Motion in One Dimension
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- Joel Lawrence
- 6 years ago
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1 Moon n One Dmenson CHAPTER OUTLINE Poson, Veloy, nd Speed Insnneous Veloy nd Speed 3 Aeleron 4 Moon Dgrms 5 One-Dmensonl Moon w Consn Aeleron 6 Freely Fllng Objes 7 Knem Equons Dered rom Clulus ANSWERS TO QUESTIONS Q I I oun 5 s beween lgnng nd under, e sound s reled 33 ms 5 s 7 km Te rns me or e lg s smller by b g 8 3 ms ms so s neglgble n omprson 5 mes, Q Yes Yes, e prle wnds up n e + regon e end Q3 Zero Q4 Yes Yes Q5 No Consder sprner runnng srg-lne re Hs erge eloy would smply be e leng o e re dded by e me ook or m o omplee e re I e sops long e wy o e s soe, en s nsnneous eloy pon would be zero Q6 We ssume e obje moes long srg lne I s erge eloy s zero, en e dsplemen mus be zero oer e me nerl, ordng o Equon Te obje mg be sonry rougou e nerl I s mong o e rg rs, mus ler moe o e le o reurn o s srng pon Is eloy mus be zero s urns round Te grp o e moon sown o e rg represens su moon, s e nl nd nl posons re e sme In n s grp, e nsnneous eloy ny me s e slope o e ure pon A n e grp, e slope o e ure s zero, nd us e nsnneous eloy me s lso zero FIG Q6 Q7 Yes I e eloy o e prle s nonzero, e prle s n moon I e eleron s zero, e eloy o e prle s unngng, or s onsn
2 Moon n One Dmenson Q8 Yes I you drop dougnu rom res, en s eleron s no zero A ommon msonepon s mmedely er e dougnu s relesed, bo e eloy nd eleron re zero I e eleron were zero, en e eloy would no nge, leng e dougnu long res n md-r Q9 No: Cr A mg e greer eleron n B, bu ey mg bo e zero eleron, or oerwse equl elerons; or e drer o B mg e rmped rd on e gs pedl n e reen ps Q Yes Consder rowng bll srg up As e bll goes up, s eloy s upwrd >, nd s eleron s dreed down < A grp o s or s suon would look lke e gure o e rg Te eleron s e slope o s grp, nd s lwys nege n s se, een wen e eloy s pose FIG Q Q () Aelerng Es Brkng Es () Crusng Es (d) Brkng Wes (e) Aelerng Wes () Crusng Wes (g) () Sopped bu srng o moe Es Sopped bu srng o moe Wes Q No Consn eleron only Yes Zero s onsn Q3 Te poson does depend on e orgn o e oordne sysem Assume e l s m ll, nd e sone rees mmum eg o m boe e op o e l I e orgn s ken s e op o e l, en e mmum eg reed by e sone would be m I e orgn s ken s e boom o e l, en e mmum eg would be 3 m Te eloy s ndependen o e orgn Sne e nge n poson s used o lule e nsnneous eloy n Equon 5, e oe o orgn s rbrry Q4 One e objes lee e nd, bo re n ree ll, nd bo eperene e sme downwrd eleron equl o e ree-ll eleron, g Q5 Tey re e sme Aer e rs bll rees s pe nd lls bk downwrd ps e suden, wll e downwrd eloy equl o Ts eloy s e sme s e eloy o e seond bll, so er ey ll roug equl egs er mp speeds wll lso be e sme Q6 W g, () 5 g 77 Te me s ler n 5 Te dsne llen s 5 g 5 Te eleon s 75, greer n 5
3 Cper 3 Q7 Aboe Your bll s zero nl speed nd smller erge speed durng e me o lg o e pssng pon SOLUTIONS TO PROBLEMS Seon Poson, Veloy, nd Speed P () 3 m s 57 5 m 9 m 6 m s 3 s *P () 57 5 m m () 5 s 5 ms F m H G I K J F yr I 7 HG K J ms 7 yr s F m H G I K J F yr I HG K J 7 yr s or n prulrly wndy mes 6 ms Te me requred mus e been 3 m 69 m mm mm yr m m F H G I K J F H G I K J yr P3 () m s 5 ms 5 m 4 s ms () 5 m m 5 4 s s ms (d) 5 m 5 m 33 7 s 4 s ms (e) P4 : For s m 8 m s () 5 m s 5 ms 4 m s 4 ms
4 4 Moon n One Dmenson P5 () Le d represen e dsne beween A nd B Le be e me or w e wlker s e ger speed n 5 m s d Le represen e longer me or e reurn rp n d d 3 m s Ten e mes re b5 m sg nd d Te erge speed b3 m sg s: Tol dsne d+ d d d d Tol me + b8 msgd b5 ms g b3 msg e5 m sj 5 e m sj 375 ms 8 ms Se srs nd nses e sme pon A W ol dsplemen, erge eloy Seon Insnneous Veloy nd Speed e j e j P6 () A ny me,, e poson s gen by 3 m s Tus, 3 s: 3 ms 3 s 7 m e j A 3 s+ : 3 ms 3 s + () Te nsnneous eloy 3 s s:, or b g e j 7 m + 8 m s + 3 m s F HG I e e j j KJ + lm lm 8 ms 3 ms 8 ms P7 () 5 s, 8 m (Pon A) 4 s, m (Pon B) m s 6 m 4 5 s ms Te slope o e ngen lne s ound rom pons C nd D bc s, C 95 mg nd bd 35 s, D g, 38 m s FIG P7 () Te eloy s zero wen s mnmum Ts s 4 s
5 Cper 5 P8 () 58 m A 5 s, e slope s 5 s 54 m A 4 s, e slope s 3 s 3 ms 8 ms 49 m A 3 s, e slope s 34 s 4 ms 36 m A s, e slope s 4 s 9 m s () 3 ms 5 s 46 ms (d) Inl eloy o e r ws zero P9 () ( 5 ) m ( ) s 5 ms ( 5 ) m 5 ( 4 ) s ms ( 5 m 5 m) () ( 5 s 4 s) (d) ( 5 m ) ( 8 7 ) + 5 s s ms FIG P9 *P One resumes e re, e re wll run or me o m 8 m 5 s 8 m s In s me, e orose n rwl dsne ms ( 5s ) 5 m
6 6 Moon n One Dmenson Seon 3 Aeleron P Coose e pose dreon o be e ouwrd dreon, perpendulr o e wll 4 ms ms 5 ms + : s P () Aeleron s onsn oer e rs en seonds, so e end, + + ms ( s ) ms Ten so s onsn rom s o 5 s And oer e ls e seonds e eloy nges o + ms + 3 ms ( 5 s ) 5 ms In e rs en seonds, ( ) ms s m Oer e ne e seonds e poson nges o And s, + + m + m s ( 5 s )+ m + + m + m s ( 5 s )+ 3 m s ( 5 s ) 6 m *P3 () dsne reled Te erge speed durng me nerl s Durng e rs qurer mle segmen, Serer s erge speed ws 5 m 3 5 s 5 s 5 4 s b35 6 m g Durng e seond qurer mle segmen, 3 4 s 55 s b37 4 m g For e rd qurer mle o e re, s 55 5 s b37 7 m g, nd durng e nl qurer mle, onnued on ne pge s 57 4 s b39 m g
7 Cper 7 Assumng 4 nd reognzng, e erge eleron durng e re ws 57 4 s ol elpsed me ( ) s 598 s P4 () Aeleron s e slope o e grp o s For < < 5 s, For 5 s < < s, For 5 s < < 5 s, (m/s ) 6 We n plo () s sown 8 ( 8 ) 6 m s 5 5 (s) 5 5 FIG P4 () For 5 s < < 5 s, 5 s, 8 m s, 5 s 8 ms ms (), 8 m s, s, 8 m s 8 ( 8 ) 8 m s d d P5 + 3, 3, d d A 3 s: () ( ) m m ( 3 6 ) ms 3 ms () m s
8 8 Moon n One Dmenson P6 () A s, 3 ( ) ( )+ 3 m m m m A 3 s, so 4 m m 3 s s 3 ms A ll mes e nsnneous eloy s ms d ( 6 ) d A s, 6 ( ) ms ms A 3 s, 63 ( ) ms 6 ms () 6 ms ms 3 s s 6 ms (d) d A ll mes ( 6 ) 6 m s (Ts nludes bo s nd 3 s) d P7 () 8 ms 6 s 3 ms Mmum pose eleron s 3 s, nd s ppromely m s (), 6 s, nd lso or > s (d) Mmum nege eleron s 8 s, nd s ppromely 5 m s Seon 4 Moon Dgrms P8 () () (d) (e) onnued on ne pge
9 Cper 9 () One wy o prsng e nswer: Te spng o e suesse posons would nge w less regulry Anoer wy: Te obje would moe w some ombnon o e knds o moon sown n () roug (e) Wn one drwng, e elerons eors would ry n mgnude nd dreon Seon 5 One-Dmensonl Moon w Consn Aeleron 3, so P9 From +, we e 97 ms + ( m ) w s 79 4 mes g P () + beomes 4 m + 8 m s ( 85 s) 8 ms 66 ms 448 ms 85 s 74 5 m s w yelds 66 P Gen m s wen 3 m ( ), nd s, 5 m, + : 5 3 ( )+ ( ) m s *P () Le be e se o mong 6 m nd be res Smlrly, + F + H G m I K J 6 m d b g 36m 4 58 m 36 s 58 8 m s F H G I K J F H G I K J F 69 8 H G I K J F m 975 H G I m s m 36 s K J ms b g 8 m b g m s m s 9 94 m s () Le be mong 8 m nd be mong 6 m b g + d b g b g 6 m 8 m + 8b5 8g m s 8 m s m s 9 36 b g m s
10 3 Moon n One Dmenson *P3 () Coose e nl pon were e plo redues e role nd e nl pon were e bo psses e buoy:, m, 3 m s,?, 35 m s,? + + : We use e qudr ormul: m + 3 m s + 35 m s 75 m s 3 m s + m 4 b b ± 3 ms ± 9 m s 4 75 ms ( m) 3 ms ± 4 ms 6 s 75 ms 35 ms or 453 s Te smller lue s e pysl nswer I e bo kep mong w e sme eleron, would sop nd moe bkwrd, en gn speed, nd pss e buoy gn 6 s + 3 ms 3 5 ms 4 53 s 4 ms P4 () Tol dsplemen re under e, ure rom o 5 s b5 ms g5 s + b5 ms g4 5 s + b5 ms g s 875m From s o 4 s, dsplemen s () ( 5 ) ms 5 s: 5 s 33 ms 5 s < < 4 s: 5 4 s 5 s: 3 ms 5 ms 5 s 4 s onnued on ne pge e j b g b g FIG P4
11 (d) () + 33 m s or 67 m s () ( s ) m s + m s ( s) () For 4 s 5 s, F HG or re under s 3 3 rom o 4 s I + KJ 5 ms 375 m ( 4 s ) + 5 m s ( 4 s) Cper 3 or e j b g m + 5 m + 5 m s 4 s + 5 m s 4 s w redues o 5 ms 5 ms m 3 b g e j ol dsplemen 875 m (e) ol elpsed me 5 s 37 5 ms P5 () Compre e poson equon o e generl orm + + o reognze m, 3 m s, nd 8 m s Te eloy equon, +, s en 3 ms 8 ms Te prle nges dreon wen, w ours 3 8 me s: s Te poson s F 3 + H G I K J F 3 H G I m 3 m s s 4 m s s K J 56 m 8 From + +, obsere wen, e me s gen by Tus, wen e prle reurns o s nl poson, e me s 3 ms 3 8 ms 4 nd e eloy s F 3 H G I 3 ms 8 ms s K J 3 ms 4 s 8
12 3 Moon n One Dmenson *P6 Te me or e Ford o slow down we nd rom Is me o speed up s smlrly m s ms d 35 ( m ) 979 s + 75 ms Te wole me s mong less n mmum speed s 699 s + 5 s s 8 s Te Meredes rels ms m d b g wle e Ford rels m 6 m, o ll bend by 558 m 6 m 958 m P7 () m s, 5 m s, + so 5, + so ( ) ( 5 ) Tus m nd s A s eleron e plne would oersoo e runwy: No P8 () Tke e boom o e ll were, 3 m s, m s Use ese lues n e generl equon + + o nd wen s n seonds + 3 ms + ms m 3 To nd n equon or e eloy, use + 3 ms + ms ( 3 ) m s e j, Te dsne o rel beomes mmum, m, wen (urnng pon n e moon) Use e epressons ound n pr () or o nd e lue o wen s s mmum lue: From ( 3 ) m s, wen 5 s Ten m m m 3 ( 3 )( 5 ) ( 5 ) 5
13 Cper 33 P9 In e smulneous equons: R S T U R V we e S W T + + U V W 56 ms ( 4 s) 6 4 m + ( 4 s) So subsung or ges 6 4 m + 56 m s ( 4 s )+ ( 4 s) Tus 4 9 ms ms ( s) 3 m s P3 Tke ny wo o e sndrd our equons, su s subsue no e oer: Tus R S T U V Sole one or, nd W Bk n problem 9, 6 4 m ( 4 s ) 56 m s ( 4 s) P3 () e 58 j 6 4 m 49 4 m 4 s s m s 4 F + I 63 H G 58 3 ms 36KJ m
14 34 Moon n One Dmenson P3 () Te me kes e ruk o re m s s ound rom + Solng or yelds ms ms s ms Te ol me s us s + s + 5 s 35 s Te erge eloy s e ol dsne reled dded by e ol me ken Te dsne reled durng e rs s s F I HG K J( ) + m W beng or s nerl, e dsne reled durng e ne s s Te dsne reled n e ls 5 s s + ( )( )+ 4 m 3 F + I HG K J( ) 5 5 m Te ol dsne m, nd e erge eloy s gen by m s P33 We e 4 m s, 6 6 m s, 5 m () + : 5 m ms + 6 ms 9 s + d : 6 4 e6 ms j e msj ( ) ( 5 m) 5 ms
15 Cper 35 *P34 () 8 + : 3 ms + ( 4 m) 6 3 ms 8 m ms We mus nd seprely e me or speedng up nd e me or osng: d e j : m m s 5 67 s d e j : 6 m 3 ms + 3 ms 5 s ol me s *P35 () Along e me s o e grp sown, le nd m Ten + ges + m m m m Te dsplemen beween nd m s Te dsplemen beween m nd s Te ol dsplemen s + + m + m + m + F HG m m m m () For onsn nd, s mnmzed by mmzng m o m Ten F mn HG I K J I K J (e) Ts s relzed by ng e sero moor on ll e me (d) We mmze by leng m ppro zero In e lm (e) Ts nno be ned beuse e eleron mus be ne
16 36 Moon n One Dmenson *P36 Le e glder ener e pooge w eloy nd moe w onsn eleron For s moon rom enry o e, d + d d d d d () Te speed lwy roug e pooge n spe s gen by F s + d d H G I K J + + nd s s no equl o d unless s d d F d Te speed lwy roug e pooge n me s gen by + H G I K J equl o d s deermned boe nd s s P37 () Tke nl nd nl pons op nd boom o e nlne I e bll srs rom res, Ten, 5 m s, 9 m ms 9 m 3 ms d e j e 6 s ms j () (d) Tke nl nd nl pons e boom o e plnes nd e op o e seond plne, respeely: 3 m s,, 5 m + ges 3 ms 5 ( m) 3 ms Tke e nl pon e boom o e plnes nd e nl pon 8 m long e seond: 3 m s, 8 m, 3 m s + 3 ms + 3 ms 8 m 4 m s 5 ms d b g e j
17 Cper 37 P38 Tke e orgnl pon o be wen Sue noes e n Coose e orgn o e -s Sue s r For er we e s, s 3 m s, s m s so er poson s gen by s() s+ s+ s 3 ms + ms For e n, 55 m, 5 m s, nd () m s + To es or ollson, we look or n nsn wen bo re e sme ple: From e qudr ormul ± ( 5 ) 4( 55) 3 6 s or 4 s Te smller lue s e ollson me (Te lrger lue ells wen e n would pull ed gn e eles ould moe roug e oer) Te wrek ppens poson 55 m + 5 m s ( 4 s ) m *P39 As n e lgebr soluon o Emple 8, we le represen e me e rooper s been mong We grp nd r rooper 5 (km) 5 5 r pole oer Tey nerse 3 4 (s) 3 s FIG P39
18 38 Moon n One Dmenson Seon 6 Freely Fllng Objes P4 Coose e orgn y, e srng pon o e bll nd ke upwrd s pose Ten y,, nd g 98 m s Te poson nd e eloy me beome: nd y y + : y g 98 m s e + : g 98 m s () s: y 98 ms ( s ) 49 m s: y 9 8 ms ( s ) 9 6 m 3 s: y 9 8 ms ( 3 s ) 44 m s: 98 ms ( s ) 98 ms s: 9 8 ms ( s ) 9 6 ms 3 s: 9 8 ms ( 3 s ) 9 4 ms P4 Assume r ressne my be negleed Ten, e eleron ll mes durng e lg s due o gry, g 98 m s Durng e lg, Go wen mle ( 69 m) up nd en mle bk down Deermne s speed jus er lun by onsderng s upwrd lg: d e jb g + y y : 98 ms 69 m 78 ms Hs me n e r my be ound by onsderng s moon rom jus er lun o jus beore mp: y y + : 78 ms 98 ms Te roo desrbes lun; e oer roo, 36 s, desrbes s lg me Hs re o py my en be ound rom b gb g $ py re 76 $s 36 s $ s We e ssumed e workmn s lg me, mle, nd dollr, were mesured o reedg preson We e nerpreed up n e sky s reerrng o e ree ll me, no o e lun nd lndng mes Bo e keo nd lndng mes mus be seerl seonds wy rom e job, n order or Go o sure o resume work j
19 P4 We e y g + + y 49 ms 8 ms + 3 m Cper 39 Solng or, 8 ± Usng only e pose lue or, we nd 79 s P43 () y y + : 4 ( 5 ) ( 49 )( 5 ) nd m s upwrd + ( 9 8)( 5) 4 68 m s 468 m s downwrd P44 Te bll srs rom res nd lls w downwrd eleron o 98 m s (due o gry) Tus, n s wll ll dsne o y g 4 9 ( ) ms s m Ts dsne s bou we e dsne beween e ener o e bll nd s op edge 8 m Tus, Dd wll be unsuessul *P45 () From y + w, we e m y ( 3 ) 98 ms 7 s Te nl eloy s + 98 ms ( 7 s ) ms () Te me ke or e sound o e mp o re e speor s sound y 3 m 676 s, 34 m s sound so e ol elpsed me s ol 7 s s 3 s
20 4 Moon n One Dmenson P46 A ny me, e poson o e bll relesed rom res s gen by y g A me, e poson o e bll rown erlly upwrd s desrbed by y g Te me w e rs bll s poson o y s ound rom e rs equon s g, w yelds To requre e seond bll e poson o y s me, use e seond g equon o obn F I g g H G Ts ges e requred nl upwrd eloy o e seond gkj bll s g P47 () g: wen 3 s, g 98 m s Tereore, y y + y y *P48 () Consder e upwrd lg o e rrow g 9 8 ms ( 3 s ) 9 4 ms b g ms s m y y y b ms g + e 98 msj + y y m s y 9 6 ms d 5 m y Consder e wole lg o e rrow y y + y+ y + b ms g+ e 98 ms j Te roo reers o e srng pon Te me o lg s gen by ms 49 ms 4 s P49 Tme o ll 3 m s ound rom Eq w, 3 98 m m s, 78 s () W e orse gllopng m s, e orzonl dsne s 78 m 78 s
21 Cper 4 P5 Tke downwrd s e pose y dreon () Wle e womn ws n ree ll, y 44,, nd g3 s Tus, y s gng ll 3 s Her eloy jus beore mp s: + g + 3 s ( 3 s ) 96 s Wle rusng e bo, 96 s,, nd y 8 n 5 Tereore, 96 s y 5 ( ) 37 () Tme o rus bo: y y 5 ( ) s P5 y 3 3 : A s, y m nd 3 s, or 37 3 s upwrd or 33 s dy A y d 9 36 m s I e eloper releses smll mlbg s me, e equon o moon o e mlbg s yb yb + g ( ) 98 Seng y b, Solng or, (only pose lues o oun), 796 s *P5 Consder e ls 3 m o ll We nd s speed 3 m boe e ground: y y + y+ y 3 m + y 5 s + 98 m s 5 s y 3 m + m 6 ms 5 s e j Now onsder e poron o s ll boe e 3 m pon We ssume srs rom res y y + y y y b 6 ms g + e 9 8 msj d y 6 m s y 86 m 9 6 ms Is orgnl eg ws en 3 m m 38 m
22 4 Moon n One Dmenson Seon 7 Knem Equons Dered rom Clulus d P53 () J onsn d d Jd z J d J+ bu wen so Tereore, J+ d d d d z zb g d J+ d J + + bu wen, so nd J + + d d d d z z d J + + d 3 J F HG I K J 3 wen, so 3 Tereore, J J+ J + +J J J F HG I K J + J J + Rell e epresson or : J + + So J + Tereore, + J
23 Cper 43 P54 () See e grps e rg Coose A 3 s, ( ) 8 3 A 5 s, m + 8 m s ( s ) 8 m A 7 s, 8 m + ( ) 8 m s s 36 m 8 ms For < < 3 s, For 3< < 5 3 s s, 67 ms 6 ms () For 5 s < < 9 s, 4 4 s ms (d) A 6 s, 8 m + 6 m s ( s ) 34 m (e) A 9 s, 36 m + 8 m s ( s ) 8 m FIG P54 d d 7 5 P55 () d d d Tke Ten d ms + 3 ms z z d d e ms ms e j e j Te bulle espes wen, ms + 3 ms s 3 3 s () New ms + 9 ms 45 ms (d) m + 35 m 9 m j
24 44 Moon n One Dmenson d P56 3, 5 m s d Solng or, d 3 d z d 3 d + 3 or 3 z Wen, 3 s Addonl Problems *P57 Te dsne e r rels onsn eloy,, durng e reon me s r Te me or e r o ome o res, rom nl eloy, er e brkes re ppled s nd e dsne reled durng s brkng perod s F + I F + I K J HG KJ HG Tus, e ol dsne reled beore omng o sop s ssop + r F HG I K J *P58 () I r s dsne ssop r (See e soluon o Problem 57) rom e nerseon o leng s wen e lg urns yellow, e dsne e r mus rel beore e lg urns red s ssop + s r + s Assume e drer does no elere n n emp o be e lg (n eremely dngerous pre!) Te me e lg sould remn yellow s en e me requred or e r o rel dsne onsn eloy Ts s r s s lg r + + W s 6 m, 6 km, m s, nd r s, 6 km ms lg s e F jhg 78 ms km I KJ + F H G 6 m km 6 km 78 ms I KJ 63 s
25 Cper 45 *P59 () As we see rom e grp, rom bou 5 s o 5 s Ael s rusng onsn pose eloy n e + dreon From 5 s o s, Ael eleres n e + dreon reng op speed o bou 7 m/ Around s, e engneer pples e brkes, nd e rn, sll relng n e + dreon, slows down nd en sops 35 s Jus er 35 s, Ael reerses dreon ( beomes nege) nd sedly gns speed n e dreon 5 3 FIG P59() 4 (s) Te pek eleron beween 45 nd 7 m/ s gen by e slope o e seepes ngen o e ersus ure n s nerl From e ngen lne sown, we nd ( 55 45) m slope m s 98 m s ( 5) s () Le us use e e re under e ersus ure equls e dsplemen Te rn s dsplemen beween nd s s equl o e re o e gry sded regon, w we e ppromed w seres o rngles nd rengles re + re + re + re + re b5 m g5 s + b5 m g5 s + b6 m g s + 5 s b m g + s b7 m 6 m g 4 m s s b g FIG P59() 4 (s) Now, e end o our lulon, we n nd e dsplemen n mles by onerng ours o seonds As 3 6 s, 4 m s 36 s F H G I K J s 67 m
26 46 Moon n One Dmenson *P6 Aerge speed o eery pon on e rn s e rs r psses Lz: 86 m s ms Te rn s s s s nsnneous speed lwy roug e 5 s me Smlrly, lwy roug e ne s, e speed o e rn s 86 m 78 ms Te me requred or e speed s o nge rom 573 m/s o 78 m/s s 5 ( s )+ ( 3 s ) s so e eleron s: 78 ms 573 ms 3 s 6 ms P6 Te re o r grow s eloy nd e re o s nrese s n eleron Ten 4 mm d nd F mm d H G I 3 K J Te nrese n e leng o e r (e, dsplemen) w durng me o 5 w 35 d s or 48 mm + 4 mm d 35 d + 3 mm d w 35 d 5 w b g b g P6 Le pon be ground leel nd pon be e end o e engne burn Le pon be e ges pon e roke rees nd pon 3 be jus beore mp Te d n e ble re ound or e pse o e roke s moon b g so m s ( o ) ( 4 ) gng s ( o ) ( ) ( 9 8) gng 98 gng s Ts s e me o mmum eg o e roke b g ( o 3) ( 9 8) gng 8 8 s 735 m FIG P6 () ol s 73 km ol onnued on ne pge
27 Cper 47 () nl 84 ms Lun 8 +4 # End Trus +4 # Rse Upwrds #3 Fll o Er P63 Dsne reled by moors 5 m s Dsne reled by polemn m s () nerep ours wen 5, or 5 s ( oer ) m s 3 m s () ( oer ) m s 5m P64 Are A s rengle Tus, A w Are A s rngulr Tereore A b b g Te ol re under e ure s nd sne A A + A + b g A A FIG P64 A + Te dsplemen gen by e equon s: +, e sme resul s boe or e ol re
28 48 Moon n One Dmenson P65 () Le be e dsne reled eleron unl mmum speed s reed I s s eed n me we n use e ollowng ree equons:, + nd + Te rs wo ge F I HG K J 4 b g F HG I K J For Mgge: 8 4 For Judy: m s 383 m s Mgge: m s Judy: m s () A e s-seond mrk + 6 Mgge: m Judy: m Mgge s ed by 6 m P66 m s 5 m s m nd F F + HG H G F I I KJ + I KJ HG s Tol me KJ 9 s 55 s
29 Cper 49 P67 Le e bll ll 5 m I srkes speed gen by + 98 ms ( 5 m) + : nd s soppng s desrbed by 5 4 m s d b g e j ms + m m s + 47 ms m Is mmum eleron wll be lrger n e erge eleron we esme by mgnng onsn eleron, bu wll sll be o order o mgnude ~ 3 ms *P68 () + + We ssume e pkge srs rom res 45 m m s ( 45 m) 98 ms 544 s ms ( 5 8 s ) 3 m dsne llen 3 m () speed ms 5 8 s 5 8 ms e j (d) Te remnng dsne s 45 m 3 5 m 3 5 m Durng deeleron, 5 8 m s,, 3 5 m + : 58 ms + ( 35 m) 58 m s 35 m ms 95 3 ms upwrd
30 5 Moon n One Dmenson P69 () y ( ) ( 4 9 ) 5 ) 49 ( ) Only e pose roo s pyslly menngul: y + nd 3 s subsue 5 ( )+ ( 98 )( ) : 3 s er e rs sone s rown 5 3 m s downwrd () + +( 98 )( 3 ) 34 m s downwrd ( 9 8)( ) 34 8 m s downwrd P7 () d ( ) 98 d ± ( 4 9)( 8 ) ± s so d m 98 Ignorng e sound rel me, d ( 9 8)( 4) 8 m, n error o 68% P7 () In wlkng dsne, n me, e leng o rope s only nresed by snθ Te pk ls re () snθ boy boy + d d d F + d d d H G I boy K J boy boy boy d boy, bu d boy d d F I HG KJ boy boy boy + 3 boy, (d) boy, FIG P7
31 P7 6 m, boy ms snθ boy boy + Howeer, boy : b () s m s g boy 4 boy From problem 7 boe, + e s m s 67 j Cper 5 FIG P7() boy boy boy FIG P7 P73 () We requre wen + s e j e j e j k k s k ms ms + 83 s k k k k e jb g e jbg k 546 s k 4 9 ms 5 46 s 73 m () k 4 9 ms 5 46 s 6 7 ms s 3 5 ms 6 46 s 6 ms
32 5 Moon n One Dmenson P74 Tme (s) Heg (m) (m) (s) (m/s) mdp me (s) TABLE P74 FIG P74 eleron slope o lne s onsn 63 ms 63 ms downwrd
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34 54 Moon n One Dmenson P44 No; see e soluon P6 6 m s P46 Te seond bll s rown speed g P48 () 5 m; 4 s P5 () 96 s; 37 3 s upwrd; () 33 s P5 38 m P54 () nd see e soluon; () 4 m s ; (d) 34 m; (e) 8 m P56 s P6 () 4 s; 73 km; () 84 m s P64 + ; dsplemens gree P66 55 s; 9 s P68 () 544 s; 3 m; () 5 8 m s; (d) 95 3 m s upwrd P7 () 64 m; 68% P7 see e soluon P74 see e soluon; 63 m s P58 () see e soluon; 63 s
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