Solution: (a) Add the lengths: m 2.5 m 15 m. Insight: The distance traveled is always positive, but the displacement can be negative.
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1 homework oluion. Picure he Problem: The ball i pued in he poiive direcion and hen he negaive direcion. Sraegy: The diance i he oal lengh of rel, and he diplacemen i he ne change in poiion. Soluion: (a) Add he lengh: 10.5 m.5 m 15 m (b) Subrac x i from x f o find he diplacemen. x x x f i 10 0 m 10 m Inigh: The diance reled i alway poiive, bu he diplacemen can be negaive. 4. Picure he Problem: You walk in boh he poiive and negaive direcion along a raigh line. Sraegy: The diance i he oal lengh of rel, and he diplacemen i he ne change in poiion. Soluion: (a) Add he lengh: mi mi.65 (b) Subrac x i from x f o find he diplacemen. x x x f i mi 0.75 mi Inigh: The diance reled i alway poiive, bu he diplacemen can be negaive. 1. Picure he Problem: The radio we propagae in a raigh line. Sraegy: The ime elaped i he diance divided by he erage peed. The diance o he Moon i mi. We mu double hi diance becaue he ignal rel here and back again. Soluion: Divide he diance by he erage peed: d mi mi/.57 Inigh: The ime i lighly horer han hi becaue he given diance i from he cener of he Earh o he cener of he Moon, bu preumably any radio communicaion would occur beween he urface of he Earh and Moon. When he radii of he wo phere i aken ino accoun, he ime decreae o Picure he Problem: You rel 8.0 km on foo and hen an addiional 16 km by car, wih boh diplacemen along he ame direcion.
2 Sraegy: Fir find he oal ime elaped by dividing he diance reled by he erage and divide by he oal ime elaped o find he erage peed. value and olve for he car peed. Soluion: 1. Ue he definiion of erage peed o deermine he oal ime elaped. d Se ha erage peed o he given km 1.1 h km/h. Find he ime elaped while in he car: h 0.84 h 0. h. Find he peed of he car: d 16 km 50 km/h 0. h Inigh: Thi problem illurae he limiaion ha ignifican figure occaionally impoe. you keep an exra figure in he oal elaped ime (1.09 h) you ll end up wih he ime elaped for he car rip a 0.5 h, no 0., and he peed of he car i 64 km/h. Bu he rule of ubracion indicae we only know he oal ime o wihin a enh of an hour, o we can only know he ime pen in he car o wihin a enh of an hour, or o wihin one ignifican digi. If 0. Picure he Problem: You rel in a raigh line a wo differen peed during he pecified ime inerval. Sraegy: Deermine he erage peed by fir calculaing he oal diance reled and hen dividing i by he oal ime elaped. Soluion: 1. (a) Becaue he ime inerval are he ame, you pend equal ime a 0 m/ and 0 m/, and your erage peed will be equal o 5.0 m/.. (b) Divide he oal diance by he ime elaped: m/ 0.0 m/ 10.0 min m/ Inigh: The erage peed i a weighed erage according o how much ime you pend reling a each peed. 1. Picure he Problem: You rel in a raigh line a wo differen peed during he pecified ime inerval. Sraegy: Deermine he diance reled during each leg of he rip in order o plo he graph. Soluion: 1. (a) Calculae he diance reled in he fir leg: d m/ 1.5 min 60 /min 1080 m. Calculae he diance reled in he d 0 m/.5 min 0 m econd leg:. Calculae he diance reled in he hird d 15 m/.5 min 60 /min 50 m leg:
3 4. Calculae he oal diance reled: d d1 d d 0 m 5. Draw he graph: 6. (b) Divide he oal diance by he ime elaped: d d d m 7.5 min 60 /min 7.4 m/ Inigh: The erage peed i a weighed erage according o how much ime you pend reling a each peed. Here you pend he mo amoun of ime a re, o he erage peed i le han eiher 1 m/ or 15 m/.. Picure he Problem: You rel in a raigh line a wo differen peed during he pecified ime inerval. Sraegy: Deermine he erage peed by fir calculaing he oal diance reled and hen dividing i by he oal ime elaped. Soluion: 1. (a) The diance inerval are he ame bu he ime inerval are differen. You will pend more ime a he lower peed han a he higher peed. Becaue he erage peed i a ime weighed erage, i will be le han 5.0 m/.. (b) Divide he oal diance by he ime elaped: d d d d 1 d d m/ mi 10.0 mi 10.0 mi 0.0 m/ 0.0 m/ Inigh: Noice ha in hi cae i i no neceary o conver mile o meer in boh he numeraor and denominaor becaue he uni cancel ou and lee m/ in he numeraor. 4. Picure he Problem: The given poiion funcion indicae he paricle begin reling in he negaive direcion bu i acceleraing in he poiive direcion. Sraegy: Creae he x-veru- plo uing a preadhee, or calculae individual value by hand and kech he curve uing graph paper. Ue he known x and informaion o deermine he erage velociy. To find he erage peed, we mu find he oal diance ha he paricle rel beween 0 and 1.0, and hen divide by 1.0.
4 Soluion: 1. (a) Ue a preadhee or imilar program o creae he plo hown a righ. Noe ha he erage velociy over he fir econd of ime i equal o he lope of a raigh line drawn from he origin o he curve a = 1.0. A ha ime he poiion i.0 m.. (b) Find he erage velociy from = 0 o = 1.0 : v x 5 m/ 1.0 m/ m 1.0. (c) Find he ime a which x = 0: 0 5 m/ m/ 5 m/ m/ The ime a which he paricle urn around i half he ime found in ep. Find x a he urnaround ime: 5. A = 1, he paricle i a x = m, o i ha reled an addiional 0.08 m afer urning around. Find he erage peed: x 5 m/ 5 6 m/ m m. m/ 1.0 Inigh: The inananeou peed i alway he magniude of he inananeou velociy, bu he erage peed i no alway he magniude of he erage velociy. For inance, in hi problem he paricle reurn o x = 0 afer 1.67, a which ime i erage peed i 4.17 m m/, bu i erage velociy i zero becaue x = Picure he Problem: The given poiion funcion indicae he paricle begin reling in he poiive direcion bu i acceleraing in he negaive direcion. Sraegy: Creae he x-veru- plo uing a preadhee, or calculae individual value by hand and kech he curve uing graph paper. Ue he known x and informaion o deermine he
5 erage peed and velociy. Soluion: 1. (a) Ue a preadhee o creae he plo:. (b) Find he erage v velociy from = 0.5 o = 0.45 : m/ 0.45 m/ 0.45 m/ 0.5 m/ x m/. (c) Find he erage v velociy from = 0.9 o = 0.41 : m/ 0.41 m/ 0.41 m/ 0.9 m/ x m/ 4. (d) The inananeou peed a = 0.40 will be cloer o 0.56 m/. A he ime inerval become maller he erage velociy i approaching 0.56 m/, o we conclude he erage peed over an infinieimally mall ime inerval will be very cloe o ha value. Inigh: Noe ha he inananeou velociy a 0.40 i equal o he lope of a raigh line drawn angen o he curve a ha poin. Becaue i i difficul o accuraely draw a angen line, we uually reor o mahemaical mehod like hoe illuraed above o deermine he inananeou velociy.. Picure he Problem: The runner accelerae uniformly along a raigh rack. Sraegy: The change in velociy i he erage acceleraion muliplied by he elaped ime. Soluion: 1. (a) Muliply he acceleraion by v v0 a 0 m/ 1.9 m/.0.8 m/ he ime:. (b) Muliply he acceleraion by he ime: v v0 a 0 m/ 1.9 m/ m/ Inigh: World cla priner he op peed over 10 m/, o hi ahlee in' bad, bu i ook him a whole 5. econd o ge up o peed. He hould work on hi acceleraion! 4. Picure he Problem: The airplane low down uniformly along a raigh runway a i rel oward he ea. Sraegy: The erage acceleraion i he change of he velociy divided by he elaped ime. Aume ha ea i in he poiive direcion.
6 Soluion: he ime: 1. Divide he change in velociy by a v m/ m/. We noe from he previou ep ha he acceleraion i negaive. Becaue ea i he poiive direcion, negaive acceleraion mu be oward he we. Inigh: In phyic we almo never alk abou deceleraion. acceleraion. Inead, we call i negaive
7 5. Picure he Problem: The car rel in a raigh line due norh, eiher peeding up or lowing down, depending upon he direcion of he acceleraion. Sraegy: Ue he definiion of acceleraion o deermine he final velociy over he pecified ime inerval. Soluion: 1. (a) Evaluae equaion v v0 a 18.1 m/ 1.0 m/ m/ norh -7 direcly:. (b) Evaluae equaion -7 direcly: v v0 a 18.1 m/ 1.15 m/ m/ nor Inigh: In phyic we almo never alk abou deceleraion. Inead, we call i negaive acceleraion. In hi problem ouh i conidered he negaive direcion, and in par (b) he car i lowing down or undergoing negaive acceleraion. 4. Picure he Problem: The paricle rel in a raigh line in he poiive direcion while acceleraing in he poiive direcion (peeding up). Sraegy: Ue he conan acceleraion equaion of moion o find he iniial velociy. Soluion: Solve equaion -7 v0 v a 9.1 m/ 6.4 m/ m/ for v 0 : Inigh: A expeced he iniial velociy i le han he final velociy becaue he paricle i peeding up.
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