Kinematics in two dimensions


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1 Lecure 5 Phsics I Kinemaics in wo dimensions Course websie: hp://facul.uml.edu/andri_danlo/teaching/phsicsi Lecure Capure: hp://echo36.uml.edu/danlo13/phsics1fall.hml , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
2 Ouline Chaper 3: Secions Vecor kinemaics Projecile moion , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
3 Vecor Kinemaics Kinemaics in more han one dimension Preiousl described 1D displacemen as Δ, where moion could onl be posiie or negaie. In more han 1D, displacemen is a ecor , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
4 Displacemen 1 1 (1D ) ĵ î r displacemen (in uni ecor noaion): r r iˆ ˆ j r iˆ In wo or hree dimensions, he displacemen is a ecor: ( 1 )ˆ i ( 1) ˆj ˆj r r , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
5 Aerage Veloci 1 Aerage eloci is he displacemen diided b he elapsed ime r r 1 1 r , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics As Δ and Δr become smaller and smaller, he aerage eloci approaches he insananeous eloci.
6 Insananeous eloci The insananeous eloci indicaes how fas he objec moes and he direcion of moion a each insan of ime 1 r r , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
7 Insananeous acceleraion 1 Aerage acceleraion r The insananeous acceleraion is in he direcion of 1,and is gien b: 1 a d d r , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
8 Using uni ecors dr d d d d d iˆ ˆj kˆ iˆ ˆ j d d z kˆ a d d d iˆ ˆj d d d z kˆ , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
9 Eample Problem Gien posiion as a funcion of ime, find insananeous eloci and insananeous acceleraion a 3s ( 3s) (8 3)ˆ i ( 3) kˆ 4ˆ i 6kˆ a( ) 8ˆ i kˆ Acceleraion is consan , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
10 Projecile moion A projecile is an objec moing in wo dimensions under he influence of Earh's grai; is pah is a parabola , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
11 Projecile moion X and Y moions are compleel independen Work he problem as wo onedimensional problems Each dimension obe differen equaions , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
12 D Moion wih consan acceleraion B spliing he equaions of moion ino componen form, we can sole problems one direcion a a ime There is onl one parameer, which connecs X and Y moion: ime r r , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics a 1 a
13 Deparmen of Phsics and Applied Phsics , Fall 13, Lecure 5 ) ( 1 a a a No forces in direcion. Air resisance is negleced Projecile moion X direcion a is consan!!!!
14 Deparmen of Phsics and Applied Phsics , Fall 13, Lecure 5 Projecile moion Y direcion ) ( 1 a a a g g a g a g if hen if hen ) ( 1 g g g ) ( 1 g g g
15 Projecile Moion: Concep Tes 1 A helicoper moing a a consan horizonal eloci o he righ drops a package when a posiion A. Which of he marked rajecories is closes o ha obsered b a saionar person on he ground? , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics obserer
16 Eample (Rescue Helicoper) Helicoper fling horizonall a 7m/s wans o drop supplies on mounain op m below. How far in adance (horizonal disance) should he package be dropped? g Draw diagram, choose coordinae ssem Knowns and unknowns Diide equaions ino and Sole, noing ha in he and calculaions he common parameer is he ime ineral , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
17 Eample (Rescue Helicoper) 7 m/s a? o o a g m g 1 g ( m) 6. 39s 9.8m / s g common parameer is ime 1 g Sole equaions o find Plug ino equaions o find g( ) 7(6.39) m 447m g , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
18 ConcepTes From he same heigh (and a he same ime), one ball is dropped and anoher ball is fired horizonall. Which one will hi he ground firs? Dropping 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 heigh
19 ConcepTes From he same heigh (and a he same ime), one ball is dropped and anoher ball is fired horizonall. Which one will hi he ground firs? Dropping 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 heigh Boh of he balls are falling ericall under he influence of grai. The boh fall from he same heigh. Therefore, he will hi he ground a he same ime. The fac ha one is moing horizonall is irrelean remember ha he and moions are compleel independen!!
20 Eample (Golf Ball) A golf ball is hi wih iniial eloci a an angle θ aboe he horizonal. Draw diagram and choose coordinae ssem Fill in knowns g θ cosθ sinθ a g a , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
21 Con. Eample (Golf Ball). A golf ball is hi wih iniial eloci a an angle θ aboe he horizonal. Find: he ime of fligh (how long he ball is in he air) This depends onl on he componen equaions, as he moion in direcion sops he fligh. Since boh and are zero a he beginning/end when he ball was hi The second is he ime of fligh 1 and when he ball landed g , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
22 Con. Eample (Golf Ball) A golf ball is hi wih iniial eloci a an angle θ aboe he horizonal. Find: Range (how far does ball rael on fla ground) ime of fligh Range when Use consan eloci o calculae how far ball raels horizonall during ime of fligh (Range) Consan eloci R cosθ , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics g sin θ 1 or θ 45 sinθ g o sinθ cosθo R g Range Maimum Range o sin θ g
23 Con. Eample (Golf Ball) A golf ball is hi wih iniial eloci a an angle θ aboe he horizonal. Find: rajecor (heigh as a funcion of posiion ) Since common parameer is ime, eliminae o ge () Equaion of parabola , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
24 ConcepTes 3. Puns I Which of he hree puns has he longes hang ime? A B C D) all hae he same hang ime h The ime in he air is deermined b he erical moion! Because all of he puns reach he same heigh, he all sa in he air for he same ime. g( ) o o 1 a
25 ConcepTes 4 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 ges hi firs? The fligh ime is fied b he moion in he direcion. The higher an objec goes, he longer i sas in fligh. The shell hiing submarine # goes less high, herefore i sas in fligh for less ime han he oher shell. Thus, submarine # is hi firs. A B C) boh a he same ime
26 Relaie Moion: Clicker Quiz obserer A helicoper moing a a consan horizonal eloci o he righ drops a package when a posiion A. Which of he marked rajecories is closes o ha obsered b a person on he helicoper? , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
27 Soling Problems Inoling Projecile Moion 1. Read he problem carefull, and choose he objec(s) ou are going o analze.. Draw a diagram. 3. Choose an origin 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 moing wih consan acceleraion g. 5. Eamine he and moions separael. 6. Lis known and unknown quaniies. Remember ha neer changes, and ha a he highes poin. 7. Plan how ou will proceed. Use he appropriae equaions; ou ma hae o combine some of hem , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
28 Thank ou See ou on Monda , Fall 13, Lecure 5 Deparmen of Phsics and Applied Phsics
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