Motion in Two Dimensions

Transcription

1 Phys 1 Chaper 4 Moon n Two Dmensons adzyubenko@csub.edu hp:// 005, 014 A. Dzyubenko 004 Brooks/Cole 1

2 Dsplacemen as a Vecor The poson of an objec s descrbed by s poson ecor, r The dsplacemen s defned as he change n objec s poson r r f r

3 Aerage Velocy The aerage elocy durng he me neral s defned as he dsplacemen dded by he me neral r s a ecor quany dreced along r Independen of pah aken Depends only on he nal and fnal poson ecors 3

4 Insananeous Velocy The nsananeous elocy s defned as he lm of he aerage elocy as D approaches zero: r lm 0 dr d The drecon a any pon s along he lne angen o he pah a ha pon The magnude of he nsananeous elocy ecor s called speed, whch s a scalar quany 4

5 Acceleraon The aerage acceleraon s defned as he change n he nsananeous elocy ecor dded by he me neral durng whch ha change occurs: f a f The nsananeous acceleraon s defned as he lmng alue of he rao / as approaches zero: d a lm 0 5 d

6 Ways an Objec Mgh Accelerae The magnude of he elocy (he speed) can change The drecon of he elocy can change Een hough he magnude s consan Boh he magnude and he drecon of he elocy ecor can change smulaneously 6

7 Quck Quz Consder he followng conrols n an auomoble: gas pedal, brake, seerng wheel. The conrols n hs ls ha cause acceleraon of he car are (a) all hree conrols (b) he gas pedal and he brakes (c) only he brake (d) only he gas pedal 7

8 Two-Dmensonal Moon wh Consan Acceleraon The moon of a parcle mong n he xy plane s descrbed by he poson ecor r r x ˆ + yˆj The elocy of a parcle s dr d dx d a a ˆ + ˆ x a j consan y ˆ + dy d ˆj ˆ ˆ x + y j xf yf x y + a x + a y 8

9 Velocy as a Funcon of Tme f ( + a ) ˆ + ( + a ) ( ˆ + ˆj ) + ( a ˆ + a ˆj ) x x y x x y y y ˆj + a f The elocy of a parcle equals he ecor sum of s nal elocy and addonal elocy a acqured a me as a resul of consan acceleraon 9

10 10 Poson Vecor Poson Vecor as a Funcon of Tme as a Funcon of Tme 1 1 a y y a x x y y f x x f ( ) ( ) ( ) ˆ ˆ 1 ˆ ˆ ˆ ˆ a a y x y x y x j j j j r ˆ 1 ˆ a y a x y y x x f

11 Poson Vecor as a Funcon of Tme, con r r + + f 1 a The poson ecor s he ecor sum of () orgnal poson r, () a dsplacemen arsng from he nal elocy and () a dsplacemen ½ a resulng from he consan acceleraon of he parcle 11

12 1 Componen Form of Knec Equaons Componen Form of Knec Equaons 1 f a r r a y y a x x y y f x x f + + a a y y yf x x xf f a + Two-dmensonal moon a a consan acceleraon s equalen o wo, x- and y-, ndependen moons

13 Projecle Moon Projecle moon: an objec may moe n boh he x- and y- drecons smulaneously Assumpons of projecle moon The free-fall acceleraon g s consan and s dreced downward The effec of ar ressance s neglgble The pah of a projecle (s rajecory) s always a parabola g 13

14 Rules of Projecle Moon The x- and y-drecons of moon can be reaed ndependenly The x-drecon s unform moon a x 0 The y-drecon s free fall a y - g The nal elocy can be broken down no s x- and y-componens 14

15 Projecle Moon 15

16 Some Deals abou Rules of Projecle Moon x-drecon a x 0 x cos θ cons x f ( cosθ ) x Ths s he only operae equaon n he x-drecon snce he elocy s unform n ha drecon 16

17 More Deals Abou he Rules y-drecon a y - g free fall problem y sn θ ake he pose drecon as upward unformly acceleraed moon, so he knec equaons all hold 17

18 Projecle Trajecory s a Parabola Sole he followng equaons smulaneously x y f f ( cosθ ) y + 1 a y x f cosθ ( sn θ ) From (1): subsue n () y g cos θ ( an θ ) x x 1 g (1) () y ax bx he equaon of a parabola ha passes hrough he orgn 18

19 Poson Vecor of Projecle as a Funcon of Tme r r + + f 1 g r 0 If here were no acceleraon, g 0, he projecle would connue o moe along a sragh pah n he drecon of 19

20 Quck Quz As a projecle hrown upwards moes n s parabolc pah, a wha pon along s pah are he elocy and acceleraon ecors for he projecle perpendcular o each oher? (a) nowhere (b) he hghes pon (a) he launch pon 0

21 Maxmum Hegh of a Projecle Noe ha a maxmum hegh h y 0 + a f 0 sn θ g A sn θ g A h Subsue n he equaon for a moon n he y-drecon ( sn θ ) sn θ g 1 sn θ g h sn θ g 1

22 Horzonal Range of a Projecle R s he horzonal range Toal me of flgh: B A A R x ( cosθ ) B ( cosθ ) sn θ g A sn θ cosθ. g sn θ sn θ cosθ R sn g B θ

23 Max Horzonal Range? R sn θ g R R when max sn θ 1 g θ? 45 o 3

24 Angle Dependence of Horzonal Range complemenary angles Complemenary alues of Θ (Θ 1 + Θ 90º) resul n he same alue of a horzonal range R of he projecle 4

25 Quck Quz Rank he launch angles for he fe pahs wh respec o he me of flgh, from he shores me of flgh o he longes 5

26 Some Varaons of Projecle Moon Objec may be fred horzonally The nal elocy s all n he x-drecon x and y 0 All he general rules of projecle moon apply 6

27 Some Varaons con. Follow he general rules for projecle moon Noce he orgn chosen: he numercal alue of y f has a negae sgn! 7

28 Unform Crcular Moon An objec raelng n a crcle, een hough moes wh a consan speed, wll hae an acceleraon The cenrpeal acceleraon s due o he change n he drecon of he elocy a f f 8

29 Cenrpeal Acceleraon Cenrpeal refers o cener-seekng The acceleraon n unform crcular moon s always perpendcular o he pah and pons oward he cener of he crcle a r r Perod T s he me of one complee reoluon T a c r πr r r 9

30 Perod T s he me of one complee reoluon Angular Speed The angle for one complee reoluon s π radans Angular speed: rad/s ω π T T πr ω π π r r ωr ω a c r r

31 Quck Quz Whch of he followng correcly descrbes he cenrpeal acceleraon ecor for a parcle mong unformly n a crcular pah? (a) consan and always perpendcular o he elocy ecor for he parcle (b) consan and always parallel o he elocy ecor for he parcle (c) of consan magnude and always perpendcular o he elocy ecor for he parcle (d) of consan magnude and always parallel o he elocy ecor for he parcle 31

32 Moon Along an Arbrary Cured Pah Velocy changes boh n drecon and n magnude The oal acceleraon ecor changes n drecon and n magnude from pon o pon 3

33 Moon Along an Arbrary Cured Pah, con The oal acceleraon ecor a can be resoled no wo componens : a radal componen a r along he radus of he model crcle a angenal componen a perpendcular o hs radus a a r + a 33

34 Tangenal Acceleraon The angenal acceleraon componen causes he change n he speed of he parcle d a d The drecon of a : he same as f s ncreasng oppose f s decreasng 34

35 Radal Acceleraon The radal acceleraon componen arses from he change n drecon of he elocy ecor a r a c r r s he radus of curaure of he pah a he pon a queson he negae sgn: he cenrpeal acceleraon s oppose o he radal un ecor r A a gen speed, a r s large when he radus of curaure s small 35

36 Toal Acceleraon n Terms of Un Vecors Defne un ecors: r s lyng along he radus ecor and dreced radally ouward from he cener of he crcle Θ s angen o he crcle (a ecor!) a a + a r d θˆ d r rˆ a a + a r 36

37 Quck Quz A parcle moes along a pah and s speed ncreases wh me. In whch of he followng cases are s acceleraon and elocy ecors perpendcular eerywhere along he pah? (a) he pah s crcular (b) he pah s sragh (c) he pah s a parabola (d) neer 37

38 Relay of Moon How obseraons of moon made by dfferen obserers n dfferen frames of reference are relaed each oher he relae elocy of he wo frames of reference The man s walkng on he mong belway 38

39 Relae Velocy I may be useful o use a mong frame of reference nsead of a saonary one I s mporan o specfy he frame of reference, snce he moon may be dfferen n dfferen frames of reference 39

40 Relae Velocy Two obserers mong relae o each oher generally do no agree on he oucome of a measuremen 40

41 Gallean Coordnae Transformaon Reference frame S s mong relae o S wh consan elocy 0 A 0 he orgns of he frames S and S concde n space fxed frame A me r s he parcle poson ecor relae o S r s he parcle poson ecor relae o S or r r 0 r r + 0 mong frame 41

42 Gallean Velocy Transformaon r r 0 Dfferenae wh respec o me dr d dr d 0 The elocy of a parcle measured n a fxed frame of reference S can be relaed o he elocy of he same parcle measured n a mong frame S by 0 or + 0 where 0 s he elocy of S relae o S 4

43 Relae Acceleraon The acceleraon of he parcle measured by an obserer n one frame of reference s he same as measured by any oher obserer mong wh consan elocy relae o he frs frame 43 0 Dfferenae wh respec o me d d 0 d d d d d d Because 0 s consan 0 0 a a

44 Solng Relae Velocy Problems The paern of subscrps can be useful n solng relae elocy problems + be br re Wre an equaon for he elocy of neres n erms of he eloces you know, machng he paern of subscrps 44