( ) ( ) ( ) ( ) ( ) ( ) j ( ) A. b) Theorem

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1 b) Theoe The u of he eco pojecon of eco n ll uull pependcul (n he ene of he cl poduc) decon equl o he eco. ( ) n e e o The pojecon conue he eco coponen of he eco. poof. n e ( ) ( ) ( ) e e e e e e e e e o o o n n n j j j n n j j j n Eple. D eco pce of oened egen ( ) ( ) ( ) ( ) j j α α β α n co co co co,n α α α β j

2 KINETICS OF PRTICLE 7. Pcle (.no) - n objec hng non zeo nd he hpe of pon (zeo ze nd no nenl ucue). In ou uon we ppoe el objec b pcle. Fo he nlonl oon of n objec we cn ue h he objec pcle hng he of he objec nd plced he cene of of he objec. 8. Poon (., 3., 4.) - eco qun oced wh confguon of he unee. Fo pcle, 3-D eco qun. z n oened egen nd d of nube cn be gned P (oophcll) o he poon of pcle. The ee w o decbe he poon of pcle o gn n oened egen fo he efeence pon o he O pcle. B noducng coodne e, one cn gn d of nube. + j + zk z [,, ]

3 9. (., 4.) Defnon of dplceen (eco) z O () P In genel, he poon (eco) ( ) of pcle depend on e. The dffeence n poon (eco) of pcle wo dffeen nnce nd clled he dplceen (eco) of he pcle, (, ) n he e,. nel ( ) (, ) ( ) ( ). (4.) Defnon of (nnneou) eloc The e, whch pcle chngng poon (eco), clled he eloc (eco) of he pcle. d d ( ) d d (Snce he dffeenl dplceen (eco) of pcle lw long he jeco, eloc lw ngen o he jeco) Fo e dependen qun, decbed b funcon f (), df f () f ( ) f '() l d o clled he e of chnge of he qun nn. 3

4 . (4.) Defnon of (nnneou) cceleon The e, whch pcle chngng eloc (eco), clled he cceleon (eco) of he pcle. d d d d ( ) d d d d. (B.6) Mhecl uppleen dee of eco. ) The dee of eco funcon of one ble The dee of funcon f of one ble ξ, funcon f' defned b he followng equon f ( ) ( ξ + ξ) f ( ξ) f ' ξ l ξ ξ b) Dffeenl df of funcon. The nfnel chnge df n he lue of he funcon f due o he nfnel chnge dξ of he guen clled he dffeenl of he funcon df ( ) f' ξ d ξ c) Dee o of dffeenl (heoe) f ' ( ξ) df dξ d) Scl coponen of eco dee (heoe). d ( + h) ( ) [ ( + h), ( + h),...] [ ( ), ( ),...] l d h l h h l h ( + h) ( ) ( + h) ( ) h, h,... h d d d, d,... 4

5 Eple (4.3). Pojecle oon - eloc nd cceleon. The genel funcon epeenng he poon of pojecle nn qudc funcon of e: nn he eloc of he fllng pcle : ( ) [,, z ] + [,, ] + [,, g] z ( ) [,, ] + [,, g] z The cceleon nn h e ndependen lue: ( ) [,, g] (Fo eueen he gnude of fee fll cceleon on Eh g 9. 8 ) g g g () g () g 5

6 3. Inee elon beween poon, eloc nd cceleon. (heoe) If he eloc of he pcle known nn nd he cceleon of he pcle known ll nnce ' beween nd, nn he pcle h eloc ( ) ( ) + ( ' ) d' If he poon of he pcle known nn nd he eloc of he pcle known ll nnce ' beween nd, nn he pcle h he poon ( ) ( ) + ( ' ) d' poof. ) Le f () be connuou funcon nd F () f (), hen f ( ) d F( ) F( ) (f fundenl heoe of clculu) Theefoe F ( ) F( ) + f ( ) d 6

7 7 4. (B.7) Mhecl uppleen - negl. ) Defne negl of funcon wh one ble. The defne negl of funcon f(ξ) oe n nel b, defned ( ) ( ) ξ ξ ξ ξ ξ b ~ f l d f b) Scl coponen of eco negl (heoe). ( ) ( ) ( ) [ ] ( ) ( ) d,... d,,... ~ l, ~ l,... ~, ~ l ~ l d b b b f ( ) ξ ξ b

8 3. (4.,4.3) Moon wh conn cceleon - oon n whch he cceleon of pcle doe no depend on e. ( ) Concluon. In h oon, he eloc of he pcle lne funcon of e ( ) + d' + Veco clled he nl eloc, nd epeen he eloc of he pcle he efeence nn. The poon of he pcle ong wh conn cceleon qudc funcon of e. ( ) + + 'd' + + Veco clled he nl poon, nd epeen he poon of he pcle he efeence nn. 8

9 Eple. Dung Wold W I, he Gen hd gun clled Bg Beh h w ued o hell P. The hell hd n nl peed of.7 k/ (gnude of eloc) n nl nclnon of 55 o he hozonl. ) How f w dd he hell h? b) How long w n he? c) Wh he ngle, whch he hell h he gound? d) Wh he u lude of he hell? Le' chooe conenen coodne e nd e efeence. I ugge h we conde he poon of he gun he efeence pon. We cn chooe he un eco n uch w h he hell oe n he -plne. ue lo h he ho he efeence fo he e cle. Wh hee upon he nl poon nd eloc e follow. α nl poon [,, ( ) ] nl eloc co α,n α, nl peed 7 nclnon ngle α55 [ ( )] We cn ppoe he oon of he hell b oon, g,. wh conn cceleon [ ( )] We know he nl eloc of he hell nd cceleon n nn (fe he ho). Inegng cceleon we cn fnd he eloc of he hell n nn: 9

10 ( ) + d' + We lo know he nl poon, heefoe we cn fnd he poon n nn: ( ) + + ' d' + + ) If we know when cen een ke plce, we cn nhng bou he hell' poon, eloc, nd ohe eled qune h e (nn). Theefoe le' f fnd when he hell h he gound. When h hppen ( ), he ecl coponen of he poon eco h zeo lue: + + g n α Mhecll, we he wo oluon o 7 55 n n o α 84 g 9. 8 The f oluon epeen he nn of he ho, nd ndeed he hell w gound leel h e. The econd oluon coepond o he nn he hell h he gound. h e, he poon of he hell w: o o ( 84) [,, ( ) ] + 7 co55,n 55,( ) [ 77,,() ]k [ ], 9.8, 84 + ( ) ( 84) 3

11 The nge (gnude of he hozonl dplceen) heefoe 77k k 77 k b) We know when he gun w ho nd when he hell h he gound. The dffeence n e beween hee wo nnce epeen he e (nel) when he hell w n he : c) The ngle equl o he ngle beween he eloc eco nd he -decon ( ). e, he eloc w [ ] +, 9.8,( ) o o ( 84) + 7 co55,n 55,( ) Theefoe [ 975, 393, ( ) ] o [ 975, 393, ( ) ] o [,, ] 975, 393, ( ) θ cco cco [ ] The hell h he gound he e ngle w ho d) We know how o fnd he poon n nn. To fnd he u lude we he o fnd he u of he funcon g ( ) + + decbng he -coponen of he poon (conen wh ou upon). o 3

12 he u, he dee of funcon zeo. We cn ue h fc. ' g ( ) The hell he u lude e g n α g Theefoe he u lude ( ) n α n g α + g n g α ( n α) g 7 n o 99k (We could lo fnd he u lude fo he condon h eloc hozonl he u.) 3

13 4. (4.) Speed ) (Defnon) The gnude of eloc clled peed. Eple. [,, ] o + ( ) + b) (Theoe) Speed equl o he e whch he pcle coe he "dnce" (long he ph) d d d o d ( ) o d d d d dl d Concluon. The lengh of he pcle' ph equl o he negl of peed oe he e. l dl ph ()d Eple. π ( [ coπ,n π] ( ) π[ n π,coπ] ( ) π ( n π) + ( coπ) π ( ) d π π 33

14 5. (4.) ege lue of phcl qune ) The ege lue of funcon f (ξ) oe n nel,b nube gned follow: f ( b ), b f ( ξ) b dξ b) The ege eloc of pcle (wh epec o e). ( ) D ( ) ( ) ( ) D -( ) ( ) d ( ) ( ) c) The ege cceleon of pcle (wh epec o e). ( ) d ( ) ( ) 34

15 6. (.) The eco poduc ) Defnon B θ B The eco poduc B of wo eco eco C, he gnude of whch C Bn θ (whee θ he ngle beween he ulpled eco), nd he decon of whch pependcul o he plne foed b he ulpled eco, followng he gh-hnd ule. b) Ipon heoe ) In Cen e j k B z ; B B B z B B B, B B, B B z z z z ) B ( B ) ) ( B + C) ( B) + ( C) d ) ( B) d db B + dξ dξ dξ B C o C B o B ) ( ) ( ) ( )C 35

16 7. (4.4) Unfon ccul oon of pcle wo denonl oon wh conn peed long cclul ph clled unfo ccul oon. In Cen e wh θ ω he ogn he cene, wo w cl coponen of ech of he followng qune, poon, z eloc nd cceleon, e honc funcon of e: ( ) [ co ω,n ω,] ( ) ω[ n ω, coω, ] w ( ) ω [ coω,n ω,] ω Veco w clled he ngul eloc of he pcle. In unfo ccul oon he ngul eloc of he pcle conn. ω ( ) ω (The ngul eloc of pcle eco w, pependcul o he plne foed b he eloc nd poon eco, wh he gnude equl o he e of chnge n he decon of he poon eco ω d θ.) d Concluon. If n unfo ccul oon n objec pefo f eoluon pe un e, he gnude of ngul eloc ω πf. If one eoluon ke e T, hen he gnude of ngul eloc ω π T ) 36

17 Eple. e.5 n du oe conn e of e/n. Fnd he peed nd cceleon of ll one lodged n he ed on he oue edge of he e. w Soluon. Snce he peed of he pcle conn, he elon beween peed nd he dnce l elled b he pcle n he gen e peod h ple fo: () l d We cn fnd h lengh decl fo he nfoon n he poble (nube f of eoluon pe nue) nd he elon beween he du nd he ccufeence of ccle. () l π f Fo hee wo equon we fnd peed: πf π Soluon. Fo he nfoon gen n he poble, we cn decl deene he gnude of ngul eloc of he one ( ) ω π d / n 56d / n d / nd ele o he peed of he one ω d / The gnude of cceleon eled o boh he peed of he pcle nd he du of he ccle.5 ω

18 8. (4.6) Gllen nfoon z z' k j O k' j' ' R O' ' Noce h n genel ' Moon lw ele. Thee e oee uon h decbed n cen efeence fe (deened b he efeence pon nd he choce of he be eco) nd nece o epeen n dffeen efeence fe. Th clled nfoon. R + ' ' ( ) R( ) ' ( ' ) ( ) ( ) ( ) +!!! We wll l ou condeon o uon n whch ', j' j nd k ' k dr nd he ele eloc u con. of he d efeence pon doe no depend on e. We wll lo ue h poble o eue e dencll n boh efeence fe. '. The poon of pcle n one efeence fe O eled o he poon of he pcle n he ohe efeence fe O'. ( ) u ' ( ' ) + The boe elon clled Gllen nfoon. Concluon. ( ) u + ' ( ) ( ) ' ( ) ' 38

19 Eple. Decbe he oon of ll one lodged n he ed of ollng e. ' u R ω ' ' We cn el decbe he oon of he pcle n he ped efeence fe ' ' n ω,co ω, nd he oon of he ped efeence fe (n he unped efeence fe) u ω',,. Ung he Gllen nfoon, we cn decbe he oon n he unped efeence fe ω',, + ' n ω,co ω, ' ω + n ω,co ω, Noce h he cceleon of he one h he e cl coponen n boh efeence fe. d ( ) ' [ ω + n ω,coω,] ω ' [ n ω,coω,] ω ' d 39

20 PRTICLE DYNMICS 9. (5.no) The oon of bod ffeced b ohe bode peen n he unee. Th nfluence clled n necon. Thee e few w o decbe h necon b hecl odel: foce F (eco), oque (eco), wok W (cl), powe P (cl), pule J (eco), he Q (cl). Dependng on he uon, one o he ohe of he decpon oe conenen. Of coue he pedcon u gee, heefoe he u be neeled. 3. (5.3) Ine nd Ine he nul endenc of n objec o oe wh conn eloc. M cl qun gned o he nel pope of bod,.e. ence o chnge of oon. 3. (5.) Newon' f lw If pcle doe no nec wh ohe bode beng condeed (we h ll bode ee zeo (eco) foce on he pcle), poble o fnd efeence fe n whch h pcle h zeo cceleon. Th efeence fe clled n nel efeence fe. 4