Chapter 4: Motion in Two Dimensions Part-1
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1 Lecue 4: Moon n Two Dmensons Chpe 4: Moon n Two Dmensons P- In hs lesson we wll dscuss moon n wo dmensons. In wo dmensons, s necess o use eco noon o descbe phscl qunes wh boh mnude nd decon. In hs chpe, we wll ben b denn dsplcemen, eloc nd cceleon s ecos n wo dmensons. Then, we wll dscuss he soluon o pojecle moon poblems n wo dmensons, such s he moon o cnnon ed e n nle, he moon o cl de jumped sh o o he moon o nucle bomb dopped om he heh. In he s secon, some denons e en. In he second secon, deons o he equons o moon n wo-dmensons e shown. The equons o he unom speed ccul moon nd he non unom speed ccul moon e deed n he hd secon. In he ls secon, he ele moon n wo dmensons s conned. Anlcl nd numecl emples e soled he end o ech secon. Dsplcemen, Veloc nd Acceleon n -Dmensons As we menoned n -dmenson, he eco nue o eloc nd cceleon s ken no ccoun b he sn pose o nee o he qun. In -dmensons we mus use componens o spec eloc o cceleon eco. Th s he meel deence, n equons, whch m be enouh o mke somehn dcul! I hee s eco ln n he X-Y plne, cn be wen s componen n he X-decon dded o componen n he Y-decon. Le A be wo-componen eco n he X-Y plne. Then s wen s A A ˆ A ˆ j whee A nd A e he X nd Y componens o he eco A See Fue 4.-b. Fue 4.: Componens o eco Fom F. b, we see h A Acos nd Asn A, nd lso A A I hee wo ecos, hee wll be hd one whose mnude nd decon s ound b he eco opeon on he ohes. Le us cll C be he esuln eco o he ddon o A wh B. Then, s wen s; A
2 Lecue 4: Moon n Two Dmensons A C C C whee A ˆ A nd B B ˆ B C ˆ C A B A B ˆ A B C A B nd C A B. See Fue 4. o he epesenon. Fue 4.: Veco ddon, esuln eco nd he componens I s seen lso n Fue 4. h he esuln eco epesens chne n poson o n objec om pon A o pon B. So, he esuln eco s dsplcemen eco o n objec h moes om A o B. Dsplcemen In Fue 4.3, n objec s nll poson me pon A. Some me le,, he objec s poson pon B. The dsplcemen eco o he objec s en b: 4. ˆ j B A ˆ Fue 4.3: Dsplcemen eco
3 Lecue 4: Moon n Two Dmensons Aee Veloc Usn he esul o dsplcemen, we cn nd he ee eloc o he objec beween me nels: Tol Dsplcemen Elpsed Tme 4. Genell, he nl condons e ssumed o be pon. We wll use hs noon o he nl condon e h pon. As wh he -dmensonl denon, ee eloc s ndependen o he ph beween he end pons. Insnneous Veloc As s menoned n he peous chpe, he nsnneous eloc s en b d lm 4.3 d Aee Acceleon I s he chne n eloc oe he chne n me: 4.4 The decon o he cceleon s n he decon o he eco /., nd s mnude s Insnneous Acceleon As we he ollowed beoe, Insnneous cceleon s clculed b kn shoe nd shoe me nels,.e. when, hen; d lm 4.5 d Noe: pcle cn ccelee n deen ws:. The mnude o cn chne n me, whle he decon o moon ss he sme.. The mnude o,, cn s consn, whle he decon o moon chnes. Ths onl hppens n moe hn one dmenson. 3. Boh nd he decon o cn chne. Moon n D wh Unom Consn Acceleon We know h
4 Lecue 4: Moon n Two Dmensons 4 In cse o unom cceleon, he ee cceleon would be equl o he nsnneous cceleon. Snce he denon o ee cceleon s s boe, hen he nsnneous cceleon m be wen s 4.7 Fom h equon, we cn we he eloc equon s. 4.8 As he moon s n -dmensonl spce, he un ecos lon nd -s wll be î nd ĵ, especel. So kn he ecnul componens o cceleon een s unom! nd he nl eloc, we e j ˆ ˆ. 4.9 Ths s he equon o eloc o n objec wh unom cceleon n -dmensonl moon. Noe h he sub-ndces nd show he nl lues o he pmees lon he -s nd -s. The poson eco o he objec s wen b usn Eq. 4., 4. Snce he eloc o he objec nceses unoml, hen we cn we 4. s we he done n he peous Chpe. Replcn hs esul no he Eq. 4., hen we e he poson eco o he objec n ems o eloc nd cceleon: 4. nd snce he nl eloc s en s n Eq. 4.8, hen he esul o he poson eco n ems o he nl lues cn be en s. 4.3 I s noed h ll bles e n -dmenson. Fo smplc hs equon m be wen s ollows b compn he coecen o î nd ĵ. Snce 4.4 hen 4.5
5 Lecue 4: Moon n Two Dmensons whee he sub-ndces nd show he nl lues o he pmees lon he -s nd -s, n. Pojecle Moon As we know well, he pojecle moon s pcul knd o dmensonl moon. Fsl, we wll mke he ollown ssumpons: The onl oce pesen s he oce due o. The mnude o he cceleon due o s 9.8m / s. We choose coodne ssem n whch he pose -s pons up pependcul o he eh's suce. Ths denon es us h ˆ j 9.8m / s nd. The oon o he eh does no ec he moon. Inl Condons: We choose he coodne ssem so h he pcle lees he on, me wh n nl eloc o. The Pocedue o Soln Pojecle Moon Poblems e s ollowns:. We wll sepe he moon no he hozonl p nd ecl p.. Then we wll consde ech p sepel usn he ppope equons. The equons o moon, o ech componen, become:. -moon =; I s seen om Eq. 4.9 h he -componen o he eloc wll onl he nl lue. So, ˆ 4.6 cos ˆ cons. ˆ cos ˆ b. -moon ˆ j 9.8m / s ; 4.6b I s seen h onl he onl oce s ppled on he moon o he bod ; hen he -componen o he eloc wll be en s sn 4.7 ˆ sn j ˆ j 4.7b We cn lso we he eloc equon h s me-ndependen s we he done n peous Chpe
6 Lecue 4: Moon n Two Dmensons Noe h hs esul s obousl jus mnude. The decon o he eloc m be decded b he lues o he ls em, poson eco--, o he equon. 3. Fnll, we wll sole he esuln ssem o equons o he unknown qunes. I s lso woh o pcce on he esuls obned boe. I he me ble n Eq. 4.6b s nseed no he Eq. 4.7b, hen s seen h he poson equon o he objec becomes s pbolc equon. n 4.9 cos I mens h he ph o he objec n he pojecle moon s pbolc: b c 4. In he pojecle moon, hee wll be nl dsnce on he -s he objec es. I he Eq. 4.9 s soled o when =, hen he ne o pojecle s ound h sn R 4. I s obous h 45 o he m ne, R m. The m heh h he objec eches cn be ound b usn he Eq Snce he ecl eloc wll be when he objec eches he hhes heh, hen, nd sn, hen sn sn m s ound, see Fue. m, nll 4. Fue 4.4: The pojecle moon The ol me o ll hese moon s ound b usn Eq Snce he eloc o he objec s zeo me when eches he hhes pon bu he hozonl eloc hs some mnude, hen we cn we ˆ j sn 4.3 hen sn 4.4 6
7 Lecue 4: Moon n Two Dmensons Ths s he me epesson o he objec h hs eched he hhes pon, A, whee he ecl componen o he eloc s zeo. I s seen h kes some me o h objec lln om pon A o B snce he pojecle does pbolc ph. So, he ol me o he pojecle o ech om o B n pbolc ph s sn T 4.5 Emples nd Poblems Queson 4.: A bombdmen c hn eloc o 8m/h lees s bomb wh 3 nle downwd n hozonl lne. The hozonl dsnce beween he pon he bomb leed nd he pon whee hs he ound s 7m. Fnd he heh whch he c lees he bomb nd b Fnd he lh me o he bomb. Soluon 4.: We e en: 8m / h 8.5m / s, 3, 7m, The pon whee he bomb s leed s ssumed o be pon. Then ˆ ˆ, sn sn ˆ j j j nd cos cos I we eplce n he equon o heh, hen j ˆ j ˆ sn ˆ, j hen j ˆ sn cos cos The soluon o he s equon o he unknown,, es us h 9 Ths mens h he bomb oes downwd ecll. To nd he me o he bomb o ech he ound s 7m. 5s cos 8.5m / scos3 7
8 Lecue 4: Moon n Two Dmensons Queson 4.: Assume h ou oun sse swns on ope boe he locl swmmn hole on ho d n he summe hold See Fue 4.5. She les o o he ope when he nl eloc s.5 m/s n nle o 35. boe he hozonl. I he lh n kes o. s, how hh boe he we ws she when she le o o he ope? Soluon 4.: We e en:.5m / s 35 Flh. s Fsl, we should nd he eloc ecos n nd - decons: sn.5sn35 cos ˆ.5cos35ˆ.76m / s ˆ j.679m / sˆ A he end o he lh me, she wll ene no he we. Snce he nl heh s hen he ol dsplcemen n he ecl poson s jus j ˆ j ˆ sn ˆ j hen Fue 4.5: Swnn l on ope boe we sn ˆ Flh j ˆ Flh j Whee he eloc eco s ken + snce he moon s 35 upwd nll. Then j ˆ Flh Flh j m s s ˆ.76 /. j 9.8m / s.s hen he nl heh s ound s ˆ j 4.64m ˆ j I should be noed h he esul mens h he moon s n - ecl decon! ˆ 8
9 Lecue 4: Moon n Two Dmensons P- Ccul Moon n Two Dmensons Unom Speed A pcle mon lon ccul ph consn speed s sd o be n unom ccul moon. As he pcle moes ound he ccle, s nul poson on he ccle chnes. So ben nen ech poson, he eloc eco s pependcul o he poson eco. Snce he speed o he pcle s consn o he s cse so h he mnude o he eloc s consn; bu he decon o he eloc eco s chnn om one poson o nohe poson s me oes. Theeoe such pe o moon hs n cceleon whose mnude emns consn bu decon chnes om one poson o nohe one. Ths cceleon s clled cenul cceleon. The deence beween cenpel nd cenul cceleons s que smple - cenul oces do no es whle cenpel cceleons do. As wh mos smple semens, hee s e del moe o undesndn hs ssue hn smpl memozn whch o he cceleons does o does no es. To undesnd he cenpel cceleon nd he cous cenul cceleon, le's s emne he wods cenpel nd cenul. cen s deed om he Ln cen menn "cene." pel s deed om he Ln peee menn "seek." ul s deed om he Ln uee menn "o lee" s n ue. So, lell, he cenpel cceleon s "cene-seekn" oce. The cous cenul cceleon s, lell, non-esn "cene-leen" cceleon. Le us mpose on pon-mss objec he condon h s on ccul ph n me. The eco eloc o hs objec s lws nen o he ccle heeoe chnes decon n me, s he objec moes lon he ccle. Consequenl, he ccul moon s cceleed moon, smpl becuse he decon o eco eloc chnes, een s mnude speed emns consn. The poblem now becomes o nd ou wh oce enees he cceleon h keeps he objec mon on ccle. Consde n nl objec poson A nd poson B whee he objec eches e me nel Fue 6. The eloc ecos A nd B e shown n he Fue s nd, especel. The c Fue 6: Dm o pon-mss objec mon lon ccul ph. o ccle eled b he objec n he me nel s clled s. Foce smplc, n hs emple, he mnude o eloc speed o be consn. Such moon o n objec on ccle, wh consn speed, s clled "unom ccul moon". An, such moon s n "cceleed moon" jus becuse he decon o he eco eloc chnes. Snce he objec dsplces s poson wh me, s dsplcemen chnes b n moun o n me. The e o chne o hs nul dsplcemen wh espec o me s en b w 4.6 nd hs chne s clled nul eloc. Fo he lm condon; d w lm w. 4.7 d Snce he nle chnes om o n me T, hen he nul eloc cn be wen s 9
10 Lecue 4: Moon n Two Dmensons w. T whee he T s he peod o he pcle h kes me o eoluon. So, he eoluon o second s wen s. T Ths s he equenc o he pcle obn ound cene. Then he nul eloc cn be wen s w. The lne eloc o he objec s wen s, s s w w. 4.8 w nd so h he elon beween he nul eloc nd he lne eloc s en b he equon w. 4.9 I should be noed h he lne eloc does no chne n decon; onl chnes n nul posons. Then s concluded h he lne eloc hs mnude o w bu s decon eco n emns consn. Becuse o hs, hee s jus mnude-elon beween w nd. Snce he cceleon,, s he chne n eloc oe me: 4.3 We noce h cceleon s eco,, mulpled b scl, /. So, he decon o he cceleon wll be he sme s he decon o he chne n eloc,. So wh s he eco "chne o eloc"? The "chne" n n phscl qun s dened s he nl qun mnus he nl qun. So, he chne n eloc s he nl eco eloc mnus he nl eco eloc: 4.3 Fue 7: Eluon o To nd he eco we phcll subc eco om eco Fue 7. Noe: o cl, we he moed he pons o on o boh ecos nd o common pon, ou o Fue 6. Remembe, snce eco s dened onl b mnude nd decon, s pon o on s elen. Fue 7 shows he eco n ed colo. We cn now nspo bck no he nl Fue Fue 6, pesen s mnude nd decon s en n Fue 7. Ae eecun hs opeon, we cll he Fue 8. So, we cn dw n Fue 8 ll hee ecos - blue, F- een, nd - ed pllel o ech ohe Hee, he F s he oce h s he subjec o ne Chpe.
11 Lecue 4: Moon n Two Dmensons I should be noced he decon o ll hese hee ecos; he e ll deced owd he cene o he ccle nd e, heeoe, "cene-seekn" o cenpel. Deon o n Anlc Epesson o Now, n ode o dee n nlc epesson o, we use he ollown pope o wo "sml" sosceles nles, s show n Fue 9: Fo he nles n Fue 6 nd Fue 7, hs elonshp s: Fue 8: Sme s Fue 6, shown he ecos F,, nd. Fue 9: The elon beween he sdes o "sml" sosceles nles. om whch: s 4.33 Dde hs equon b he me nel : hen s Ths epesson epesens he mnude o cenpel cceleon. I we wn o epess hem s ecos, becomes: ˆ 4.36 whee -h epesens he "un eco" lon nd s obous h he cceleon hs mnus sn n decon. The cenpel cceleon s lon he nwd dus. I cn be seen n Fue 7 h s becomes smlle nd smlle hen s decon becomes nwd dus decs no he on!. In he Cesn coodnes, he bles cn be undesood moe esl s seen n Fue. s 4.3 Fue : Posonl eco dm o pon-mss objec mon lon ccul ph nd s nd - s componens.
12 Lecue 4: Moon n Two Dmensons The locon o he objec n momen h mens he equon o moon - ele o he cene o oon s en b ˆ cos ˆ sn 4.37 usn he Equon 4.6, hen cos wˆ sn w 4.38 whee nd j, wh he lle hs, e he un ecos n he nd -decons. The Eq s he equon o moon o he bod n he Cesn coodnes. The objec's eloc s esl ound b kn he dee o s locon wh espec o me: d d cos wˆ sn w d d wsn wˆ wcos w 4.39 Ths eloc s lws nen o he ccle o equlenl, s lws pependcul o poson eco,, nd The objec's cceleon s esl ound b kn he dee o s eloc wh espec o me: d d w w w cos wˆ cos wˆ d d wsn wˆ w sn w sn w wcos w 4.4 I s obousl seen h he decon o he objec's cceleon s oppose,.e., deced owds he cene o moon. Emples nd Poblems Queson 4.3: Wh s he cenpel cceleon o he selle obn 64km boe he Equo kes 98mn o one eoluon ound he eh? eh =637km Noe h he esul wll he hs heh! Soluon 4.3: We e en: h 64km, eh 637km 98mn 858s Snce he dus whch he selle obs s he dsnce o he selle o he cene o he eh, hen he ol dsnce o o he obn cene s 6 h eh 64 km 637km 7km 7. m hen he cceleon o he selle s
13 Lecue 4: Moon n Two Dmensons s 4 6 m 8.m / s Queson 4.4: An objec obn unoml ound cene, hn dus o km, speds nle n.s.. Wh s he lne eloc o hs objec? b. Wh s he cceleon o he objec? c. Fnd he poson eco n nd componens o h objec o =s. Soluon 4.4: We e en:,.s, km Snce he objec dsplces deee n.second, hen * / 36 w w d / sec.. 36 Then he lne eloc w km* d / sec m / s 36 The cceleon s m / s The poson eco s en b cos wˆ sn w hen cos ˆ sn Fo he second, he poson eco componens e: cos *ˆ sn * ˆ So, =984.8m nd =73.65m 3
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