# Kinematics in two Dimensions

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1 Lecure 5 Chaper 4 Phsics I Kinemaics in wo Dimensions Course websie: hp://facul.uml.edu/andri_danlo/teachin/phsicsi PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics

2 Toda we are oin o discuss: Chaper 4: Moion in Two Dimensions: Secion 4.1 Projecile moion: Secion 4. PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics

3 Moion in Two Dimensions (Vecor Kinemaics) We use he ecor mahemaics o consider moion in more han one dimension. Preiousl described 1D displacemen as Δ, where moion could onl be posiie or neaie. In more han 1D, displacemen are D ecors r r r r r() () PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics

4 Displacemen in wo dimensions Posiion of an objec r ( )ˆ i ( ) ĵ î Rabbi s pah r 1 1 r r r r r 1 displacemen (in uni ecor noaion): r In wo dimensions, he displacemen is a ecor: r iˆ r iˆ 1 1 ( 1 )ˆ i ( 1) 1 PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics

5 r 1 Insananeous Veloci in wo dimensions 1 Tanen 1 r r r r r Aerae eloci is he displacemen diided b he elapsed ime r r1 1 r As Δ and Δr become smaller and smaller, he aerae eloci approaches he insananeous eloci. lim r d r d The insananeous eloci indicaes how fas he objec moes and he direcion of moion a each insan of ime dr d dr d d iˆ d d d iˆ PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics

6 Insananeous acceleraion in wo dimensions r r Aerae acceleraion a The insananeous acceleraion is in he direcion of,and is ien b: lim 1 d d 1 1 a d r d a d d iˆ d d d iˆ d d d PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics

7 () r Posiion of an objec r ( )ˆ i ( ) Insananeous eloci dr d d iˆ iˆ d d d () a Insananeous acceleraion d d iˆ d d d d d iˆ d r,, a wrien in erm of iˆ, ˆ, j kˆ PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics

8 Eample A rabbi runs across a parkin lo on which a se of coordinae aes has, sranel enouh, been drawn. The coordinaes as funcions of ime are ien The, coordinaes are componens of he rabbi s posiion ecor ( ) (4 1) ( ) 3 r( ) ( )ˆ i ( ) r ( ) (4 1)ˆ i 3j ˆ PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics

9 Projecile moion A projecile is an objec moin in wo dimensions under he influence of Earh's rai; is pah is a parabola. () r () a a?,? PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics

10 r r a B spliin he equaions of moion ino componen form, we can sole problems one direcion a a ime 1 a a a f ( ) Veloci equaion o Posiion equaion 1 a No ime equaion a (1) () (3) f ( ) Veloci equaion o Posiion equaion 1 a No ime equaion a (1) () (3) There is onl one parameer, which connecs X and Y moion: ime PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics X and Y moions are compleel independen Work he problem as wo one-dimensional problems

11 Equaions in he X - direcion No forces in direcion (air resisance is neleced), as a resul, no acceleraion is consan!!!! f ( ) Veloci equaion o Posiion equaion 1 a No ime equaion a (1) () (3) a = So, from here, we can e onl one equaion. PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics

12 Deparmen of Phsics and Applied Phsics PHYS.141 Lecure 5 Danlo Equaions in he Y- direcion a = a = if hen if hen ) ( 1 ) ( 1 1 a f (3) No ime equaion Posiion equaion a o ) ( Veloci equaion () (1)

13 Rescue Helicoper Helicoper flin horizonall a 7m/s wans o drop supplies on mounain op m below. How far in adance (horizonal disance) should he packae be dropped? Draw diaram, choose coordinaes Knowns and unknowns Diide equaions ino and Sole, noin ha in he and calculaions he common parameer is he ime ineral PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics

14 ConcepTes 1 From he same heih (and a he same ime), one ball is dropped and anoher ball is fired horizonall. Which one will hi he round firs? Droppin he Ball I A) he dropped ball B) he fired ball C) he boh hi a he same ime D) i depends on how hard he ball was fired E) i depends on he iniial heih OR Boh of he balls are fallin ericall under he influence of rai. The boh fall from he same heih. Therefore, he will hi he round a he same ime. The fac ha one is moin horizonall is irrelean remember ha he and moions are compleel independen!! 1

15 Eample (Golf Ball) A olf ball is hi wih iniial eloci a an anle θ aboe he horizonal. Draw diaram and choose coordinae ssem Fill in knowns cos sin a a PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics

16 Con. Eample (Golf Ball). A olf ball is hi wih iniial eloci a an anle θ aboe he horizonal. Find: he ime of flih (how lon he ball is in he air) This depends onl on he -componen equaions, as he moion in direcion sops he flih. o o 1 a Since boh and are zero a he beinnin/end = when he ball was hi PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics o 1 ( o ) The second is he ime of flih 1 and = when he ball landed

17 Con. Eample (Golf Ball) A olf ball is hi wih iniial eloci a an anle θ aboe he horizonal. Find: Rane (how far does ball rael on fla round) ime of flih Rane when Use consan -eloci o calculae how far ball raels horizonall durin ime of flih (Rane) Consan eloci R PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics sin 1 or 45 cos R sin o R ma o Rane sin coso Maimum Rane o sin

18 Con. Eample (Golf Ball) A olf ball is hi wih iniial eloci a an anle θ aboe he horizonal. Find: rajecor (heih as a funcion of posiion ) o 1 Since common parameer is ime, eliminae o e () Equaion of parabola A B PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics

19 ConcepTes. Puns I Which of he hree puns has he lones han ime? A B C D) all hae he same han ime h The ime in he air is deermined b he erical moion! Because all of he puns reach he same heih, he all sa in he air for he same ime. ( ) o o 1 a

20 ConcepTes 3 Puns II A baleship simulaneousl fires wo shells a wo enem submarines. The shells are launched wih he same iniial eloci. If he shells follow he rajecories shown, which submarine es hi firs? The flih ime is fied b he moion in he -direcion. The hiher an objec oes, he loner i sas in flih. The shell hiin submarine # oes less hih, herefore i sas in flih for less ime han he oher shell. Thus, submarine # is hi firs. A B C) boh a he same ime

21 Solin Problems Inolin Projecile Moion 1. Read he problem carefull, and choose he objec(s) ou are oin o analze.. Draw a diaram. 3. Choose an oriin and a coordinae ssem. 4. Decide on he ime ineral; his is he same in boh direcions, and includes onl he ime he objec is moin wih consan acceleraion. 5. Eamine he and moions separael. 6. Lis known and unknown quaniies. Remember ha neer chanes, and ha = a he hihes poin. 7. Plan how ou will proceed. Use he appropriae equaions; ou ma hae o combine some of hem. PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics

22 PHYS.141 Lecure 5 Danlo Deparmen of Phsics and Applied Phsics Thank ou See ou on Monda

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