Physics 101 Lecture 4 Motion in 2D and 3D

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Phsics 11 Lecure 4 Moion in D nd 3D Dr. Ali ÖVGÜN EMU Phsics Deprmen www.ogun.com

Vecor nd is componens The componens re he legs of he righ ringle whose hpoenuse is A A A A A n ( θ ) A Acos( θ) A A A nd A A 1 Asin( θ ) θ n A ( A ) ( A ) A A or θ n 1 A A Or, Februr 5-8, 13

Moion in wo dimensions Kinemic ribles in one dimension n Posiion: () m n Veloci: () m/s n Accelerion: () m/s Kinemic ribles in hree dimensions n Posiion: r ( ) iˆ j ˆ zkˆ m n Veloci: ( ) iˆ ˆ j zkˆ m/s n Accelerion: ( ) iˆ ˆj kˆ m/s All re ecors: he direcion nd mgniudes z z k j i Februr 5-8, 13

Posiion nd Displcemen In one dimension In wo dimensions n n Δ 1 ( 1 ) - 3. m, ( ) 1. m Δ 1. m 3. m 4. m Posiion: he posiion of n objec is described b is posiion ecor r () - -lws poins o pricle from origin. Displcemen: Δ r ( iˆ ( )ˆ i ( 1 Δiˆ Δj ˆ ( ) 1 ( 1) Δr r r 1 ˆ) j ( iˆ 1 ) ˆj 1 1 ˆ) j Δr r r 1 Februr 5-8, 13

Aerge & Insnneous Veloci Aerge eloci g Δ iˆ Δ Insnneous eloci lim dr d Δ Δ ˆj Δr lim Δ g d d g dr d is ngen o he ph in - grph; iˆ g, Δr Δ d iˆ ˆj iˆ d g, ˆj ˆj Februr 5-8, 13

Moion of Turle A urle srs he origin nd moes wih he speed of 1 cm/s in he direcion of 5 o he horizonl. () Find he coordines of urle 1 seconds ler. (b) How fr did he urle wlk in 1 seconds? Februr 5-8, 13

Moion of Turle Noice, ou cn sole he euions independenl for he horizonl () nd ericl () componens of moion nd hen combine hem X componens: Y componens: cos 5 9.6 cm/s sin 5 4.3 cm/s Disnce from he origin: d Δ Δ Δ 9.6 cm Δ 4.3 cm 1. cm Februr 5-8, 13

Aerge & Insnneous Accelerion Aerge ccelerion g Δ Δ iˆ Δ Δ ˆj iˆ g, g g, Δ Δ ˆj Insnneous ccelerion lim Δ lim Δ g d d d d d d d iˆ ˆj iˆ d ˆj The mgniude of he eloci (he speed) cn chnge The direcion of he eloci cn chnge, een hough he mgniude is consn Boh he mgniude nd he direcion cn chnge Februr 5-8, 13

Summr in wo dimension Posiion Aerge eloci d Insnneous eloci d Δr dr d ( ) lim iˆ Δ d d Accelerion r ( ) iˆ g Δ ( ) lim Δ r(), ( ), nd ( ) j ˆ Δr Δ d d d d Δ iˆ Δ d d d d Δ Δ d iˆ d ˆj iˆ re no necessril sme direcion. ˆj d d d iˆ g, d d ˆj ˆj iˆ d d d g, ˆj ˆj Februr 5-8, 13

Februr 5-8, 13 Moion in wo dimensions Moions in ech dimension re independen componens Consn ccelerion euions Consn ccelerion euions hold in ech dimension n beginning of he process; n where nd re consn; n Iniil eloci iniil displcemen ; 1 r r 1 ) ( 1 ) ( j i ˆ ˆ j i ˆ ˆ j i r ˆ ˆ

Februr 5-8, 13

Projecile Moion -D problem nd define coordine ssem: - horizonl, - ericl (up ) Tr o pick, Horizonl moion Vericl moion Horizonl:, consn eloci moion Vericl: -g -9.8 m/s, Euions: Horizonl Vericl f i i 1 g 1 1 ( ) ( Februr ) 5-8, 13

Projecile Moion X nd Y moions hppen independenl, so we cn re hem seprel Horizonl g 1 g Vericl Tr o pick, Horizonl moion Vericl moion Horizonl:, consn eloci moion Vericl: -g -9.8 m/s nd re conneced b ime () is prbol Februr 5-8, 13

Projecile Moion -D problem nd define coordine ssem. Horizonl: nd ericl: -g. Tr o pick,. Veloci iniil condiions: n cn he, componens. n is consn usull. cosθ n chnges coninuousl. sinθ Euions: Horizonl Vericl g 1 g Februr 5-8, 13

Februr 5-8, 13 Iniil condiions ( ):, cosθ nd sinθ Horizonl moion: Vericl moion: Prbol; n θ nd θ 9? Trjecor of Projecile Moion 1 g g cos n g θ θ

Februr 5-8, 13 Iniil condiions ( ):, cosθ nd sinθ, hen Wh is R nd h? Horizonl Vericl 1 g g g R sin sin cos θ θ θ g g sin θ 1 g g h h h g h sin θ g g g h g 1 g

Projecile Moion Vrious Iniil Angles Complemenr lues of he iniil ngle resul in he sme rnge R sin φ g n The heighs will be differen The mimum rnge occurs projecion ngle of 45 o Februr 5-8, 13

Summr Posiion Aerge eloci d Insnneous eloci d Δr dr d ( ) lim iˆ Δ d d Accelerion r ( ) iˆ g Δ ( ) lim Δ j ˆ Δr Δ d d d d Δ iˆ Δ d d d d d iˆ d ˆj iˆ r(), ( ), nd ( ) re no necessril in he sme direcion. Δ Δ ˆj d d d iˆ g, d d ˆj ˆj iˆ d d d g, ˆj ˆj Februr 5-8, 13

Februr 5-8, 13 If pricle moes wih consn ccelerion, moion euions re Projecile moion is one pe of -D moion under consn ccelerion, where, -g. Summr j i j i r i i i i i i f f f ˆ ) ( )ˆ ( ˆ ˆ 1 1 j i j i i i f f f ˆ ) ( )ˆ ( ˆ ˆ ) ( i 1 r r i i f

Februr 5-8, 13

Februr 5-8, 13

Februr 5-8, 13

Emple: 1 During olcnic erupions, chunks of solid rock cn be blsed ou of he olcno; hese projeciles re clled olcnic bombs. The figure below shows cross secion of M. Fuji, in Jpn. From he en A o he foo of he olcno B, he ericl disnce is h 3.3km nd horizonl disnce is d 94m. Neglecing ir resisnce, () clcule he ime of fligh, nd (4 P) (b) clcule he iniil speed of he projecile. (P) Emple: A moie sun drier on moorccle speeds horizonll off 5m high cliff. If he moorccle will lnd 9m from he bse of he cliff, (ignore n kind of fricion or resisnce) () Find he ime of fligh, (b) Find is iniil speed in -direcion, (c) Find is ccelerion ecor jus before hiing he ground. Februr 5-8, 13

Emple: 3 A cliff dier is bou o jump down cliff of heigh 35.m, he boom of he cliff here is 5m wide rock bnk ne o he se. Clcule he minimum horizonl iniil eloci he cliff jumper hs o push off. (No iniil eloci componen in direcion) Februr 5-8, 13

Problem:1 A projecile is fired n iniil eloci of 35. m/s n ngle of 3. degrees boe he horizonl from he roof of building 3. m high, s shown. Find ) The mimum heigh of he projecile b) The ime o rise o he op of he rjecor c) The ol ime of he projecile in he ir d)the eloci of he projecile he ground e)the rnge of he projecile Problem: A plne drops pckge of supplies o pr of eplorers. If he plne is reling horizonll 4 m/s nd is 1 m boe he ground. Where does he pckge srike he ground? Februr 5-8, 13

Problem:3 Februr 5-8, 13

Problem:4 Februr 5-8, 13

Problem:5 Februr 5-8, 13

Problem:6 Februr 5-8, 13

Problem:7 A wh iniil speed mus he bskebll pler in Figure hrow he bll, ngle u 37 boe he horizonl, o mke he foul sho? The horizonl disnces re 3 cm nd, 44cm nd he heighs re cm nd. 3 cm. Februr 5-8, 13

Uniform circulr moion Consn speed, or, consn mgniude of eloci Moion long circle: Chnging direcion of eloci Februr 5-8, 13

Circulr Moion: Obserions Objec moing long cured ph wih consn speed n Mgniude of eloci: sme n Direcion of eloci: chnging n Veloci: chnging n Accelerion is NOT zero n Ne force cing on he objec is NOT zero n Cenripel force F m ne Februr 5-8, 13

Uniform Circulr Moion Cenripel ccelerion Δ Δr r Δ Δr Δ Δ Δ r Δ r r so, r Δr r Direcion: Cenripel Δ A i f i r i Δr Δ f - i O r f R B f Februr 5-8, 13

Uniform Circulr Moion Veloci: n Mgniude: consn n The direcion of he eloci is ngen o he circle Accelerion: n Mgniude: c r n direced owrd he cener of he circle of moion Period: n ime inerl reuired for one complee reoluion of he pricle T c πr c r Februr 5-8, 13

Problem:8 The bobsled rck conins urns wih rdii of 33 m nd 4 m. Find he cenripel ccelerion ech urn for speed of 34 m/s. Epress nswers s muliples of. Februr 5-8, 13

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