/s.6 s a 9.6 m/s Insigh: There are seeral oher ways of finding hese speeds, including graphical analysis. Try measuring he slope of he graph a he launch poin and he poin a which he plug his he bell o find he iniial and final speeds. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 34

35 Chaper : One-Dimensional Kinemaics 88. Picure he Problem: Chesnu A is dropped from res. When i has fallen.5 m, chesnu B is hrown downward wih an iniial speed B,. Boh nus land a he same ime afer falling. m. Sraegy: Firs find he ime i akes for nu A o fall.5 m using he equaion of moion for posiion as a funcion of ime and acceleraion (Equaion -). Also find he ime required for nu A o fall he enire. m. Subrac he firs ime from he second o find he ime ineral oer which nu B mus reach he ground in order o land a he same insan as nu A. Then use Equaion - again o find he iniial elociy B, required in order for nu B o reach he ground in ha ime. Soluion:. Find he ime i akes for nu A o fall.5 m by soling Equaion - for and seing A, =. Nu B hrown B, =?.5 m A, Nu A branch ground x.5 m.74 s g 9.8 m/s. m Boh land. Find he ime i akes for nu A o fall he enire. m: A,oal x. m.48 s g 9.8 m/s 3. Subrac he imes o find he ime oer B,oal A,oal A, s.74 s which nu B mus reach he ground: 4. Sole Equaion - for B, : B, B, x g B,oal B,oal.5 m/s m/s. m 9.8 m/s.74 s.74 s Insigh: In his problem we kep an addiional significan figure han is warraned in seps,, and 3 in an aemp o ge a more accurae answer in sep 4. Howeer, if you choose no o do so, differences in rounding will lead o an answer of m/s. The specified.5 m drop disance for nu A limis he answer o wo significan digis, and because he answer is righ beween and m/s, i could correcly go eiher way. 89. Picure he Problem: A rock acceleraes from res sraigh downward and lands on he surface of he Moon. Sraegy: Employ he relaionship beween acceleraion, displacemen, and elociy (Equaion -) o find he final elociy. Soluion: Sole Equaion - for elociy : a x.6 m/s.5 m. m/s Insigh: On Earh he rock would be raeling 4.95 m/s, bu he weaker graiy on he Moon acceleraes he rock only abou one-sixh as much as would he Earh s graiy. 9. Picure he Problem: An eleaor in he Taipei skyscraper acceleraes o is maximum speed. Sraegy: Because ime informaion is neiher gien nor requesed, he ime-free equaion for elociy in erms of displacemen (Equaion -) is he bes choice for finding he displacemen. Soluion: Sole Equaion - for displacemen: 6 m/s x 6 m a. m/s Insigh: The obseraory eleaors in Taipei were he world s fases when insalled, whisking passengers from he fifh floor o he 89 h -floor obseraory, a disance of 369 m, in only 37 seconds. The 6.83 m/s maximum speed is equialen o 37.7 mi/h. The passengers are reaed o a memorable ride! Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 35

36 Chaper : One-Dimensional Kinemaics 9. Picure he Problem: Waer pouring hrough an ancien Srai of Gibralar acceleraes downward and impacs he waer surface below. Sraegy: Employ he relaionship beween acceleraion, displacemen, and elociy (Equaion -) o find he heigh from which he waer mus fall so ha is final elociy jus before landing is 34 m/s. Soluion: Sole Equaion - for elociy x: 34 m/s x 59 m 5.9 km g 9.8 m/s Insigh: This heigh corresponds o 3.7 miles or oer 9, fee! Wih air resisance, howeer, an een higher aliude would be required o obain speeds his grea. 9. Picure he Problem: A juggler hrows a ball sraigh upward, i briefly comes o res, and falls downward, reurning o he juggler s hand. Sraegy: By symmery he oal ime of fligh is exacly wice he amoun of ime elapsed as he ball falls from res from is maximum heigh. Use his obseraion, ogeher wih he equaion for posiion as a funcion of ime (Equaion -) o find he maximum heigh of he ball aboe he juggler s hand. Soluion:. Sole Equaion - for x, assuming ha he final posiion x and iniial speed : x x a x g x g x 9.8 m/s.98 s. m. Subsiue alues o find he maximum heigh: Insigh: You can show ha he ball lef he juggler s hand wih an upward elociy of 4.8 m/s, or abou mi/h. 93. Picure he Problem: Ball A is dropped from res a he edge of a roof, and a he same insan ball B is hrown upward from he ground wih an iniial elociy sufficien o reach he original locaion of ball A. Sraegy: Use an undersanding of elociy and acceleraion o answer he concepual quesion. Le upward be he posiie direcion. Soluion:. The elociy of ball A is negaie because i is falling downward.. The acceleraion of ball A is negaie because graiy acs in he downward direcion. 3. The elociy of ball B is posiie because i is raeling upward. 4. The acceleraion of ball B is negaie because graiy acs in he downward direcion. Insigh: Acceleraion is he rae of change of elociy, so acceleraion can be zero when he elociy has a large magniude (for example, a car raeling along a highway a consan speed), and he elociy can be zero when he acceleraion has a large magniude (for example, a ball a he op of is erical fligh). The acceleraion of ball B is always downward, een when is elociy is upward. 94. Picure he Problem: Two balls are released simulaneously. Ball A is dropped from res bu ball B is hrown upward wih an iniial elociy. Sraegy: Use a correc inerpreaion of moion graphs o answer he concepual quesions. Recall ha he slope of a elociy-ersus-ime graph is he acceleraion. Soluion:. (a) The speed of ball A sars a zero and hen increases linearly wih a slope of 9.8 m/s. The graph ha corresponds o ha descripion is plo 3.. (b) The speed of ball B sars a and hen decreases linearly wih a slope of 9.8 m/s, equal in magniude bu opposie in direcion o he slope of ball A s plo. The graph ha corresponds o ha descripion is plo. Insigh: Een if ball B were fired upward a an exremely high speed, is elociy-ersus-ime graph would sill be linear wih a slope of 9.8 m/s, bu he line would begin ery high on he speed axis of he graph. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 36

37 Chaper : One-Dimensional Kinemaics 95. Picure he Problem: A plo of posiion s. ime yields informaion abou he aerage elociy of an objec. Sraegy: Use a correc inerpreaion of posiion-ime plos o answer he concepual quesions. Recall ha he slope of a posiion-ersus-ime graph is he elociy. Soluion:. (a) The aerage speed of he objec is he magniude of he slope of a line beween he wo poins. The line beween poins and (dark solid line) has a seeper slope han he line beween poins and 3 (ligh solid line). Therefore, he aerage speed for he ime ineral beween poins and is greaer han he aerage speed for he ime ineral beween poins and 3.. (b) The aerage elociy of he objec is he slope of a line beween he wo poins. The line beween poins and 4 (dark dashed line) has a smaller slope han he line beween poins 3 and 4 (ligh dashed line). Therefore, he aerage elociy for he ime ineral beween poins and 4 is less han he aerage elociy for he ime ineral beween poins 3 and 4. Insigh: The negaie slopes of he wo solid lines indicae he elociy of he objec is negaie for hose ime inerals. Howeer, he quesion in par (a) asked abou he speed, no he elociy, hence only he magniude of he slopes was considered. 96. Picure he Problem: A package falls sraigh downward, acceleraing for. seconds before impacing air bags. Sraegy: Find he disance he package will fall from res in. seconds by using Equaion -. Use he known acceleraion and ime o find he elociy of he package jus before impac by using Equaion -7. Finally, use he known iniial and final elociies, ogeher wih he disance oer which he package comes o res when in conac wih he air bags, o find he sopping acceleraion using Equaion -. Soluion:. (a) Find he disance he package x falls from res in. s using Equaion -: g. (b) Find he elociy jus 9.8 m/s. s 4 m g before impac using Equaion -7: 3. (c) Sole Equaion - for a: land 9.8 m/s. s m/s 48 mi/h! m/s a x.75 m 3 m/s 33 Insigh: Increasing he sopping disance will decrease he sopping acceleraion. We will reurn o his idea when we discuss impulse and momenum in Chaper 9. g Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 37

38 Chaper : One-Dimensional Kinemaics 97. Picure he Problem: A plo of elociy s. ime yields informaion abou he acceleraion of an objec. Sraegy: Use a correc inerpreaion of moion graphs o answer he concepual quesions. Recall ha he slope of a elociy-ersus-ime graph is he acceleraion. Soluion:. (a) The acceleraion of he objec is he slope of he elociyersus-ime graph: a 4. m/s m/s. s. (b) The displacemen is he area under he elociyersus-ime cure: 3. The final posiion is he iniial posiion plus he displacemen: 4. (c) Use he known acceleraion, iniial elociy, and iniial posiion o find he final posiion a = 5. s: x area of boom recangle area of riangle. s.5 m/s. s 4. m/s.5 m x x x. m.5 m 4.5 m x x a. m.5 m/s5. s 4. m/s 5. s x 64.5 m Insigh: Equaion - can also be used o find he final posiion in par (b), insead of deermining he area under he elociy-ersus-ime graph: x x a. m.5 m/s. s 4. m/s. s 4.5 m. 98. Picure he Problem: A golf ball rolls in a sraigh line, decreasing is speed a a consan rae unil i comes o res. Sraegy: You could find he (negaie) acceleraion by using Equaion - and he known iniial and final elociies and he disance raeled. Then employ Equaion - again using he same acceleraion, bu soling for he required o go he longer disance. Insead, we ll presen a way o calculae he same answer using a raio, which will also be useful o calculae he iniial speed needed o make he pu oer he new 6.-f disance. Soluion:. (a) Calculae he raio of iniial elociies based upon Equaion -:. Now sole for, he iniial speed make, needed o make he 3.5-f pu: 3. (b) Employ he same raio o find he iniial speed for he new 6.-f pu: a x a x x miss, a x a x x make, make make make miss x miss make make, miss, xmiss f x miss 3.5 f.54 m/s.77 m/s 6. f.54 m/s.9 m/s new new, miss, xmiss f Insigh: Calculaing raios can ofen be a conenien and simple way o sole a problem. In his case a hree-sep soluion became wo seps when we calculaed he raio, and furhermore we neer needed o coner fee o meers because he unis cancel ou in he raio. Learning o calculae raios in his manner is a aluable skill in physics. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 38

39 Chaper : One-Dimensional Kinemaics 99. Picure he Problem: Afer is release by a glaucous-winged gull, a shell rises sraigh upward, slows down, and momenarily comes o res before falling sraigh downward again. Sraegy: Find he exra aliude aained by he shell due o is upward iniial elociy upon release, and add ha alue o.5 m o find he maximum heigh i reaches aboe ground. The ime-free equaion for elociy in erms of displacemen (Equaion -) can be employed for his purpose. The ime he shell spends going up and he ime i spends going down can each be found from he known heighs and speeds (Equaions -7 and -). Then he speed upon landing can be deermined from he known ime i spends falling (Equaion -7). Le upward be he posiie direcion hroughou he soluion o his problem. Soluion:. (a) The moion of he shell is influenced only by graiy once i has been released by he gull. Therefore is acceleraion will be 9.8 m/s downward from he momen i is released, een hough i is moing upward a he release.. (b) Use Equaion -, seing he final speed =, o find he exra aliude gained by he shell due o is iniial upward speed, and add i o he.5 m: 3. (c) The ime he shell raels upward is he ime i akes graiy o bring he speed o zero (Equaion -7): x x 5. m/s max.5 m.5 m max g 9.8 m/s.5 m.38 m 3.9 m 5. m/s 9.8 m/s g.53 s 4. The ime he shell raels down is goerned by he disance and he acceleraion (Equaion -): x x g x g x g 3.9 m 9.8 m/s.68 s 5. The oal ime of fligh is he sum: oal up down s. s 6. (d) The speed of he shell upon impac is gien by he acceleraion of graiy and he fall ime (Equaion -7): g 9.8 m/s.68 s 6.5 m/s 6.5 m/s Insigh: There are a ariey of oher ways o sole his problem. For insance, i is possible o find he final elociy of 6.5 m/s in par (d) by using Equaion - wih 5. m/s and x.5 m wihou using any ime informaion. Try i for yourself!. Picure he Problem: Liquid from a syringe squirs sraigh upward, slows down, and momenarily comes o res before falling sraigh downward again. Sraegy: Find he ime of fligh by exploiing he symmery of he siuaion. If i akes ime for graiy o slow he liquid drops down from heir iniial speed o zero, i will ake he same amoun of ime o accelerae hem back o he same speed. They herefore reurn o he needle ip a he same speed wih which hey were squired. Use his fac ogeher wih Equaion -7 o find he ime of fligh. The maximum heigh he drops achiee is relaed o he square of, as indicaed by Equaion -. Soluion:. (a) Calculae he ime of fligh for.5 m/s, using Equaion -7:. (b) Calculae he maximum heigh for.5 m/s, using Equaion -:.5 m/s g g g 9.8 m/s.3 s.5 m/s x. m g g g 9.8 m/s Insigh: The symmery of he moion of a freely falling objec can ofen be a useful ool for soling problems quickly. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 39

40 Chaper : One-Dimensional Kinemaics. Picure he Problem: The rajecories of a ho-air balloon and a camera are shown a righ. The balloon rises a a seady rae while he camera s speed is coninually slowing down under he influence of graiy. The camera is caugh when he wo rajecories mee. Sraegy: The equaion of moion for posiion as a funcion of ime (Equaion -) can be used o describe he balloon, while he equaion for posiion as a funcion of ime and acceleraion (Equaion -) can be used o describe he camera s moion. Se hese wo equaions equal o each oher o find he ime a which he camera is caugh. Then find he heigh of he balloon a he insan he camera is caugh. Soluion:. Wrie Equaion - for he balloon: xb xb, b. Wrie Equaion - for he camera: x g c c, 3. Se xb x and sole for : c x g b, b c, x g b, c, b 4. Muliply by and subsiue he numerical alues:.5 m 3. m/s 9.8 m/s 5. Apply he quadraic formula and sole for. The larger roo corresponds o he ime when he camera would pass he balloon a second ime, on is way down back o he ground b b 4ac a or. s 6. Find he heigh of he balloon a ha ime: x x b b, b.5 m. m/s.6 s 3. m Insigh: If he passenger misses he camera he firs ime, she has anoher sho a i afer. s (from he ime i is hrown) when he camera is on is way back oward he ground. Tha is he meaning of he second soluion for.. Picure he Problem: The heigh-ersus-ime plo of a rock on a disan plane is shown a righ. The rock sars wih a high elociy upward, slows down and momenarily comes o res afer abou 4. seconds of fligh, and hen falls sraigh down and lands a abou 8. seconds. Sraegy: The equaion of moion for posiion as a funcion of ime and acceleraion (Equaion -) can be used o find he acceleraion from he second half of he rajecory, where he rock falls 3 m from res and lands 4. seconds laer. Once acceleraion is known, he iniial elociy can be deermined from Equaion -7. Le upward be he posiie direcion. Soluion:. (a) Sole Equaion - for acceleraion, assuming a he peak of is fligh and he rock falls 3 m in 4. s:. (b) Find he iniial elociy using Eq. -7, concenraing on he firs half of he fligh ha x 3 m a 3.8 m/s a 3.8 m/s 4. s a ends wih = a he peak: a 3.8 m/s 4. s 5 m/s Insigh: There are seeral oher ways of finding he answers, including graphical analysis. Try measuring he slope of he graph a he launch poin and he poin a which he rock lands o find he iniial and final elociies. Those alues (abou ±5 m/s) can hen be used o find he acceleraion. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 4

41 Chaper : One-Dimensional Kinemaics 3. Picure he Problem: A squid emis a je of waer, propelling iself forward wih consan acceleraion, hen coass o res wih consan (negaie) acceleraion. Sraegy: The magniude of he acceleraion can be deermined from he posiion as a funcion of ime equaion (Equaion -) and he gien informaion. During he firs par of he squid s moion he iniial elociy is zero, and during he second par he final elociy is zero. Soluion:. (a) Find he acceleraion during he firs par of he squid s moion, noing ha x :. (b) Find he squid s elociy a he end of he x a x.79 m a.4 m/s.7 s a firs par of is moion: 3. The ime elapsed during he second par of he squid s moion is found by subracion: 4. The disance raeled during he second par of he squid s moion is found by subracion: 5. Calculae he squid s acceleraion during he second par of is moion: x.4 m/s.7 s. m/s oal.4.7 s.3 s x xoal x.4.79 m.4 m x a a a.3 s.4 m. m/s.3 s 9. m/s Insigh: Noice ha he answer o par (b) can also be deermined wihou finding he squid s elociy. Insead, work backward and preend he squid sars from res and coers.4 m in.3 s. Then a x.4 m.3 s 9.5 m/s, which is he same resul o wihin rounding error, alhough you mus recognize ha he acceleraion is negaie, no posiie. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 4

42 Chaper : One-Dimensional Kinemaics 4. Picure he Problem: A ball falls sraigh downward from res a an iniial heigh h. Sraegy: The problem requires ha he ime o fall he final 3/4 h from res is. s. Find he elociy a ¾ h aboe he ground using Equaion -. Use Equaion - along wih ha iniial elociy and he ime elapsed o deermine h. Then he oal ime of fall can be found using Equaion - again, his ime wih an iniial elociy of zero. Soluion:. (a) Find he elociy of he ball afer falling a disance ¼ h: 4 gx g h gh ¾ h h. Now inser ha elociy as he iniial elociy for he remaining porion of he x g fall ino Equaion -: 3. The ime is. s as gien in he problem saemen. Rearrange he aboe equaion and square boh sides o ge a quadraic equaion: h gh g h g h g g h h g h g h g h g h h g gh h 9.8 m/s. s 9.8 m/s. s h.8h Now apply he quadraic formula for h: b b 4ac h.8, 9.6 m a 5. (b) Use Equaion - again o find he oal ime of fall: h 9.6 m g 9.8 m/s. s Insigh: The firs roo in sep 4 (.8 m) is hrown ou because he oal fall ime from ha heigh would be less han. s, bu he ball is supposed o be in he air for longer han. s. Noice i akes half he oal fligh ime o fall he firs quarer of he fall disance, and half o fall he final hree quarers. 5. Picure he Problem: A ski gloe falls sraigh downward from res, acceleraes o a maximum speed under he influence of graiy, hen deceleraes due o is ineracion wih he snow before coming o res a a deph d below he surface of he snow. Sraegy: We can find he maximum speed of he gloe from is iniial heigh and he acceleraion of graiy by using Equaion -. The same equaion can be applied again, his ime wih a zero final speed insead of zero iniial speed, o find he acceleraion caused by he snow. Le downward be he posiie direcion. Soluion:. (a) Sole Equaion - for, assuming :. (b) Use Equaion - o find gh gh h ad ad gh a g d he acceleraion caused by he snow: 3. The negaie sign on he acceleraion means he gloe is acceleraed upward during is ineracion wih he snow. Insigh: In Chaper 5 we will analyze he moion of objecs like his gloe in erms of force ecors. This moion can also be explained in erms of energy using he ools inroduced in Chapers 7 and 8. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 4

43 Chaper : One-Dimensional Kinemaics 6. Picure he Problem: A ball rises sraigh upward, passes a power line, momenarily comes o res, and falls back o Earh again, passing he power line a second ime on is way down. Sraegy: The ball will reach he peak of is fligh a a ime direcly beween he imes i passes he power line. The ime o reach he peak of fligh can be used o find he iniial elociy using Equaion -7, and he iniial elociy can hen be used o find he heigh of he power lines using Equaion -. Soluion:. Find he ime a which he ball reaches is maximum aliude:.75 s.5.75 s peak line up line down line up peak. s. Find he iniial elociy using Equaion -7: g 9.8 m/s. s m/s peak 3. Find he heigh of he power line using Equaion -: x g line up line up x m/s.75 s 9.8 m/s.75 s 5.5 m Insigh: As is ofen he case, here are seeral oher ways o sole his problem. Try seing he heighs a.75 s and.5 s equal o each oher and soling for. Can you hink of ye anoher way? 7. Picure he Problem: A ball appears a he boom edge of he window, rising sraigh upward wih iniial speed. I raels upward, disappearing beyond he op edge of he window, comes o res momenarily, and hen falls sraigh downward, reappearing some ime laer a he op edge of he window. In he drawing a righ he moion of he ball is offse horizonally for clariy. Sraegy: Le = correspond o he insan he ball firs appears a he boom edge of he window wih speed. Wrie he equaion of posiion as a funcion of ime and acceleraion (Equaion -) for when he ball is a he op edge (posiion ) in order o find. Use o find he ime o go from posiion o he peak of he fligh (Equaion -7). Subrac.5 s from ha ime o find he ime o go from posiion o he peak of he fligh. The ime elapsed beween posiions and 3 is wice he ime o go from posiion o he peak of he fligh. The ime from posiion o he peak can be used o find h from Equaion -. Soluion:. (a) Wrie Equaion - for posiions and, and sole for :. Find he ime o go from posiion o he peak of he fligh using Equaion -7: d g d g.5 m 9.8 m/s.5 s 5.4 m/s.5 s 5.4 m/s, p.55 s g 9.8 m/s 3. Subrac.5 s o find he ime o go, p, p,.55.5 s.3 s from posiion o he peak of he fligh: 4. The ime o reappear is wice his ime: 5. (b) The heigh h can be found from, p and Equaion -, by considering he ball.3 s.6 s,3, p h g h dropping from res a he peak o posiion 3:, p 9.8 m/s.3 s.44 m Insigh: As usual here are oher ways o sole his problem. Try finding he elociy a posiion and use i ogeher wih he acceleraion of graiy and he aerage elociy from posiion o he peak o find and h.,3 h d 3 Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 43

44 Chaper : One-Dimensional Kinemaics 8. Picure he Problem: The lunar lander falls sraigh downward, acceleraing oer a disance of 4.3 f before impacing he lunar surface. Sraegy: Use he gien acceleraion and disance and he ime-free equaion of moion (Equaion -) o find he elociy of he lander jus before impac. Use he known iniial and final elociies, ogeher wih he disance of he fall, o find he ime elapsed using Equaion -. Soluion:. Find he elociy jus before impac using Equaion -:. Sole Equaion - for fall : ax land.5 f/s.6 m/s 3.8 f/m 4.3 f 6.78 f/s x 4.3 f.8 s fall fall land f/s Insigh: An alernaie sraegy would be o sole Equaion - as a quadraic equaion in. Assuming he lander fee had lile in he way of shock absorbers, he lander came o res in a disance gien by he amoun he lunar dus compaced underneah he fee. Supposing i was abou cm, he asronaus experienced a brief deceleraion of 6 m/s = g! Bam! 9. Picure he Problem: The lunar lander falls sraigh downward, acceleraing oer a disance of 4.3 f before impacing he lunar surface. Sraegy: Use he gien acceleraion and disance and he ime-free equaion of moion (Equaion -) o find he elociy of he lander jus before impac. Soluion: Find he elociy jus before impac using Equaion -: ax land.5 f/s.6 m/s 3.8 f/m 4.3 f 6.78 f/s Insigh: The iniial speed made lile difference; if you se you ll noe ha land 6.76 f/s.. Picure he Problem: The lunar lander falls sraigh downward, acceleraing oer a disance of 4.3 f before impacing he lunar surface. Sraegy: The lander has an iniial downward elociy and acceleraes downward a a consan rae. Use he knowledge ha he elociy-ersus-ime graph is a sraigh line for consan acceleraion o deermine which graph is he appropriae one. Soluion: Graph B is he only one ha depics he speed increasing linearly wih ime. Insigh: Graph D would be an appropriae depicion of he aliude ersus ime graph.. Picure he Problem: We imagine ha he asronaus increase he upward hrus, giing he lunar lander a small upward acceleraion. Sraegy: The lander has an iniial downward elociy and acceleraes upward a a consan rae. This means he lander s speed would decrease a a consan rae. Use he knowledge ha he elociy-ersus-ime graph is a sraigh line for consan acceleraion o deermine which graph is he appropriae one. Soluion: Plo C is he only one ha depics he speed decreasing linearly wih ime. Insigh: The aliude-ersus-ime graph in his case would cure upward much like plo A bu would hae an iniially negaie slope like plo D. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 44

45 Chaper : One-Dimensional Kinemaics. Picure he Problem: The rajecories of he speeder and police car are shown a righ. The speeder moes a a consan elociy while he police car has a consan acceleraion, excep he police car is delayed in ime from when he speeder passes i a x =. Sraegy: The equaion of moion for posiion as a funcion of ime and elociy (Equaion -) can be used o describe he speeder, while he equaion for posiion as a funcion of ime and acceleraion (Equaion -) can be used o describe he police car s moion. Se hese wo equaions equal o each oher and sole he resuling equaion o find he speeder s head-sar x. shs Soluion:. Wrie Equaion - for he speeder, wih = corresponding o he insan i passes he police car:. Wrie Equaion - for he police car: 3. Se xp x and sole for x : s shs xs xshs s x a p a x p p shs s shs p s shs 53 m 3.8 m/s 5 s 5 m/s5 s x a x Insigh: This head sar corresponds o abou. seconds (erify for yourself, and/or examine he plo) so he police officer has o be ready o sar he chase ery soon afer he speeder passes by! 3. Picure he Problem: The rajecories of he speeder and police car are shown a righ. The speeder moes a a consan elociy while he police car has a consan acceleraion. Sraegy: The equaion of moion for posiion as a funcion of ime and elociy (Equaion -) can be used o describe he speeder, while he equaion for posiion as a funcion of ime and acceleraion (Equaion -) can be used o describe he police car s moion. Se hese wo equaions equal o each oher and sole he resuling equaion for he acceleraion of he police car. Soluion:. Wrie Equaion - for he speeder, wih = corresponding o he insan i passes he police car:. Wrie Equaion - for he police car: x s x a p s p 3. Se xp x and sole for s a : p a a p s s p 5 m/s 7. s 4.3 m/s Insigh: A faser acceleraion of he police car would allow i o cach he speeder in less han 7. s. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 45

46 Chaper : One-Dimensional Kinemaics 4. Picure he Problem: The rajecory of a bag of sand is shown a righ. Afer release from he balloon i rises sraigh up and comes momenarily o res before acceleraing sraigh downward and impacing he ground. Sraegy: Because he iniial elociy, acceleraion, and aliude are known, we need only use Equaion - o find he final elociy. Soluion:. (a) Because he upward speed of he sandbag is he same, i will gain he same addiional m in aliude as i did in he original Example -. Therefore he maximum heigh will be equal o 3 m.. (b) Apply Equaion - o find he final elociy: ax 6.5 m/s 9.8 m/s 3. m 5 m/s Insigh: Anoher way o find he final elociy jus before impac is o allow he sandbag o fall from res a disance of 3 m. Try i! 5. Picure he Problem: A bag of sand has an iniial downward elociy when i breaks free from he balloon, and is acceleraed by graiy unil i his he ground. Sraegy: Because he iniial elociy, acceleraion, and aliude are known, we need only use Equaion - o find he final elociy. The ime can hen be found from he aerage elociy and he disance. Soluion:. (a) Apply Equaion - o find he final : ax 4. m/s 9.8 m/s 35. m 6.5 m/s. Use Equaion - o find he ime: x x 35 m m/s.3 s 3. (b) Apply Equaion - again o find a x = 5 m: ax 4. m/s 9.8 m/s 5 35 m m/s Insigh: Anoher way o find he descen ime of he bag of sand is o sole Equaion - using he quadraic formula. Try i! Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 46

47 Chaper : One-Dimensional Kinemaics Answers o Concepual Quesions 6. Can you drie your car in such a way ha he disance i coers is (a) greaer han, (b) equal o, or (c) less han he magniude of is displacemen? In each case, gie an example if your answer is yes, explain why no if your answer is no. (a) Yes. If you drie in a complee circle your disance is he circumference of he circle, bu your displacemen is zero. (b) Yes. The disance and he magniude of he displacemen are equal if you drie in a sraigh line. (c) No. Any deiaion from a sraigh line resuls in a disance ha is greaer han he magniude of he displacemen. 7. CE Predic/Explain You drie your car in a sraigh line a 5 m/s for minues, hen a 5 m/s for anoher minues. (a) Is your aerage speed for he enire rip more han, less han, or equal o m/s? (b) Choose he bes explanaion from among he following: I. More ime is required o drie a 5 m/s han a 5 m/s. II. Less disance is coered a 5 m/s han a 5 m/s. III. Equal ime is spen a 5 m/s and 5 m/s. Yes. For example, your friends migh hae backed ou of a parking place a some poin in he rip, giing a negaie elociy for a shor ime. 8. In 99 Zhuang Yong of China se a women s Olympic record in he -meer freesyle swim wih a ime of seconds. Wha was her aerage speed in m/s and mi/h? No. If you hrow a ball upward, for example, you migh choose he release poin o be y =. This doesn change he fac ha he iniial upward speed is nonzero. 9. A finch rides on he back of a Galapagos oroise, which walks a he saely pace of.6 m/s. Afer. minues he finch ires of he oroise s slow pace, and akes fligh in he same direcion for anoher. minues a m/s. Wha was he aerage speed of he finch for his.4-minue ineral? Ignoring air resisance, he wo gloes hae he same acceleraion. Soluions o Problems and Concepual Exercises. In 99 Zhuang Yong of China se a women s Olympic record in he -meer freesyle swim wih a ime of seconds. Wha was her aerage speed in m/s and mi/h? Picure he Problem: The swimmer swims in he forward direcion. Sraegy: The aerage speed is he disance diided by elapsed ime. Soluion: Diide he disance by he ime: disance. m mi 36 s s = = =.83 m/s = 4.95 mi/h ime s 69 m h Insigh: The displacemen would be zero in his case because he swimmer swims eiher wo lenghs of a 5-m pool or four lenghs of a 5-m pool, reurning o he saring poin each ime. Howeer, he aerage speed depends upon disance raeled, no displacemen. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No

48 Chaper : One-Dimensional Kinemaics. Esimae how fas your hair grows in miles per hour. Picure he Problem: Your hair grows a a fixed speed. Sraegy: The growh rae is he lengh gained diided by he ime elapsed. Hair grows a a rae of abou half an inch a monh, or abou cm or. m per monh. Soluion: Diide he lengh gained by he elapsed ime: d. m mi mo d s = = = mo 69 m 3.5 d 4 h mi/h Insigh: Try conering his growh rae o a more appropriae uni such as µm/h. (Answer: 4 µm/h.) Choosing an appropriae uni can help you communicae a number more effeciely.. IP You drie in a sraigh line a. m/s for. minues, hen a 3. m/s for anoher. minues. (a) Is your aerage speed 5. m/s, more han 5. m/s, or less han 5. m/s? Explain. (b) Verify your answer o par (a) by calculaing he aerage speed. Picure he Problem: You rael in a sraigh line a wo differen speeds during he specified ime ineral. Sraegy: Deermine he aerage speed by firs calculaing he oal disance raeled and hen diiding i by he oal ime elapsed. Soluion:. (a) Because he ime inerals are he same, you spend equal imes a m/s and 3 m/s, and your aerage speed will be equal o 5. m/s.. (b) Diide he oal disance by he ime elapsed: s a = = s a (. m/s)(. min 6 s) + ( 3. m/s)( 6 s) sδ + s Δ Δ +Δ s = 5. m/s Insigh: The aerage speed is a weighed aerage according o how much ime you spend raeling a each speed. 3. IP You drie in a sraigh line a. m/s for. miles, hen a 3. m/s for anoher. miles. (a) Is your aerage speed 5. m/s, more han 5. m/s, or less han 5. m/s? Explain. (b) Verify your answer o par (a) by calculaing he aerage speed. Picure he Problem: You rael in a sraigh line a wo differen speeds during he specified ime ineral. Sraegy: Deermine he aerage speed by firs calculaing he oal disance raeled and hen diiding i by he oal ime elapsed. Soluion:. (a) The disance inerals are he same bu he ime inerals are differen. You will spend more ime a he lower speed han a he higher speed. Because he aerage speed is a ime weighed aerage, i will be less han 5. m/s.. (b) Diide he oal disance by he ime elapsed: s s a = = = d d s s a d + d d + d. mi Δ +Δ. mi. mi + +. m/s 3. m/s = 4. m/s Insigh: Noice ha in his case i is no necessary o coner miles o meers in boh he numeraor and denominaor because he unis cancel ou and leae m/s in he numeraor. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No

49 Chaper : One-Dimensional Kinemaics 4. CE Predic/Explain Two bows shoo idenical arrows wih he same launch speed. To accomplish his, he sring in bow mus be pulled back farher when shooing is arrow han he sring in bow. (a) Is he acceleraion of he arrow sho by bow greaer han, less han, or equal o he acceleraion of he arrow sho by bow? (b) Choose he bes explanaion from among he following: I. The arrow in bow acceleraes for a greaer ime. II. Boh arrows sar from res. III. The arrow in bow acceleraes for a greaer ime. Picure he Problem: Two arrows are launched by wo differen bows. Sraegy: Use he definiions of aerage speed and acceleraion o compare he moions of he wo arrows. Soluion:. (a) We can reason ha because boh arrows undergo uniform acceleraion beween he same iniial and final elociies, boh arrows mus hae he same aerage speed. If hey hae he same aerage speed, hen arrow, which mus rael a longer disance, will be acceleraed for a longer period of ime. We conclude ha he acceleraion of he arrow sho by bow is less han he acceleraion of he arrow sho by bow.. (b) As discussed aboe, he bes explanaion is III. The arrow in bow acceleraes oer a greaer ime. Saemen I is false and saemen II is rue bu is no a complee explanaion. Insigh: We could also se = in he equaion, = + aδx and sole for a: a = Δ x From his expression we can see ha for he same final elociy, he arrow ha is acceleraed oer he greaer disance Δ x will hae he smaller acceleraion. 5. IP In he preious problem, (a) does he disance needed o sop increase by a facor of wo or a facor of four? Explain. Verify your answer o par (a) by calculaing he sopping disances for iniial speeds of (b) 6 m/s and (c) 3 m/s. Picure he Problem: The car raels in a sraigh line in he posiie direcion while acceleraing in he negaie direcion (slowing down). Sraegy: Use he aerage elociy and he ime elapsed o deermine he disance raeled for he specified change in elociy. Soluion:. (a) Because he disance raeled is proporional o he square of he ime (Equaion -), or alernaiely, because boh he ime elapsed and he aerage elociy change by a facor of wo, he sopping disance will increase by a facor of four when you double your driing speed.. (b) Ealuae Equaion - direcly: x ( ) ( )( ) Δ = + = 6 + m/s 3.8 = 3 m =.3 km 3. (c) Ealuae Equaion - direcly: x ( ) ( )( ) Δ = + = 3 + m/s 7.6 = m =. km Insigh: Doubling your speed will quadruple he sopping disance for a consan acceleraion. We will learn in chaper 7 ha his can be explained in erms of energy; ha is, doubling your speed quadruples your kineic energy. 6. Suppose he car in Problem 44 comes o res in 35 m. How much ime does his ake? Picure he Problem: The car raels in a sraigh line oward he wes while acceleraing in he easerly direcion (slowing down). Sraegy: The aerage elociy is simply half he sum of he iniial and final elociies because he acceleraion is uniform. Use he aerage elociy ogeher wih Equaion - o find he ime. Soluion: Sole Equaion - for ime: Δx 35 m = = = 5.8 s ( + ) ( + m/s) Insigh: The disance raeled is always he aerage elociy muliplied by he ime. This sems from he definiion of aerage elociy. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 3

50 Chaper : One-Dimensional Kinemaics 7. Air Bags Air bags are designed o deploy in ms. Esimae he acceleraion of he fron surface of he bag as i expands. Express your answer in erms of he acceleraion of graiy g. Picure he Problem: An air bag expands ouward wih consan posiie acceleraion. Sraegy: Assume he air bag has a hickness of f or abou.3 m. I mus expand ha disance wihin he gien ime of ms. Employ he relaionship beween acceleraion, displacemen, and ime (Equaion -) o find he acceleraion. Δx (.3 m) g Soluion: Sole Equaion - for a: a= = = 6 m/s 6g ms. s/ms 9.8 m/s ( ) Insigh: The ery large acceleraion of an expanding airbag can cause seere injury o a small child whose head is oo close o he bag when i deploys. Children are safes in he back sea! 8. IP Coasing due wes on your bicycle a 8.4 m/s, you encouner a sandy pach of road 7. m across. When you leae he sandy pach your speed has been reduced by. m/s o 6.4 m/s. (a) Assuming he sand causes a consan acceleraion, wha was he bicycle s acceleraion in he sandy pach? Gie boh magniude and direcion. (b) How long did i ake o cross he sandy pach? (c) Suppose you ener he sandy pach wih a speed of only 5.4 m/s. Is your final speed in his case 3.4 m/s, more han 3.4 m/s, or less han 3.4 m/s? Explain. Picure he Problem: A bicycle raels in a sraigh line, slowing down a a uniform rae as i crosses he sandy pach. Sraegy: Use he ime-free relaionship beween displacemen, elociy, and acceleraion (Equaion -) o find he acceleraion. The ime can hen be deermined from he aerage elociy and he disance across he sandy pach. Soluion:. (a) Calculae he acceleraion:. (b) Sole Equaion - for : ( ) ( ) ( ) 6.4 m/s 8.4 m/s a = = =. m/s Δx 7. m where he negaie sign means. m/s o he eas. Δ x 7. m = = =.97 s ( + ) ( m/s) 3. (c) Examining = + aδ x (Equaion -) in deail, we noe ha he acceleraion is negaie, and ha he final elociy is he square roo of he difference beween and aδ x. Because aδ x is consan because he sandy pach doesn change, i now represens a larger fracion of he smaller, and he final elociy will be more han. m/s differen han. We herefore expec a final speed of less han 3.4 m/s. Insigh: In fac, if you ry o calculae in par (c) wih Equaion - you end up wih he square roo of a negaie ( 5.4 m/s ) number, because he bicycle will come o res in a disance Δ x = = = 6.9 m, less han he 7. m a (. m/s ) lengh of he sandy pach. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 4

51 Chaper : One-Dimensional Kinemaics 9. In a physics lab, sudens measure he ime i akes a small car o slide a disance of. m on a smooh rack inclined a an angle aboe he horizonal. Their resuls are gien in he following able. θ.. 3. ime, s (a) Find he magniude of he car s acceleraion for each angle. (b) Show ha your resuls for par (a) are in close agreemen wih he formula, a = g sin θ. (We will derie his formula in Chaper 5.) Picure he Problem: The car slides down he inclined rack, each ime raeling a disance of. m along he rack. Sraegy: The disance raeled by he car is gien by he consan-acceleraion equaion of moion for posiion as a funcion of ime (Equaion -), where x = =. The magniude of he acceleraion can hus be deermined from he gien disance raeled and he ime elapsed in each case. We can hen make he comparison wih a= gsinθ. θ. m Soluion:. Find he acceleraion from Equaion -:. Now find he alues for θ =. : 3. Now find he alues for θ =. : x x= + + a a= a = gsinθ. m a = =.7 m/s ( a = 9.8 m/s ) sin. =.7 m/s (.8 s). m a = = 3.37 m/s ( a = 9.8 m/s ) sin. = 3.35 m/s (.77 s) 4. Now find he alues for θ = 3. :. m a = = 4.88 m/s ( a = 9.8 m/s ) sin. = 4.9 m/s (.64 s) Insigh: We see ery good agreemen beween he formula a= gsinθ and he measured acceleraion. The experimenal accuracy ges more and more difficul o conrol as he angle ges bigger because he elapsed imes become ery small and more difficul o measure accuraely. For his reason Galileo s experimenal approach (rolling balls down an incline wih a small angle) gae him an opporuniy o make accurae obseraions abou free fall wihou fancy elecronic equipmen. 3. Legend has i ha Isaac Newon was hi on he head by a falling apple, hus riggering his houghs on graiy. Assuming he sory o be rue, esimae he speed of he apple when i sruck Newon. Picure he Problem: An apple falls sraigh downward under he influence of graiy. Sraegy: The disance of he fall is esimaed o be abou 3. m (abou f). Then use he ime-free equaion of moion (Equaion -) o esimae he speed of he apple. Soluion:. Sole Equaion - for, assuming he apple drops from res ( = ): = + aδ x. Le a = g and calculae : ( )( ) = 9.8 m/s 3. m = 7.7 m/s = 7 mi/h Insigh: Newon supposedly hen reasoned ha he same force ha made he apple fall also keeps he Moon in orbi around he Earh, leading o his uniersal law of graiy (Chaper ). One lesson we migh learn here is wear a helme when siing under an apple ree! Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 5

52 Chaper : One-Dimensional Kinemaics 3. Jordan s Jump Michael Jordan s erical leap is repored o be 48 inches. Wha is his akeoff speed? Gie your answer in meers per second. Picure he Problem: Michael Jordan jumps erically, he acceleraion of graiy slowing him down and bringing him momenarily o res a he peak of his fligh. Sraegy: Because he heigh of he leap is known, use he ime-free equaion of moion (Equaion -) o find he akeoff speed. Soluion: Sole Eq. - for : g x ( )( ) = Δ = 9.8 m/s 48 in.54 m/in = 4.9 m/s Insigh: Tha speed is abou half of wha champion spriners achiee in he horizonal direcion, bu is ery good among ahlees for a erical leap. High jumpers can jump een higher, bu use he running sar o heir adanage. 3. Bill seps off a 3.-m-high diing board and drops o he waer below. A he same ime, Ted jumps upward wih a speed of 4. m/s from a.-m-high diing board. Choosing he origin o be a he waer s surface, and upward o be he posiie x direcion, wrie x-ersus- equaions of moion for boh Bill and Ted. Picure he Problem: Two diers moe erically under he influence of graiy. Sraegy: In boh cases we wish o wrie he equaion of moion for posiion as a funcion of ime and acceleraion (Equaion -). In Bill s case, he iniial heigh x = 3. m, bu he iniial elociy is zero because he seps off he diing board. In Ted s case he iniial heigh x =. m and he iniial elociy is +4. m/s. In boh cases he acceleraion is 9.8 m/s. Soluion:. Equaion - for Bill: = + + = 3. m + + ( 9.8 m/s ) x = ( 3. m ) ( 4.9 m/s ) x x a. Equaion - for Ted: = + + =. m + ( 4. m/s ) + ( 9.8 m/s ) x = (. m ) + ( 4. m/s ) ( 4.9 m/s ) x x a Insigh: The differen iniial elociies resul in significanly differen rajecories for Bill and Ted. 33. Repea he preious problem, his ime wih he origin 3. m aboe he waer, and wih downward as he posiie x direcion. Picure he Problem: Two diers moe erically under he influence of graiy. Sraegy: In boh cases we wish o wrie he equaion of moion for posiion as a funcion of ime and acceleraion (Equaion -). Here we ll ake he origin o be a he leel of Bill s board aboe he waer, Ted s diing board o be a +. m, and he waer surface a +3. m. Downward is he posiie direcion so ha he acceleraion is 9.8 m/s. In Bill s case, he iniial heigh x =. m and his iniial elociy is zero because he seps off he diing board. In Ted s case he iniial heigh is x =+. m and he iniial elociy is 4. m/s (upward). Soluion:. Equaion - for Bill: = + + =. m + + ( 9.8 m/s ) x = ( 4.9 m/s ) x x a. Equaion - for Ted: = + + =. m + ( 4. m/s ) + ( 9.8 m/s ) x = (. m) + ( 4. m/s ) + ( 4.9 m/s ) x x a Insigh: The differen iniial elociies resul in significanly differen rajecories for Bill and Ted. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 6

53 Chaper : One-Dimensional Kinemaics 34. IP Sanding side by side, you and a friend sep off a bridge a differen imes and fall for.6 s o he waer below. Your friend goes firs, and you follow afer she has dropped a disance of. m. (a) When your friend his he waer, is he separaion beween he wo of you. m, less han. m, or more han. m? (b) Verify your answer o par (a) wih a calculaion. Picure he Problem: You and your friend boh accelerae from res sraigh downward, bu a differen imes. You sep off he bridge when your friend has fallen. m, and your friend his he waer while you are sill in he air. Sraegy: Firs find he ime i akes for your friend o fall. m using he equaion of moion for posiion as a funcion of ime and acceleraion (Equaion -). Subrac ha ime from.6 s o find he ime elapsed beween when you jump and when your friend his he waer. Use Equaion - and he imes found aboe o find he posiions of you and your friend a he ime your friend lands. Then deermine he separaion beween he known posiions. Soluion:. (a) Because your friend has a greaer aerage speed han you do during he ime beween when you jump and your friend lands, he separaion beween he wo of you will increase o a alue more han. m.. (b) Find he ime i akes o fall. m from Equaion - wih = : ( ) Δx. m = = =.64 s g 9.8 m/s 3. Find he disance your friend fell in.6 s: x g ( )( ) = = 9.8 m/s.6 s = 3 m friend 4. Find he disance you fell in he shorer ime: x g ( ) ( )( ) you. m = = 9.8 m/s.6.64 s = 4.5 m 5. Find he difference in your posiions: S = xfriend xyou = m = 8 m Insigh: Because of her head sar, your friend will always hae a higher aerage elociy han you, and he separaion beween you and her will coninue o increase he longer you boh fall. you jump. m bridge waer S? friend lands =.6 s 35. In a well-known Jules Verne noel, Phileas Fogg raels around he world in 8 days. Wha was Mr. Fogg s approximae aerage speed during his adenure? Picure he Problem: Phileas Fogg raels in a sraigh line all he way around he world. Sraegy: The aerage speed is he disance diided by elapsed ime. We will esimae ha Mr. Fogg raels a disance equal o he equaorial circumference of he Earh. This is an approximaion, because his pah was mos likely much more complicaed han ha, bu we were asked only for he approximae speed. Soluion: Find he circumference of he Earh: d πr π ( ) = = = m 4. m 7 disance 4. m Diide he disance by he ime: s = = = 5.8 m/s ime 8 d 4 h/d 36 s/h Insigh: This speed corresponds o abou 3 mi/h and is faser han humans can walk. Giing ime for sleeping, eaing, and oher delays, Mr. Fogg needs a relaiely fas means of rael. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 7

54 Chaper : One-Dimensional Kinemaics 36. You jump from he op of a boulder o he ground.5 m below. Esimae your deceleraion on landing. Picure he Problem: You jump off a boulder, accelerae from res sraigh downward and land, bending your knees so ha your cener of mass comes o res oer a shor erical disance. Sraegy: Employ he relaionship beween acceleraion, displacemen, and elociy (Equaion -) o find your final elociy jus before landing. Then esimae he disance your cener of mass will moe afer your fee conac he ground, and use ha disance o esimae your deceleraion rae. Soluion:. Sole Equaion - for elociy : a x ( )( ). Esimae your cener of mass moes downward abou.5 m afer your fee conac he ground and you bend your knees ino a crouching posiion. Sole Equaion - for acceleraion: = + Δ = m/s.5 m = 5.4 m/s ( ) ( ) 5.4 m/s a = = = = Δy.5 m 9 m/s 3. Insigh: When a gymnas lands from an een higher aliude, she migh ry o bend her knees een less in order o impress he judges. If she lands from an aliude of 3. m and bends her knees so her cener of mass moes only. m, her acceleraion is 5g! g 37. CE A he edge of a roof you drop ball A from res, and hen hrow ball B downward wih an iniial elociy of. Is he increase in speed jus before he balls land more for ball A, more for ball B, or he same for each ball? Picure he Problem: Two balls are released from he edge of a roof. Ball A is dropped from res bu ball B is hrown downward wih an iniial elociy. Sraegy: Use he definiion of acceleraion o answer he concepual quesion, keeping in mind he aerage speed of ball B is greaer han he aerage speed of ball A. Soluion: The wo balls fall he same disance bu ball B has he greaer aerage speed and falls for a shorer lengh of ime. Because each ball acceleraes a he same rae of 9.8 m/s, ball A acceleraes for a longer ime and he increase in speed is more for ball A han i is for ball B. Insigh: If ball B were fired downward a an exremely high speed, i would reach he ground wihin a ery shor ineral of ime and is speed would hardly change a all. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 8

55 Chaper : One-Dimensional Kinemaics 38. IP A youngser bounces sraigh up and down on a rampoline. Suppose she doubles her iniial speed from. m/s o 4. m/s. (a) By wha facor does her ime in he air increase? (b) By wha facor does her maximum heigh increase? (c) Verify your answers o pars (a) and (b) wih an explici calculaion. Picure he Problem: A youngser bounces sraigh up and down on a rampoline. The child rises sraigh upward, slows down, and momenarily comes o res before falling sraigh downward again. Sraegy: Find he ime of fligh by exploiing he symmery of he siuaion. If i akes ime for graiy o slow he child down from her iniial speed o zero, i will ake he same amoun of ime o accelerae her back o he same speed. She herefore lands a he same speed wih which she ook off. Use his fac ogeher wih Equaion -7 o find he ime of fligh. The maximum heigh she achiees is relaed o he square of, as indicaed by Equaion -. Soluion:. (a) Because he ime of fligh depends linearly upon he iniial elociy, doubling will increase her ime of fligh by a facor of.. (b) Because he ime of fligh depends upon he square of he iniial elociy, doubling will increase her maximum aliude by a facor of (c) The ime of fligh for =. m/s, using Eq. -7: 4. The ime of fligh for = 4. m/s : 5. The maximum heigh for =. m/s, using Eq. -: 6. The maximum heigh for 4. m/s ( ) (. m/s) = = = = = g g g ( ) 4. m/s = = = g 9.8 m/s = : ( 4. m/s).8 s 9.8 m/s (. m/s) ( ).4 s Δ x = = = = =. m g g g 9.8 m/s Δ x = = =.8 m g 9.8 m/s ( ) Insigh: The reason he answer in sep 6 is no exacly four imes larger han he answer in sep 5 is due o he rounding required by he fac ha here are only wo significan digis. If you recalculae using. m/s and 4. m/s, he answers are.4 and.86 m, respeciely. 39. IP A popular enerainmen a some carnials is he blanke oss (see phoo, p. 39). (a) If a person is hrown o a maximum heigh of 8. f aboe he blanke, how long does she spend in he air? (b) Is he amoun of ime he person is aboe a heigh of 4. f more han, less han, or equal o he amoun of ime he person is below a heigh of 4. f? Explain. (c) Verify your answer o par (b) wih a calculaion. Picure he Problem: The person is hrown sraigh upward, slows down, and momenarily comes o res before falling sraigh downward again. Sraegy: Find he ime of fligh by exploiing he symmery of he siuaion. If i akes ime for graiy o slow he person down from her iniial speed o zero, i will ake he same amoun of ime o accelerae her back o he same speed. I herefore akes he same amoun of ime for her o rise o he peak of her fligh han i does for her o reurn o he blanke. Use his fac ogeher wih Equaion - wih = (corresponding o he second half of her fligh, from he peak back down o he blanke) o find he ime of fligh. The ime aboe and below 4. f can be found using he same equaion. Soluion:. (a) The ime of fligh can be found from Equaion -: ( ) Δ x 8. f.35 m/f = down = = =.64 s g 9.8 m/s. (b) The person s aerage speed is less during he upper half of her rajecory, so he ime she spends in ha porion of her fligh is more han he ime she spends in he lower half of her fligh. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No 9

56 Chaper : One-Dimensional Kinemaics 3. (c) The ime she spends aboe 4. f is he same ime of her fligh if her maximum heigh were 4. f: ( ) Δx 4. f.35 m/f = = =.87 s g 9.8 m/s aboe 4. The ime spen below 4. f is he below = oal aboe = s =.77 s remaining porion of he oal ime of fligh: Insigh: The symmery of he moion of a freely falling objec can ofen be a useful ool for soling problems quickly. 4. Referring o Concepual Checkpoin 5, find he separaion beween he rocks a (a) =. s, (b) =. s, and (c) = 3. s, where ime is measured from he insan he second rock is dropped. (d) Verify ha he separaion increases linearly wih ime. Picure he Problem: The wo rocks fall sraigh downward along a similar pah excep a differen imes. Sraegy: Firs find he ime elapsed beween he release of he wo rocks by finding he ime required for he firs rock o fall 4. m, using he equaion of moion for posiion as a funcion of ime and acceleraion (Equaion -). The posiions as a funcion of ime for each rock can hen be compared o find a separaion disance as a funcion of ime. Soluion:. (a) Find he ime required for rock A o fall 4. m: ( ) Δ x 4. m = = =.93 s g 9.8 m/s 4. Le represen he ime elapsed from he insan rock B is dropped. The posiion of rock A (Equaion -) is hus: ( ) x = + g + = g + g + g A The posiion of rock B (Equaion -) is: x = + g = g B 5. Find he separaion beween he rocks: Δ x = xa xb = ( g + g 4 + g 4 ) g Δ x = g4 + g 4 = ( 9.8 m/s ) (.93 s) + ( 9.8 m/s )(.93 s) 6. Find x 7. (b) Find x ( ) Δ x = 8.86 m/s + 4. m Δ for =. s: x ( )( ) Δ = 8.86 m/s. s + 4. m =.9 m Δ for =. s: x ( )( ) Δ = 8.86 m/s. s + 4. m = m Δ for =. s: ( )( ) 8. (c) Find x Δ x = 8.86 m/s 3. s + 4. m = 3 m 9. (d) The linear dependence of Δ x upon can be erified by examining he equaion deried in sep 5. Insigh: The only way for rock B o cach up o rock A would be for rock B o be hrown downward wih a large iniial speed. In ha case he separaion becomes Δ x= ( 8.86 m/s B, ) + 4. m, which decreases o zero as long as is B, greaer han 8.86 m/s. Copyrigh 7 Pearson Educaion, Inc. All righs resered. This maerial is proeced under all copyrigh laws as hey currenly exis. No