Ch.4 Motion in 2D. Ch.4 Motion in 2D


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1 Moion in plne, such s in he sceen, is clled 2dimensionl (2D) moion. 1. Posiion, displcemen nd eloci ecos If he picle s posiion is ( 1, 1 ) 1, nd ( 2, 2 ) 2, he posiions ecos e 1 = = 2 2 Aege eloci eco V g = = In he limi b > 0, we he he insnneous eloci eco s Obecies To undesnd he eco nue of 2D moion To genelize 1D moion o 2D moion To descibe specil 2D moion: poecile moion To undesnd unifom cicul moion The displcemen eco is defined s he chnge in posiion = 2 1 = ( 2 1 ) i^ + ( 2 1 ) o = Wh if we diide b, e.g., = 3 i^ 1 (m), i would ell how fs he posiion chnges, wih sense of diecion, i.e. he eloci eco V = lim > 0 = d d d d = V V V = d/d, V = d/d : he  nd  componens of eloci. Noe h eloci is in he diecion of, which in he limi of > 0 is ngen o he ph 2 Speed = V + V 2 Checkpoin 1 Checkpoin 1 Checkpoin 1 A picle is moing in cicul ph s shown. In picul insn, is eloci is found o be V = 2 i^ 2 (m/s) A picle is moing in cicul ph s shown. In picul insn, is eloci is found o be V = 2 i^ 2 (m/s) A picle is moing in cicul ph s shown. In picul insn, is eloci is found o be V = 2 i^ 2 (m/s) Assume he moion is clockwise, which qudn is he picle in h insn? Dw he eloci eco on he digm. Assume he moion is clockwise, which qudn is he picle in h insn? Dw he eloci eco on he digm. Moe he eloci eco unil i is ngen o he ph, which is he 1 s qudn Assume he moion is clockwise, which qudn is he picle in h insn? Dw he eloci eco on he digm. Whee is he picle if he moion is coune clockwise? Moe he eloci eco unil i is ngen o he ph, which is he 1 s qudn In n cse, wh is he speed? 2 2 m/s 1
2 2. Acceleion eco (Ag) Acceleion should be defined s = g = = chnge in eloci ime inel The diecion of cceleion is in he sme diecion s If eloci (no speed!) chnges, in eihe mgniude o diecion, hen is no zeo, nd cceleion is no zeo. Fo emple, V 1 = 2 i^ 2 (m/s), V 2 = 2 2 (m/s) Hs mgniude chnged? Diecion? Is cceleion zeo? Insnneous cceleion Fom g = = In he limi > 0, insnneous cceleion is gien b d = d = d d whee = d d nd = d d e he  nd componens Since = d/d, = d/d, so = d 2 /d 2, = d 2 /d 2 Undesnding cceleion nd eloci Q: Is i possible o moe wih consn speed, nd e cceleion is no zeo? chnge in eloci Acceleion = ime inel Emple: Acceleion of c ounding cone wih consn speed cceleion is owd cene Hs eloci chnged? Mg o Di? Dw eco digm so h = Wh is he diecion of? cceleion? 3. Moion wih consn cceleion eco Since cceleion is gien b = fo i o be consn eco, boh is componens mus be consn, i.e. = consn, nd = consn Q: Think bck o Ch.2, wh equions descibe moion in 1D when is consn? = 0 + = ½ 2 Wh equions fo moion long Wh bou when is consn? is when is consn? = 0 + = ½ 2 0, 0 = iniil pos & el = 0 + = ½ 2 0, 0 = iniil pos & el Noe: Moions long  nd  diecions eole in ime independenl! 2D moion is sme s 1D moion, onl wice. Checkpoin 2 (1) = , = (2) = , = (3) = 2 2 ^ i (4 + 3) (4) = (4 32) 3 ^ ^ Diecl compe wih = ½ 2 = i.e., 0 = 2 m, 0 = 4 m/s ½ = 3, =  6 m/s 2 Posiion is in mees, nd ime in seconds. Ae he nd cceleion componens consn? Is cceleion eco consn? Fo he cses whee ou nswes e es, wh e he iniil posiion nd eloci ecos? 4. Poecile moion If we e o wie down giionl cceleion in he uni eco noion, we he = i^ g ^ 0 i.e., = 0, nd =  g, boh consn, g = 9.8 m/s 2 Moion unde gi lone is known s poecile moion. Subsiuing nd ino he genel equions of moion fo consn cceleion, = 0 + = ½ 2 = 0 + = ½ 2 we obin specific equions of moion fo poecile moion s = 0 = = 0 g = ½g 2 0, 0 = iniil  & pos 0, 0 = iniil  & el In ems of iniil speed 0 nd ngle θ 0, 0 = 0 cosθ 0, 0 = 0 sinθ 0 = 2
3 Undesnding poecile moion Mhemicll, equions of moion fo poecile moion e = 0 = 0 g = = ½ g 2 1. The hoizonl eloci emins consn, s hee is no ccel. As esul, posiion chnges linel in ime. 2. The eicl eloci deceses he e g, nd eicl posiion chnges qudicll in ime. 3. Moions long  nd  diecions eole in ime independenl! Poecile moion is sme s 1D moion, onl wice. Tems, discuss wh he iems men o ou, especill iem 3 bou independen moion. Gie n emple if possible. An emple Q: Two coins e ossed wo diffeen ws: One is dopped fom es, nohe is flung sidews hoizonll. Will hei eicl moion be he sme o diffeen? i.e., will he eicl moion be ffeced b he hoizonl moion o will he be independen of ech ohe? Conclusion fom demo: The moion is he sme in boh cses, independen of moion. Conesel, he moion is no ffeced b he moion, i.e., moing independenl of moion. Leding o he fmous deecie obseion h bulle fied hoizonll coss leel field nd elling hundeds of mees will hi he gound he sme ime s book dopped he sme heigh nd elesed he sme ime. A numeicl emple A bll olling speed 3.3 m/s flls off he edge of he ble. The heigh of he ble is 1.0 m boe he floo. A wh ime does he bll hi he floo? How f hoizonll does i lnd fom he edge? Ides: Duing flling, he moion is pue poecile moion. If we cn sole fo ime, nd since = 0 is consn, he hoizonl nge cn be found b =  0 = 0. How o find? S wih iniil condiions nd giens Iniil condiions: 0 = 3.3 m/s, 0 = 0. Since he chnge in eicl posiion is 1.0m s gien (noe he negie sign since decesed), we cn use =  0 = 0 ½ g 2 = ½ g 2 o ge = sq( 2 /g), o = 0.45 s. = 0 = 1.5 m Is his he cul ph lengh? Undesnding poecile moion Gphicll, equions of moion fo poecile moion = 0 = 0 g = = ½ g 2 Skech he eloci nd posiion gphs slope = g Undesnding poecile moion The ph, o eco, of poecile moion in  plne pbol Wh so simil o ? Dw eloci ecos seel poins long he ph. Bek hem down o  nd  componens. A: nd linel eled Veloci ecos long he ph Compe ou gphs. Did s consn? Did decese consn e g? slope = 3
4 Rnk he ime A cem ngeine psses windows 1,2,3. Rnk he ime i spends pssing ech window, gees fis. Wh? Answe: 3 > 2 > 1 Rnk he ime Now on he downwd ph he cem ngeine psses windows 4,5,6. Agin nk he ime i spends pssing ech window, gees fis. Wh? Answe: 4 = 5 = 6 Reson: The hoizonl widh is he sme, nd he hoizonl eloci is consn. Anohe numeicl emple A sone is hown cliff of heigh h wih n iniil speed of 42 m/s dieced n iniil ngle of 60 deg. I sikes poin A 5.5 s le. Find he heigh h nd he mimum heigh H boe he gound. Ides: Gien iniil speed nd ngle, we cn find 0 = 0 sinθ 0 = 36.4 m/s. Tking he lunch poin o be oigin, iniil 0 =0, finl posiion is h. i.e., =h. Thus h = ½ g 2 = 51.8 m. How o find H, which is he op of fligh? If we knew he ime A he op, eicl eloci is zeo, = 0 g =0, o = 0 /g = 3.71 s. So plugging he ime ino H = = ½ g 2 = 67.6 m. Q: How o find he hoizonl posiion of poin A? Ides. Moe numeicl emples A plne fling 198 km/h nd heigh h=500 m eleses life es. Wh is he ngle Φ fom line of sigh in ode o hi he ge? Ides? Afe elese, i will ech he ge if in he ime i kes flling heigh h, i els hoizonl disnce of. n Φ = / h Need need Giens: 0 =0, 0 = 198 km/h = 55 m/s, =500 m =  0 = 0 ½ g 2, = sq (2 /g) = 10.1 s = 0 = 55*10.1 = m Φ = n( / h) = 48 o Rnge of poecile moion Time of fligh: =  0 = 0 ½ g 2, when i his gound, =0, 0= 0 ½ g 2, = 2 0 /g Rnge: = 0 = /g 0 = 0 cosθ 0, 0 = 0 sinθ 0 = 2 02 sinθ 0 cosθ 0 /g = 02 sin(2θ 0 )/g = R Tig ideni: 2 sinθ cosθ = sin(2θ) M nge sin(2θ 0 ) = 1, o θ 0 =45 o This is cnned fomul. Don use in quiz o es wihou poof. R 5. Unifom cicul moion A picle moes in sigh line wih consn speed. Is he eloci consn? Is he cceleion zeo? If picle moes in cicle wih consn speed, is he eloci consn? Is he cceleion zeo? No, becuse diecion chnges ll he ime. Since he eloci eco chnges, hee mus be cceleion. Now conside he elociies he op & side Dw eco digm so h = o Wh is he diecion of? cceleion? owd he cene 4
5 Flling owd he cene One w o hink bou cicul moion is h he picle flls coninuousl owd he cene fom sigh line. Cenipel cceleion The mgniude of cenipel cceleion is Is diecion is owd he cene, i.e., pependicul o he eloci. = 2 Le be dius of cicle, s be he fllen disnce owd he cene wihin smll ime. How e, s, eled? s =dius ( + s) 2 = 2 + () 2 o s + s 2 = 2 + () 2 If is smll, s 2 will be e in. Neglec s 2, we he 2 s = () 2, o s = ½ 2 2 Compe wih = ½ 2 in 1D, we idenif = = cceleion owd cene Esime he cenipel cceleion of he fis while full swinging ou m in cicles bou once pe second. Le =0.5 m, = 2π / = 2*3.14*0.5/1 = 3.14 m/s, = /0.5 = 19.7 m 2 /s 2 /m, i.e., m/s 2 I is bou 2g. 5
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Phsics 1 Lecue 5 Tod s Topics n Moion in D (Chp 4.14.3): n D Kinemicl Quniies (sec. 4.1) n D Kinemics wih Consn Acceleion (sec. 4.) n D Pojecile (Sec 4.3) n Epeced fom Peiew: n Displcemen eloci cceleion
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