CHAPTER 13 VECTOR-VALUED FUNCTIONS AND MOTION IN SPACE

Transcription

1 CHAPER VECOR-VALUED FUCIOS AD MOIO I SPACE. CURVES I SPACE AD HEIR AGES. x n y Ê y (x x x; i j Ê j Ê i j n j r r y x y y. x n y Ê x Ê y ; i j Ê i j Ê 4i4 j n 6i 6 j r x e n y e Ê y x ; e i e j Ê e i e j Ê i4 jn i8 j ln r 9 4. x cos n y sin Ê x y ; ( sin i(6 cos j Ê ( 4 cos i( sin j Ê 6 j n 4 i r (cos i(sin j n (sin i(cos j Ê for, ˆ i j n ˆ i j; for, ˆ j n ˆ i r 6. ˆ sin iˆ cos j n ˆ cos ˆ sin i j Ê for, ( i n ( j; for, ˆ i j n ˆ i j r 7. ( cos i(sin j n (sin i(cos j Ê for, ( in ( j; for, ˆ ij n ˆ i r 8. i j n j Ê for, ( i j n ( j; for, ( in ( j; for, ( i j n ( j Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

2 76 Chper Vecor-Vlue Funcions n Moion in Spce r r 9. r ( i j Ê ij Ê j; Spee: ( (( ; ( i( j ( Ê Direcion: i j ( ˆ i j r r $. r ( i j Ê i j Ê j ; Spee: ( ( ( ( i j ( ( Ê Š ( ; Direcion: i j Ê ( Š i j r. r ( cos i( sin j4 Ê ( sin i( cos j4 Ê ( cos i( sin j; Spee: ˆ Ɉ sin ˆ cos 4 5; Direcion: ˆ ˆ 4 Š sin Š cos Ê ˆ i j i 5 Š i r 4 r 4 r. r (sec i(n j Ê (sec n isec j Ê $ ˆ ˆ 4 i j ˆ 6 sec 6 n 6 ˆ sec 6 i j Ê ˆ 6 ˆ i j 6 sec n sec i sec n j; Spee: ˆ Ɉ sec n ˆ sec ˆ 4 ; Direcion: r. r ( ln ( i j Ê ˆ r ij Ê i j; ( Spee: ( Ɉ (( ( 6; Direcion: ( Ê ( 6 i j Š i j Š i( j( 6 4. r e i( cos j( sin Ê e i(6 sin j(6 cos Ê r e (8 cos (8 sin ; Spee: ( Ée i j [ 6 sin (] [6 cos (] 7; ( e i6 sin ( j 6 cos ( 6 6 Direcion: i Ê ( 7 Š i ( r 5. i j n Ê ( i j n ( Ê ( Ê Š n ( ; ( ( Ê cos Ê 6. iš j n j Ê ( i j n ( j Ê ( ÊŠ Š ( 4 n ( ( ; ( ( Š ( 6 6 Ê cos Ê 7. ˆ iˆ Î j n i j Ê ( j n $Î ( i Ê ( n ( 5; ( ( Ê cos Ê Î Î Î Î Ê Éˆ ˆ ˆ Ɉ ˆ ( i ( j n ( i ( j Ê ( i j n ( i j ( n ( ; ( ( Ê cos Ê Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

3 Secion. Cures in Spce n heir nens r( (sin i cos je Ê ( (cos i( sin je ; Ê ( in r( P (ßß Ê x, y, n z re prmeric equions of he nen line. r( i j Ê ( i j ; Ê 4 i j n r( P 4ßß8 Ê x 4 4, y, n z 8 re prmeric equions of he nen line r ß ß Ê. r( ln i j ln Ê ( i jln ; Ê i jn ( P x, y, n z re prmeric equions of he nen line. r( (cos isin j(sin Ê ( ( sin i(cos j( cos ; Ê ( i n r( P (ßß Ê x, y, n z re prmeric equions of he nen line. ( ( (sin i(cos j Ê ( (cos i(sin j; (i ( ( sin (cos Ê consn spee; (ii (sin (cos (cos (sin Ê yes, orhoonl; (iii counerclocwise moemen; (i yes, r( ij ( ( ( sin i( cos j Ê ( (4 cos i(4 sin j; (i ( 4 sin 4 cos Ê consn spee; (ii 8 sin cos 8 cos sin Ê yes, orhoonl; (iii counerclocwise moemen; (i yes, r( ij (c ( sin ˆ icos ˆ j Ê ( cos ˆ isin ˆ j; (i ( Ésin ˆ cos ˆ Ê consn spee; (ii sin ˆ cos ˆ cos ˆ sin ˆ Ê yes, orhoonl; (iii counerclocwise moemen; (i no, r( ijinse of ij ( ( (sin i(cos j Ê ( (cos i(sin j; (i ( ( sin ( cos Ê consn spee; (ii (sin (cos (cos (sin Ê yes, orhoonl; (iii clocwise moemen; (i yes, r( ij (e ( ( sin i( cos j Ê ( ( sin cos i( cos sin j; (i ( Éc sin ( cos 4 sin cos, Ê rile spee; (ii 4 sin sin cos 4 cos cos sin 4 Á in enerl Ê no orhoonl in enerl; (iii counerclocwise moemen; (i yes, r( ij 4. Le p i jenoe he posiion ecor of he poin,, n le, u i j n i j. hen r( p(cos u(sin. oe h (ßß is poin on he plne n n ij is norml o he plne. Moreoer, u n re orhoonl uni ecors wih u n n Ê u n re prllel o he plne. herefore, r( ienifies poin h lies in he plne for ech. Also, for ech, (cos u (sin is uni ecor. Srin he poin Š,, he ecor rces ou circle of rius n cener (ßß in he plne x y z. r Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

4 76 Chper Vecor-Vlue Funcions n Moion in Spce 5. he elociy ecor is nen o he rph of y x he poin (ß, hs lenh 5, n posiie i y y x x¹ ÐßÑ componen. ow, y x Ê y Ê Ê he nen ecor lies in he irecion of he 5 ecor i j Ê he elociy ecor is ˆ i j 5 ˆ i j 5i5j É 4 Œ 5 6. ( ( ( cos i(sin jn (sin i(cos j; ( cos sin cos Ê is mx when cos Ê,, 5, ec., n hese lues of, 4 Ê mx 4 ; is min when cos Ê,, 4, ec., n hese lues of, Ê min ; sin cos for eery Ê mx min 7. ( r r r r r r r r Ê r ris consn Ê r r r is consn 8. ( ( u ( ( u u w w u w ( w u ˆ w w u ( w u w w u r r r r r r r r r r r $ $ $ $ ( r Š Š r Š r Š r Š, since A ( A B n A ( B B for ny ecors A n B f h 9. ( u f( i( jh( Ê cu cf( ic( jch( Ê (c u c ic jc f h u cš i j c f f f f h ( fu ff( if( jfh( Ê ( fu f( f i ( f j h( f f f h f u [f( i( jh( ] f i j uf. Le u f ( if ( jf $ ( n ( i ( j $ (. hen u [f( (] i[f ( (] j[f $ ( $ (] w w w w w w Ê ( u [f( (] i[f ( (] j[f $ ( $ (] [f w ( f w ( f w ( ] [ w ( w ( w u i j i j ( ] ; $ $ u [f( (] i[f ( (] j[f $ ( $ (] w w w w w w Ê ( u [f( (] i[f ( (] j[f $ ( $ (] w w w w w w u [f ( if ( jf ( ] [ ( i ( j ( ] $ $. Suppose r is coninuous. hen lim r( r( Í lim [f( i( jh( ] Ä Ä f( i( jh( Í lim f( f(, lim ( (, n lim h( h( Í f,, n h re Ä Ä Ä coninuous. Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

5 Secion. Cures in Spce n heir nens 76 i j i j. lim [ ( (] lim f( f( f( lim f ( lim f ( r r $ lim f $ ( Ä Ä Ä Ä Ä ( ( ( $ lim ( lim ( lim $ ( Ä Ä Ä lim ( lim ( Ä r Ä r A B w w w w w w w. r( exiss Ê f ( i ( jh ( exiss Ê f (, (, h ( ll exis Ê f,, n h re coninuous Ê r( is coninuous u c 4. u C ijc wih,, c rel consns Ê i j ij 5-8. Exmple CAS commns: Mple: > wih( plos ; r := -> [sin(-*cos(,cos(+*sin(,^]; := *Pi/; lo := ; hi := 6*Pi; P := spcecure( r(, =lo..hi, xes=oxe, hicness= : isply( P, ile=5( (Secion. ; Dr := unpply( iff(r(,, ; ( Dr(; (c q := expn( r( + Dr(*(- ; := unpply( q, ; P := spcecure( (, =lo..hi, xes=oxe, hicness=, color=lc : isply( [P,P], ile=5( (Secion. ; 9-4. Exmple CAS commns: Mple: := ; := ; r := (,, -> [cos(*,sin(*,*]; Dr := unpply( iff(r(,,,, (,, ; := *Pi/; q := expn( r(,, + Dr(,,*(- ; := unpply( q, (,, ; lo := ; hi := 4*Pi; P := ULL: for in [,, 4, 6 ] o P := spcecure( r(,,, =lo..hi, hicness= : P := spcecure( (,,, =lo..hi, hicness=, color=lc : P := P, isply( [P,P], xes=oxe, ile=sprinf(9 (Secion.\n =%, ; en o: isply( [P], insequence=rue ; 5-4. Exmple CAS commns: Mhemic: (ssine funcions, prmeers, n inerls will ry he x-y-z componens for he cure re enere s lis of funcions of. he uni ecors i, j, If rph is oo smll, hihlih i n r ou corner or sie o me i lrer. Copyrih Person Eucion Inc. Pulishin s Aison-Wesley. re no insere.

6 764 Chper Vecor-Vlue Funcions n Moion in Spce Only he componens of r[] n lues for, min, n mx require lerion for ech prolem. Cler[r,,, x, y, z] r[_]={ Sin[] Cos[], Cos[] Sin[], ^} = / ; min= ; mx= 6; PrmericPloD[Elue[r[]], {, min, mx}, AxesLel Ä {x, y, z}]; [_]= r[] nline[_]= [] r[] PrmericPloD[Elue[{r[], nline[]}], {, min, mx}, AxesLel Ä {x, y, z}]; For 9 n 4, he cure cn e efine s funcion of,, n. Lee spce eween n n n. Cler[r,,, x, y, z,, ] r[_,_,_]:={cos[ ], Sin[ ], } = / ; min= ; mx= 4; [_,_,_]= D[r[,, ], ] nline[_,_,_]=[,, ] r[,, ] p=prmericplod[elue[{r[,, ], nline[,, ]}], {,min, mx}, AxesLel Ä {x, y, z}]; p=prmericplod[elue[{r[,, ], nline[,, ]}], {,min, mx}, AxesLel Ä {x, y, z}]; p4=prmericplod[elue[{r[, 4, ], nline[, 4, ]}], {,min, mx}, AxesLel Ä {x, y, z}]; p6=prmericplod[elue[{r[, 6, ], nline[, 6, ]}], {,min, mx}, AxesLel Ä {x, y, z}]; Show[GrphicsRow[{p, p, p4, p6}]]. IEGRALS OF VECOR FUCIOS; PROJECILE MOIO %. c $ 7 ( [7] i j i j i7j 4 4. (6 6 ˆ 4 6 i j c i jc4 iš 4 j Î% $Î. c(sin i( cos jsec ccos ic sin jcn Š j cî% Î Î Î% Î% Î% Î% Î% Î% 4. csec n in j sin cos [ sec n in jsin ] Î$ Î$ c c Î$ sec i ln cos j cos i (ln j 5. 4ˆ % % ln ln (5 c c ln % i j i j (ln 4 i(ln 4 j(ln 5 6. Š c sin n i i i 4 7. Š e e e c e c e e i j i j i i ln e ln ln ln 8. e ie jln ce e ice jc ln lnieilnlnln Î Î 9. ccos sin sin cos sin ˆ i j i j cos Î csin Î cos Î sin 4 4 i j ij 4 Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

7 /4 /4 Secion. Inerls of Vecor Funcions; Projecile Moion 765. csec in j sin csec isec j sin clnsec n /4 cn /4 c cos sin /4 lnš ˆ i j i jš 4 4. r ( ij i j C; r( ijc ij Ê C ij Ê r Š iš jš. r c (8 i 8 6 j 9 iˆ 9 6 $ jc; r( 9( i 9( 6 ( $ jc j Ê C j Ê r 9 iˆ 6 $ 9 j. r ˆ Î c ( ie jˆ $Î c ( ie jln ( C; $Î c r( ( ie jln ( C Ê C ij Ê r $Î ( c i e j[ ln ( ] 4. r c $ 4 ij % Š $ % i j C; r( ( $ ( i j C % $ ij Ê C ij Ê r Š iš j r r 5. ( C ; ( 8i8 j Ê ( C 8i8 j Ê C 8i8j Ê r 8 i 8 j ; r (8 i 8 j 8 i 8 j 6 C ; r ( Ê 8( i8( j6( C Ê C Ê r 8i8j 6 r r 6. ( ij (ij C ; ( Ê (ij C Ê C r Ê (ij ; r (ij Š i j C ; r( ij Ê Š i j C ij Ê C ij Ê r Š iš jš 7. ij Ê ( ij C ; he pricle rels in he irecion of he ecor (4 i( j(4 ij(since i rels in srih line, n ime i hs spee r 6 9 Ê ( ( ij C Ê ( Š iš jš 6 Ê r( Š iš jš C ; r( ij C 6 Ê r( Š iš jš Š ( ij ( ij 8. ij Ê ( ij C ; he pricle rels in he irecion of he ecor ( i( ( j( ij(since i rels in srih line, n ime i hs spee r Ê ( ( ij C Ê ( Š iš jš Ê r( Š iš jš C ; r( ij C Ê r( Š iš jš Š ( ij ( ij Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

8 766 Chper Vecor-Vlue Funcions n Moion in Spce 9. x ( cos Ê ( m ˆ m, m (84 m/s(cos 6 Ê 5 secons m (84 m/s(cos m/s. R sin n mximum R occurs when 45 Ê 4.5 m Š (sin 9 Ê (9.8(4,5 m /s 49 m/s sin (5 m/s(sin 45 (5 m/s 9.8 m/s 9.8 m/s 5 m (5 m/s(cos 45 ( sin ((5 m/s(sin m/s 678 m. ( 7. secons; R sin (sin 9 5,5. m ( x ( cos Ê 5 m (5 m/s(cos 45 Ê 4.4 s; hus, y ( sin Ê y (5 m/s(sin 45 (4.4 s 9.8 m/s (4.4 s 4 m (c ymx. y y ( sin Ê y f ( f/sec(sin f/sec Ê y 6 6 ; he ll his he roun when y Ê 6 6 Ê or Ê sec since ; hus, x ( cos Ê x ( f/sec(cos Š ( 55.4 f 9.8 m/s. ( R sin Ê m Š (sin 9 Ê 98 m s Ê 9.9 m/s; (9.9 m/s 9.8 m/s ( 6m (sin Ê sin Ê 6.87 or 4. Ê 8.4 or m/s n x 4 cm.4 m; hus x ( cos Ê.4m 5 m/s (cos Ê.8 s 8 s; lso, y y ( sin % Ê y 5 m/s (sin 8 s 9.8 m/s 8 s.6 m or.6 cm. herefore, i rops.6 cm. (4 m/s 9.8 m/s 5. R sin Ê 6, m sin Ê sin.98 Ê 78.5 or.5 Ê 9. or 5.7 ( 4 6. ( R sin sin 4 Š sin or 4 imes he oriinl rne. ( ow, le he iniil rne e R sin. hen we wn he fcor p so h p will oule he rne (p Ê sin Š sin Ê p Ê p or ou 4%. he sme percene will pproximely p sin sin oule he heih: Ê p Ê p. y sin sin sin sin sin sin mx ˆ Ê Ê sin sin sin sin Ê Š 7. he projecile reches is mximum heih when is ericl componen of elociy is zero Ê sin Ê Ê y sin Š Š. o fin he flih ime we fin he ime when he projecile lns: sin sin or. is he ime when he projecile is fire, so is he ime when he projecile sries he roun. he rne is he lue of he horizonl componen when R x cos sin cos sin. he rne is lres when sin Ê 45. R 8. When mrle A is loce R unis ownrne, we he x ( cos Ê R ( cos Ê. A cos R R Š cos Š cos h ime he heih of mrle A is y y ( sin ( sin Ê y R n Š. he heih of mrle B he sme ime cos secons is R R cos Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

9 Secion. Inerls of Vecor Funcions; Projecile Moion 767 R cos h R n R n Š. Since he heihs re he sme, he mrles collie rerless of he iniil elociy. r r 9. ( j jc n ( ( cos i( sin j Ê ( jc ( cos i( sin j C ( cos i ( sin j r ( cos i ( sin j; r [( cos i ( sin j] i ˆ j C r i j i j C i j C i j r i ˆ j Ê Ê ( cos sin n ( x y Ê [ ( cos ] ( sin ( x y Ê x y Ê (x cos y sin Ê x x cos n y y sin ( sin sin cos. he mximum heih is y n his occurs for x sin. hese equions escrie prmericlly he poins on cure in he xy-plne ssocie wih he mximum heihs on he prolic rjecories in ˆ % sin sin % sin cos erms of he prmeer (lunch nle. Eliminin he prmeer, we he x % sin % sin % % % 6 4 (y (y Ê x 4y Š y Ê x 4 y Š y 4 4 % Ê x 4 Š y, where x.. ( A he ime when he projecile his he line OR we y he n ; x [ cos ( ] n x y [ sin ( ] since R is elow leel roun. herefore le y [ sin ( ] so h n ( sin ( sin ( [ cos ( ] cos ( sin ( cos ( n Ê cos ( n sin ( Ê, which is he ime when he projecile his he ownhill slope. herefore, x [ cos ( ] ccos ( n sin ( cos (. If x is sin ( cos ( n x mximize, hen OR is mximize: [ sin ( n cos ( ] Ê sin ( n cos ( Ê n co ( Ê ( 9 Ê (9 Ê (9 of naor. ( A he ime when he projecile his OR we he y n ; x [ cos ( ] n x c sin( sin( [ cos ( ] cos ( sin ( cos ( n y [ sin ( ] Ê n Ê cos( n sin( Ê, which is he ime when he projecile his he uphill slope. herefore, x [ cos ( ] csin ( cos ( cos ( n. If x is sin ( cos ( n x mximize, hen OR is mximize: [cos ( sin ( n ] Ê cos ( sin ( n Ê co ( n Ê co ( n n ( Ê ( 9 ( 9 Ê (9 of naor. herefore woul isec naor for mximum rne uphill. Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

10 768 Chper Vecor-Vlue Funcions n Moion in Spce. 6 f/sec, 45, n x ( cos Ê 69 (6 cos 45 Ê 4.5 sec; lso y ( sin Ê y (6 sin 45 (4.5 (( f. I will e he ll 4.5 sec o rel 69 f. A h ime he ll will e 45. f in he ir n will hi he reen ps he pin.. ( (Assumin h x is zero he poin of impc: r x iy j; where x 5 cos 7 n y 4 5 sin 7 6. sin 5sin 7 sin 5sin 7 ( ymx fee, which is reche.497 secons. (c For he ime, sole y 4 5 sin 7 6 for, usin he quric formul 5 sin 7 É5 sin sec. hen he rne is ou x. 5 cos fee. ( For he ime, sole y 4 5 sin for, usin he quric formul 5 sin 7 É5 sin n.74 secons. A hose imes he ll is ou x.54 5 cos fee n x.74 5 cos fee he impc poin, or ou fee n fee from he lnin spo. (e Yes. I chnes hins ecuse he ll won cler he ne (ymx x x ( cos ( cos n y y ( sin 6.5 ( sin ; now he sho wen 7.8 f Ê Ê sec; he sho lns when y 48,65 48, Ê 6.5 (.64( Š Ê Ê É 46.6 f/sec, he shos iniil spee 5. Flih ime sec n he mesure of he nle of eleion is ou 64 (usin prorcor so h sin 64 (7.8 sin 64 (7.8 mx ( sin Ê Ê 7.8 f/sec. hen y 4. f n R sin Ê R sin f Ê he enine rele ou 7.8 f in sec Ê he enine elociy ws ou 7.8 f/sec 6. ( r x iy j; where x 45 cos 4 n y.5 45 sin 6. ( y fee, which is reche.77 secons. mx sin 45sin sin 45sin 64 (c For he ime, sole y.5 45 sin 6 for, usin he quric formul 45 sin É45 sin sec. hen he rne.585 is ou x 45 cos fee. ( For he ime, sole y.5 45 sin 6 for, usin he quric formul 45 sin É45 sin.4 n.99 secons. A hose imes he ll is ou x.4 45 cos fee from home ple n x cos fee from home ple. (e Yes. Accorin o pr (, he ll is sill fee oe he roun when i is 8 fee from home ple. r r P r 7. j Ê P n Q j Ê P Ê e e Ê Q e e e e j j C jce, where C C ; pply he iniil coniion: r ¹ cos i sin j j C Ê C cos i ˆ sin j r Ê ˆ ˆ c e cos i e ˆ sin j, r ˆ e cos i ˆ e ˆ sin j Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

11 c Secion. Inerls of Vecor Funcions; Projecile Moion 769 ˆ e cos e ˆ i Š sin j C ; pply he iniil coniion: r ˆ cos i ˆ sin j C Ê C ˆ cos iˆ sin j ˆ ˆ ˆ Ê r ˆ e cos iˆ e sin e j 8. ( r x i y j; where x ˆ 5.. e cos n y ˆ 5. e sin ˆ.. e.. ( Sole rphiclly usin clculor or CAS: A.484 secons he ll reches mximum heih of ou 4.45 fee. (c Use rphin clculor or CAS o fin h y when he ll hs rele for.6 secons. he rne is ou x.6 ˆ 5 ˆ..6 e. cos 7. fee. ( Use rphin clculor or CAS o fin h y for.689 n.5 secons, which imes he ll is ou x fee n x fee from home ple. (e Yes, he er hs hi home run since rph of he rjecory shows h he ll is more hn 4 fee oe he roun when i psses oer he fence. 9. ( r( [f( i( jh( ] [f(] i [(] j [h(] Œ f( i ( j h( r( ( [ r ( r (] cf ( i ( j h ( cf ( i ( jh ( f ( c f ( i [( ( ] j [h( h (] cf ( f ( i c ( ( j ch ( h ( f ( i f ( i ( j ( j h ( h ( r ( r ( (c Le C c ic jc. hen C r( cc f( c ( c h( $ $ c f( c ( c h( = C r( ; $ C r( cc h( c ( icc f( c h( jcc ( c f( $ $ c h( c $ ( i c $ f( c h( j c ( c f( C r( 4. ( Le u n r e coninuous on [ß]. hen lim u( r( lim [u(f( iu(( ju(h( ] Ä Ä u( f( iu( ( ju( h( u( r( Ê u ris coninuous for eery in [ß]. ( Le u n re ifferenile. hen (u r [u(f( iu(( ju(h( ] ˆ u f f( u( u Š ( u( ˆ u h i j h( u( [f( i( jh( ] u( Š i j r u u f h u r R f h f h R f f h h $ 4. ( If R ( n R ( he ienicl eriies on I, hen i j i j Ê,, Ê f ( f ( c, ( ( c, h ( h ( c Ê f ( i ( jh ( [f( c ] i[ ( c ] j[h ( c $ ] Ê R ( R ( C, where C c ic jc. $ Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

12 77 Chper Vecor-Vlue Funcions n Moion in Spce w ( Le R( e n nieriie of r( on I. hen R ( r(. If U( is n nieriie of r( on I, hen w w w U( r(. hus U( R ( on I Ê U( R( C. i j r r r 7 7 r r( 7 7 [f( 7 i( 7 jh( 7 ] 7 f( 7 7 i ( 7 7 j h( 7 7 f( ( h( (. Since ( (, we he h ( is n nieriie of r. If R is ny nieriie of r, hen R( r( 7 7 C y Exercise 4(. hen R( r( 7 7 C C Ê C R( Ê r( 7 7 R( C R( R( Ê r( 7 7 R( R(. 4. ( r x i y j; where x ˆ.8.8 e 5 cos 7.6 n y ˆ 5.8 e sin ˆ.8.8 e.8.8 ( Sole rphiclly usin clculor or CAS: A.57 secons he ll reches mximum heih of ou 4.89 fee. (c Use rphin clculor or CAS o fin h y when he ll hs rele for.8 secons. he rne is ou x.8 ˆ ˆ.8.8 e.8 5 cos fee. ( Use rphin clculor or CAS o fin h y 5 for.877 n.9 secons, which imes he ll is ou x fee n x fee from home ple. (e o; he rne is less hn 8 fee. o fin he win neee for home run, firs use he meho of pr ( o fin h y.76 n.76 secons. hen efine xw ˆ ˆ.8.76 e 5 cos w, n sole xw 8 o fin w.846 f/sec y Ê y n y ( sin Ê ( sin mx ( sin ( sin ( sin 4 mx 8 8 sin sin sin mx mx sin 4 mx Ê ( sin (8 sin 4 Ê 4 (8 sin ( sin Ê sin or sin Ê or. Since he ime i es o rech y is, hen he ime i es he projecile o rech of y is he shorer ime or hlf he ime i es o rech he mximum heih.. ARC LEGH I SPACE. r ( cos i( sin j5 Ê ( sin i( cos j5 Ê Ê( sin ( cos Š 5 4 sin 4 cos 5 ; ˆ sin ˆ cos 5 i j n Lenh c. r (6 sin i(6 cos j5 Ê ( cos i( sin j5 Ê ( cos ( sin 5 44 cos 44 sin 5 ; cos sin 5 i j n Lenh ˆ ˆ c $Î Î. r i Ê i Ê Î É ; i n Lenh ( 8 $Î 5 4. r ( i( j Ê ij Ê ( ; i j n Lenh $ Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

13 $ $ 5. r cos jsin Ê cos sin j sin cos Ê É cos sin sin cos 9 cos sin cos sin cos sin ; cos sin sin cos cos sin cos sin j ( cos j(sin, if Ÿ Ÿ, n Î Î Î Lenh cos sin cos sin sin cos Secion. Arc Lenh in Spce 77 Î 4 $ $ $ 6. r 6 i j Ê 8 i6 j9 Ê É 44% ; $ i j i j n Lenh c7 49 $Î Î 7. r ( cos i( sin j Ê (cos sin i(sin cos j Š Ê Ê(cos sin (sin cos Š (, if ; Î ˆ cos sin iˆ sin cos jš n Lenh ( 8. r ( sin cos i( cos sin j Ê (sin cos sin i(cos sin cos j ( cos i( sin j Ê ( cos ( sin if Ÿ Ÿ ; cos sin i j (cos i (sin j n Lenh ˆ ˆ 9. Le P( enoe he poin. hen (5 cos i(5 sin j n 6 5 cos 5 sin 44 Ê, n he poin is P( (5 sin ß5 cos ß4 (ß5ß4. Le P( enoe he poin. hen ( cos i( sin j5 n 44 cos 44 sin 5 Ê, n he poin is P( ( sin ( ß cos ( ß5 (ßß5. r (4 cos i(4 sin j Ê ( 4 sin i(4 cos j Ê ( 4 sin (4 cos 5 5 Ê s( 5 5 Ê Lenh s ˆ 7 5. r (cos sin i(sin cos j Ê ( sin sin cos i(cos cos sin j ( cos i( sin j Ê ( cos ( cos, since Ÿ Ÿ Ê s( 7 7 ˆ 8 Ê Lenh s( s ˆ. r e cos ie sin je Ê e cos e sin ie sin e cos je Ê Ée cos e sin e sin e cos e e e Ê s( e ln 4 4 e Ê Lenh s( s( ln 4 Š e r ( i( j (6 6 Ê ij6 Ê ( 6 7 Ê s( Ê Lenh s( s( ( 7 7 Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

14 77 Chper Vecor-Vlue Funcions n Moion in Spce 5. r Š iš j Ê ij Ê ÊŠ Š ( 4 4 Ê Lenh Š ln Š ln Š 6. Le he helix me one complee urn from o. oe h he rius of he cyliner is Ê he circumference of he se is. When, he poin P is (cos ßsin ß (ßß Ê he cyliner is unis hih. Cu he cyliner lon PQ n flen. he resulin recnle hs wih equl o he circumference of he cyliner n heih equl o, he heih of he cyliner. herefore, he recnle is squre n he porion of he helix from o is is ionl. 7. ( r (cos i(sin j (cos, Ÿ Ÿ Ê x cos, y sin, z cos Ê x y cos sin, rih circulr cyliner wih he z-xis s he xis n rius. herefore P(cos ßsin ß cos lies on he cyliner x y ; Ê P(ßß is on he cure; Ê Q( ßß Ä Ä is on he cure; Ê R( ßß is on he cure. hen PQ ij n PR i i j Ê PQ Ä Ä Ô PR i is ecor norml o he plne of P, Q, n R. hen he Õ Ø plne coninin P, Q, n R hs n equion x z ( ( or x z. Any poin on he cure will sisfy his equion since x z cos ( cos. herefore, ny poin on he cure lies on he inersecion of he cyliner x y n he plne x z Ê he cure is n ellipse. ( ( sin i(cos j(sin Ê sin cos sin sin Ê ( sin (cos (sin Ê ( j, ˆ, ( j, ˆ i j i i sin (c ( cos i(sin j(cos ; n i is norml o he plne x z Ê n cos cos Ê is orhoonl o n Ê is prllel o he plne; ( i, ˆ j, i, ˆ j ( sin (See pr ( Ê L sin (e L 7.64 (y Mhemic 8. ( r (cos 4 i(sin 4 j4 Ê ( 4 sin 4 i(4 cos 4 j4 Ê ( 4 sin 4 (4 cos 4 4 Î Î 4 Ê Lenh 4 4 % Lenh c ( r ˆ cos iˆ sin j Ê ˆ sin iˆ cos j Ê Éˆ sin ˆ cos ˆ É Ê Lenh (c r (cos i(sin j Ê ( sin i(cos j Ê ( sin ( cos ( Ê Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

15 9. npqb nqob n PQ rc (AQ since PQ lenh of he unwoun srin lenh of rc (AQ; hus x OB BC OB DP cos sin, n y PC QB QD sin cos Secion.4 Curure n orml Vecors of Cure 77. r cos sin isin cos j Ê sin cos sin icos sin cos j cos i sin j Ê É cos sin cos sin, Ê i j cos isin j. x u i y u j z u u iu ju u, so s ll lul7 7. r i j i j É Ê Ê l l ,, Ê n, 4, 8 Ê. hus L l l Usin Simpsons rule wih n n.? x. Ê L Šll4l.ll.4l4l.6ll.8l4lll.l4l.4l. l.6l4l.8lll Š CURVAURE AD ORMAL VECORS OF A CURVE. r iln (cos j Ê iˆ sin j i(n j Ê ( n sec cos sec sec, since Ê ˆ iˆ n j (cos i(sin j; ( sin i(cos j sec sec ˆ, sec Ê ( sin ( cos Ê ( sin (cos ; i j cos.. r ln (sec i j Ê ˆ sec n ij (n ij Ê ( n sec sec sec sec, since Ê ˆ n iˆ sec sec j (sin i(cos j; (cos i(sin j ˆ, sec Ê (cos ( sin Ê (cos (sin ; i j cos.. r ( i5 j Ê i j Ê ( Ê i j ; Í Ê i j $ i $ j $ $ Š Š Š Š Ì ˆ, / É Ê ; i j 4. r (cos sin i(sin cos j Ê ( cos i( sin j Ê ( cos ( sin, since ( cos i ( sin j Ê (cos i(sin j; ( sin i(cos j Ê ( sin (cos ˆ, Ê ( sin i(cos j; Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

16 774 Chper Vecor-Vlue Funcions n Moion in Spce x x w 5. (,x ¹ ¹. ow, if xj Ê x É cfwx Ê w Î Î w w Š cf x if xš cf x j. hus x i j Í f x f x w ww x Ê ¹ ¹ Ì f x fww c x fw ww Š c x Î cfw x Î Š Ë cfw x cfw Š Š x ww ww f x f x Òfw x Ó Î Òfw x Ó Š f x Î w c hus,x w ww ww f xf x f x Î Î fw x fw Š c Š c x ww f x ¹ cfw x ¹ y y sec x ( y ln (cos x Ê ˆ ( sin x n x Ê sec x Ê, sec x x cos x x c ( n x $Î sec$ x x sec x cos x, since ww (c oe h f (x n inflecion poin. Þ Þ Þ Þ Þ Þ 6. ( r f( i( j xiy j Ê xiy j Ê x y x y Ê i j Þ x Þ y Þ Þ x y Þ ÞÞÞ ÞÞÞ Þ ÞÞÞ ÞÞÞ Þ ÞÞÞ ÞÞÞ yyxxy xxyyx yyxxy Þ ÞÞÞ ÞÞÞ xxy yx Þ Þ Þ Þ Ê Þ Þ / i / j Ê / x y x y x y Þ Þ / x y Ê Þ Þ ÞÞÞ ÞÞÞ y x yxxy Þ Þ x y Þ ÞÞ Þ ÞÞ Þ ÞÞ Þ ÞÞ Þ ÞÞ Þ ÞÞ yxxy yxxy lxyyxl Þ Þ ;,. x y Þ Þ Þ Þ x y x y Þ Þ / x y Þ ÞÞ Þ cos ÞÞ sin csc csc, co $Î csc$ Þ cosh Ê Ê sinh cosh $ ÞÞ Þ sinh ÞÞ sech sech nh cosh, sech nh ( r( iln (sin j, Ê x n y ln (sin Ê x, x ; y co, y csc Ê sin (c r( n (sinh i ln (cosh j x n (sinh n y ln (cosh x sech, x sech nh ; y nh, y sech Ê sech sech w w 7. ( r( f( i( j Ê f ( i ( jis nen o he cure he poin (f( ß(; w w w w w w w w n c ( if ( j cf ( i ( j (f ( f ( ( ; n ( n ; hus, nn nre oh norml o he cure he poin n ( r( ie j Ê ie j Ê n e ijpoins owr he conce sie of he cure; n e 4e4 4e4 4 4 n 4e 4 Ê i j (c r( 4 i j Ê ij Ê n i jpoins owr he conce sie of he cure; n n n É Ê Š 4 ij n 4 4 $ 8. ( r( i j Ê i j Ê n ijpoins owr he conce sie of he cure when n n ij poins owr he conce sie when Ê ij for n % ij for % % 464 % % % $Î % $Î % $ ( From pr (, Ê i j Ê i j Ê É ˆ % $ $ ; ; Á. oes no exis, where % Š % $Î i % $Î j i % j he % cure hs poin of inflecion; so he curure, s s Ê s is, $ $ unefine. Since x n y Ê y x, he cure is he cuic power cure which is conce own for x n conce up for x. 9. r ( sin i( cos j4 Ê ( cos i( sin j4 Ê ( cos ( sin Ê ˆ cos iˆ sin 4 j Ê ˆ sin iˆ cos j n Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

17 Secion.4 Curure n orml Vecors of Cure 775 ˆ 5 5 5, Ê Éˆ sin ˆ cos Ê ( sin i(cos j;. r (cos sin i(sin cos j Ê ( cos i( sin j Ê ( cos ( sin, if Ê (cos i(sin j, Ê ( sin i(cos j ˆ, Ê ( sin (cos Ê ( sin i(cos j;. r e cos ie sin j Ê e cos e sin ie sin e cos j Ê Ée cos e sin e sin e cos e e ; cos sin sin cos sin cos cos sin Š iš j Ê Š iš j ˆ e e Ê sin cos cos sin ÊŠ Š Ê cos sin sin cos Š i Š j ;,. r (6 sin i(6 cos j5 Ê ( cos i( sin j5 Ê ( cos ( sin 5 69 Ê ˆ cos ˆ sin 5 Ê ˆ 4 i j sin i ˆ 4 cos j Ê Éˆ 4 sin ˆ 4 cos 4 Ê ( sin i(cos j;, ˆ $. r Š iš j, Ê i j Ê, since Ê % $Î $Î $Î $Î ˆ É $ i j, i j Ê i j Ê ÊŠ Š Ê ;. $Î $ $ 4. r cos isin j, Ê cos sin i sin cos j Ê É cos sin sin cos 9 cos% sin 9 sin% cos cos sin, since Ê ( cos i(sin j Ê (sin i(cos j Ê sin cos Ê (sin i(cos j ;,. cos sin cos sin ˆ 5. r iˆ cosh j, Ê iˆ sinh j Ê É sinh ˆ Écosh ˆ cosh Ê ˆ sech iˆ nh j Ê ˆ sech nh iˆ sech j ˆ sech ˆ sech ˆ. cosh Ê É sech ˆ nh ˆ sech % ˆ sech ˆ Ê ˆ nh ˆ sech i j;, 6. r (cosh i(sinh j Ê (sinh i(cosh j Ê sinh ( cosh cosh Ê Š nh i jš sech Ê Š sech iš sech nh Ê É sech% sech nh sech Ê (sech i (nh ;, sech sech. cosh ˆ Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

18 776 Chper Vecor-Vlue Funcions n Moion in Spce w ww $Î 4 x 7. y x Ê y x Ê y ; from Exercise 5(,,(x 4 x $Î w Ê, (x w w w 4 x 8 x ; hus,, (x Ê x. ow,, (x for x n, (x for &Î x so h,(x hs n solue mximum x which is he erex of he prol. Since x is he only criicl poin for,(x, he curure hs no minimum lue. 8. r ( cos i( sin j Ê ( sin i( cos j Ê ( cos i( sin j Ê i j sin cos Ê, since ;,( $ cos sin $Î w &Î, &Î w sin cos ; ( ( sin cos sin cos sin cos ( (sin sin cos ; hus,, ( Ê sin Ê, ienifyin poins on he mjor xis, or, ienifyin poins on he minor xis. Furhermore,, w ( for w n for ;, ( for n. herefore, he poins ssocie wih n on he mjor xis ie solue mximum curure n he poins ssocie wih n on he minor xis ie solue minimum curure.,,, 9., Ê ; Ê Ê Ê since,. ow, if, n if Ê, is mximum for n,( is he mximum lue of,.. ( From Exmple 5, he curure of he helix r( ( cos i( sin j,, is ; lso,. For he helix r ( ( cos i ( sin j, Ÿ Ÿ 4, n Ê, 4 % n Ê K ( y x Ê x n y, Ê r( i j Ê i j Ê 4 ; i j; i j;. hus / 4 / É 4 4,. hen K c_ Š $ c_ Š 4 Š 4 lim lim lim n lim n Ä_ 4 Ä_ 4 c c Ä_ Ä_ lim n lim n Ä_ Ä_. r i(sin j Ê i(cos j Ê (cos cos Ê ˆ É cos ˆ ; sin cos cos / cos / cos cosˆ n he cener is ˆ ˆ x y i cos j Ê sin cos sin Ê sin ; i j. hus,ˆ Ê ß Ê. r ( ln iˆ j Ê ˆ iˆ j Ê É 4 ˆ Ê i j; ˆ Ê. hus Ê i j Ê 4,,, Ê. he circle of curure is nen o he cure P(ß Ê circle hs sme nen s he cure Ê ( iis nen o he circle Ê he cener lies on he y-xis. If Á (, hen ( Ê Ê Ê since Ê Ê ˆ Ê y on oh sies of (ß Ê he cure is conce own Ê cener of circle of curure is (ß4 Ê x (y 4 4 is n equion of he circle of curure Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

19 Secion.4 Curure n orml Vecors of Cure 777 w ww. y x Ê f (x x n f (x (x $Î 4x Ê, $Î x 4 w $ ww % 4. y Ê f (x x n f (x x x x Š x$ x Ê, $Î $Î w 5. y sin x Ê f (x cos x n f (x sin x sin x sin x cos $Î x cos x Ê, ww $Î x w x ww x 6. y e Ê f (x e n f (x e x e Ê, Š ex x e ˆ ex $Î $Î 7-4. Exmple CAS commns: Mple: wih( plos ; r := -> [*cos(,5*sin(]; lo := ; hi := *Pi; := Pi/4; P := plo( [r([], =lo..hi] : isply( P, sclin=consrine, ile=7( (Secion.4 ; CURVAURE := (x,y, ->simplify(s(iff(x,*iff(y,,-iff(y,*iff(x,,/(iff(x,^+iff(y,^^(/; pp := el(curvaure(r([],,=; Uniorml := (x,y, ->expn( [-iff(y,,iff(x,]/sqr(iff(x,^+iff(y,^ ; := el( Uniorml(r([],, = ; C := expn( r( + /pp ; OscCircle := (x-c[]^+(y-c[]^ = /pp^; elf( OscCircle ; Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

20 778 Chper Vecor-Vlue Funcions n Moion in Spce P := impliciplo( (x-c[]^+(y-c[]^ = /pp^, x=-7..4, y=-4..6, color=lue : isply( [P,P], sclin=consrine, ile=7(e (Secion.4 ; Mhemic: (ssine funcions n prmeers my ry In Mhemic, he o prouc cn e pplie eiher wih perio. or wih he wor, Do. Similrly, he cross prouc cn e pplie eiher wih ery smll x (in he plee nex o he rrow or wih he wor, Cross. Howeer, he Cross commn ssumes he ecors re in hree imensions For he purposes of pplyin he cross prouc commn, we will efine he posiion ecor r s hree imensionl ecor wih zero for is z-componen. For rphin, we will use only he firs wo componens. Cler[r,, x, y] r[_]={ Cos[], 5 Sin[] } = /4; min= ; mx= ; r[_]= {r[][[]], r[][[]]} pp=prmericplo[r[], {, min, mx}]; m[_]=sqr[.] el[_]= r[] spee[_]=m[el[]] cc[_]= el[] cur[_]= m[cross[el[],cc[]]]/spee[] //Simplify unin[_]= el[]/spee[]//simplify uninorm[_]= unin[] / m[unin[]] cr= r[] + ( / cur[] uninorm[] //Simplify {,}= {cr[[]], cr[[]]} o plo he osculin circle, lo rphics pce n hen plo i, n show i oeher wih he oriinl cure. <<Grphics`ImpliciPlo` pc=impliciplo[(x + (y == /cur[], {x, 8, 8},{y, 8, 8}] rius=grphics[line[{{, }, r[]}]] Show[pp, pc, rius, AspecRio Ä ].5 AGEIAL AD ORMAL COMPOES OF ACCELERAIO. r ( cos i( sin j Ê ( sin i( cos j Ê ( sin ( cos Ê ; ( cos ( sin Ê ( cos ( sin i j Ê É É Ê (. r ( i( j Ê ij Ê ( 9 Ê ; Ê É Ê ( (. r ( ij Ê ij Ê ( 5 4 Î Ê 5 4 ( Î 4 4 Ê ( ; Ê ( Ê ( Ê É É ˆ 4 É Ê ( 9 4. r ( cos i( sin j Ê (cos sin i(sin cos j Ê (cos sin (sin cos ( 5 Î Ê 5 ( Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

21 5 5 Secion.5 nenil n orml Componens of Accelerion 779 Ê ( ; ( sin cos i( cos sin j Ê ( j Ê ( Ê É ÊŠ Ê ( ( 5. r iˆ $ jˆ $ Ê i j Ê É( % Ê Ê ( ; ij Ê ( i Ê ( Ê É Ê ( ( 6. r e cos ie sin j e Ê e cos e sin ie sin e cos j e Ê Êe cos e sin e sin e cos Š e 4e e Ê e Ê ( ; e cos e sin e sin e cos ie sin e cos e cos e sin j e e sin e cos i j e Ê ( j Ê ( Ê Š 6 Ê É ÊŠ 6 Ê ( 7. r (cos i(sin j Ê ( sin i(cos j Ê ( sin (cos Ê ( sin i(cos j Ê ˆ i j; ( cos i(sin j Ê 4 ( cos ( sin i j ˆ Ê ( cos i(sin j Ê ˆ i j; B sin cos 4 cos sin Ê Bˆ, he norml o he osculin plne; rˆ i j Ê P Š ß ß lies on he 4 4 osculin plne Ê Š x Š y (z ( Ê z is he osculin plne; is norml o he norml plne Ê Š Š x Š Š y (z ( Ê x y Ê x y is he norml plne; is norml o he recifyin plne Ê Š Š x Š Š y (z ( Ê x y Ê x y is he recifyin plne 8. r (cos i(sin j Ê ( sin i(cos j Ê sin cos Ê Š sin Š cos Ê Š cos Š sin Ê i j i j É ˆ cos sin Ê ( cos i (sin j ; hus ( j n ( i i j Ê B( j, he norml o he osculin plne; r( i Ê P(ßß lies on he osculin plne Ê (x (y (z Ê y z is he osculin plne; is norml o he norml plne Ê (x (y (z Ê y z is he norml plne; is norml o he recifyin plne Ê (x (y (z Ê x is he recifyin plne. Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

22 78 Chper Vecor-Vlue Funcions n Moion in Spce 9. By Exercise 9 in Secion.4, ˆ cos iˆ sin j 5 n ( sin i(cos j so h B i j 4 cos sin ˆ 4 cos ˆ 4 sin. Also ( cos ( sin i j i j sin cos i j Ê ( sin i( cos j Ê ( cos i ( sin j n cos sin 4 sin cos ( cos i( sin j9 Ê cos sin 9 5. hus cos sin 4 sin sin cos sin 4 9 sin 9 cos By Exercise in Secion.4, (cos i(sin j n ( sin i(cos j; hus B i j cos sin cos sin. Also ( cos i( sin j sin cos Ê sin cos i cos sin j Ê cos sin sin i sin cos cos j i j cos sin i cos sin j. hus cos sin sin cos cos sin 4 [( cos ( cos sin ( sin ( sin cos ] Ê. hus cos sin cos sin sin cos sin cos cos sin cos sin sin cos cos sin sin cos. By Exercise in Secion.4, Š iš j n Š iš j; hus i j cos sin sin cos B cos cos sin sin sin sin cos cos Š Š cos sin sin cos sin sin Š Š. Also, e cos e sin ie sin e cos j Ê ce sin cos e cos sin ice cos sin e sin cos j= e sin ie cos j i j Ê e cos sin e sin cos. hus e cos sin e sin cos i j e e sin e cos Ê 4 e 4e. hus 7 e cos sin e sin cos e sin e cos e cos sin e sin cos 4e 4. By Exercise in Secion.4, ˆ cos iˆ sin 5 j n ( sin i(cos jso i j B cos sin cos i sin j. Also, ˆ ˆ 5 ˆ 5 ˆ 5 sin cos ( cos i( sin j5 Ê ( 4 sin i(4 cos jn ( 48 cos i(48 sin j i j cos sin 5 ( cos i( sin j88 Ê 4 sin 4 cos ( cos ( sin ( 88 cos sin hus Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

23 cos sin 5 4 sin 4 cos 48 cos 48 sin Secion.5 nenil n orml Componens of Accelerion 78. By Exercise in Secion.4, i j n i j so h B / / i j. Also, i j Ê ij Ê i so h Ê 7 4. By Exercise 4 in Secion.4, ( cos i(sin jn (sin i(cos j so h B i j cos sin. Also, cos sin i sin cos j sin cos Ê cos sin i sin cos j Ê ˆ cos sin i ˆ sin cos j cos sin sin cos Ê cos sin sin cos Ê 7 cos sin sin cos ˆ ˆ 5. By Exercise 5 in Secion.4, ˆ sech iˆ nh jn ˆ nh iˆ sech j so h B i j sech nh ˆ ˆ. Also, ˆ sinh Ê ˆ cosh Ê sinh ˆ i j j j so h nh ˆ sech ˆ sinhˆ cosh ˆ Ê 7 sinh ˆ 6. By Exercise 6 in Secion.4, Š nh i jš sech n (sech i(nh so h i j B nh sech nh i j sech. Also, (sinh i(cosh j Š Š sech nh i j (cosh i(sinh j Ê (sinh i (cosh j n sinh cosh cosh sinh (sinh i(cosh jcosh sinh (sinh i(cosh j Ê sinh cosh. hus sinh cosh cosh sinh sinh cosh sinh cosh sinh cosh cosh Yes. If he cr is moin lon cure ph, hen, Á n, Á Ê Á. 8. consn Ê Ê is orhoonl o Ê he ccelerion is norml o he ph 9. ¼ Ê ¼ Ê Ê Ê is consn. (, where ( n,, Ê,. ow, from ww Exercise 5( Secion.4, we fin for y f(x x h, ; lso, f (x w f (x c (x 4x $Î $Î $Î Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

24 78 Chper Vecor-Vlue Funcions n Moion in Spce r( i jis he posiion ecor of he moin mss Ê i j Ê 4 Ê 4 ( i j. A (ß: ( i, ( jn,( Ê F m m(, m j; 7 A Š : Š Š, Š ß i j i j i j, n Š, Ê F m m(, ˆ 4 m Š i j m i m j. By we he ˆ s s s s s s, ˆ Š, ˆ, ˆ s s B. I follows h l l, ¹ ¹ lbl, ll Ê, l l ll. Ê, Ê, (since he pricle is moin, we cnno he zero spee Ê he curure is zero so he pricle is moin lon srih line,. From Exmple, n so h, Ê,, Á Ê 4. r (x A i(y B j(z C Ê AiBjC Ê Ê Ê,. Since he cure is plne cure, If plne cure is sufficienly ifferenile he orsion is zero s he followin rumen shows: w w ww ww www www r f( i( j Ê f ( i ( j Ê f ( i ( j Ê f ( i ( j w w f ( ( ww ww f ( ( f ( ( www www Ê 7 6. sin i cos j n cos i sin j sin cos cos sin sin cos ˆ cos sin cos ˆ sin w Š Ê 7 o fin he orsion: 7 ( ; w w w 7 so 7mx occurs when 7mx 7 ( Ê Ê Ê Ê since,. Also Ê 7 n Ê Ê w w w w 7. r( f( i( jh( Ê f ( i ( jh ( ; Ê h ( Ê h( C Ê r( f( i( jc n r( f( i( jc Ê f(, ( n C Ê h(. 8. From Exercise 6, ( sin i( cos j Ê Ê c( sin i( cos j ; ( cos ( sin Ê c i j i j (cos i(sin j; B sin cos cos sin sin cos B B i j Ê c( cos i( sin j Ê Ê 7 ˆ B Š Š, which is consisen wih he resul in Exercise 6. ˆ Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

25 Secion.5 nenil n orml Componens of Accelerion Exmple CAS commns: Mple: wih( LinerAler ; r := < *cos( *sin( >; := sqr(; rr := el( r, = ; := mp( iff, r, ; := el(, = ; := mp( iff,, ; := el(, = ; s := simplify(orm(, ssumin ::rel; ss := el( s, = ; := /s; := /ss ; q := mp( iff, simplify(, : := simplify(el( q/orm(q,, = ; BB := CrossProuc(, ; pp := orm(crossprouc(,,/ss^; u := simplify( Deerminn(<,, el(mp(iff,,,= >/orm(crossprouc(,,^ ; _ := el( iff( s,, = ; _n := elf[4]( pp*ss^ ; Mhemic: (ssine funcions n lue for will ry Cler[,,, ] m[ecor_]:=sqr[ecor.ecor] Prin[he posiion ecor is, r[_]={ Cos[], Sin[], }] Prin[he elociy ecor is, [_]= r[]] Prin[he ccelerion ecor is, [_]= []] Prin[he spee is, spee[_]= m[[]]//simplify] Prin[he uni nen ecor is, un[_]= []/spee[] //Simplify] Prin[he curure is, cur[_]= m[cross[[],[]]] / spee[] //Simplify] Prin[he orsion is, orsion[_]= De[{[], [], []}] / m[cross[[],[]]] //Simplify] Prin[he uni norml ecor is, unorm[_]= un[] / m[un[]] //Simplify] Prin[he uni inorml ecor is, uinorm[_]= Cross[un[],unorm[]] //Simplify] Prin[he nenil componen of he ccelerion is, [_]=[].un[] //Simplify] Prin[he norml componen of he ccelerion is, n[_]=[].unorm[] //Simplify] You cn elue ny of hese funcions specifie lue of. = Sqr[] {un[], unorm[], uinorm[]} [{un[], unorm[], uinorm[]}] {cur[], orsion[]} [{cur[], orsion[]}] {[], n[]} [{[], n[]}] o erify h he nenil n norml componens of he ccelerion ree wih he formuls in he oo: []== spee[] //Simplify n[]==cur [] spee[] //Simplify Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

26 784 Chper Vecor-Vlue Funcions n Moion in Spce.6 VELOCIY AD ACCELERAIO I POLAR COORDIAES Þ ÞÞ.... Ê, r cos Ê r sin sin Ê r cos 9 cos sin u cos u sin u cos u r r Š 9 cos cos u cos sin u r 9 cos 9 9 cos u 8 sin u 9 cos u 8 sin u r r Þ ÞÞ.... Ê, r sin Ê r cos 4 cos Ê r 4 ˆ sin 4 cos 6 sin 4 cos 4 cos u sin u 4 cos u sin u r r 6 sin 4 cos sin ur sin 4 cos u 6 sin 4 cos 4 sin u sin 6 cos u r r r sin 4 cos u sin 6 cos u 4cos 5 sin u sin 8 cos u.... Þ ÞÞ Ê, r e Ê r e e Ê r e 4 e ˆ e u ˆ e u ˆ e u ˆ e u r r ˆ 4 e ˆ e u ˆ e ˆ e u 4 e 4e u 8 e u 4e 8 e u ˆ u r r r 4. e Ê Þ ÞÞ e Ê e, r sin Ê r. cos Ê.. r sin cosu sine u cosu e sinu r r r sin sin e u sin e cos e u r sine sin u e sine cos u sin e sin ur e sin cos u sin e sin u e cos sin u r Þ ÞÞ 5. Ê Ê, r cos 4 Ê r. 8 sin 4 Ê.. r cos 4 8 sin 4u cos 4u 8sin 4u 4cos 4u r r Š cos 4 cos 4 u cos 4 8 sin 4u r cos 4 8 cos 4u sin 4u 4cos 4u sin 4u r r r GM(e GM(e GM r É r 6. e Ê Ê ; Circle: e Ê É GM r GM r Ellipse: e Ê É É Prol: e Ê É GM r Hyperol: e Ê É GM r GM r GM GM GM É r r 7. r Ê Ê which is consn since G, M, n r (he rius of ori re consn Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

27 ? A r(? r(? r( r(??? 8.? A r(? r( Ê ¹ r( ¹ ¹ r( ¹ r(? r( r(? r( A r(? r( Chper Prcice Exercises 785 ¹? r(? r( r( ¹ ¹? r( ¹ Ê lim ¹? r( ¹? Ä r ( r r r( r r Þ % % 9. Š e 4 4 r Ê Š e Š Š (from Equion 5 r r r % % % 4 r r 4 GMr r ˆ 4 GM r r GM GM r GM rgm % % Š Š Š GM r $ 4 4 r GM GM GM GM $ GM % 4 Š ˆ % 4 ˆ ˆ (from Equion Ê Ê GM hours minues secons y hour minue. r ys ys 4 6 6,558,8.4 secons.6, m 4 GM GM 4 7 ˆ c ˆ 4 G 6.676, n he mss of he sun M.99. Ê Ê Ê m illion m CHAPER PRACICE EXERCISES 7. r( (4 cos iš sin j Ê x 4 cos n y x y sin Ê ; 6 ( 4 sin iš cos j n ( 4 cos iš sin j; r( 4 i, ( j, ( 4 i; rˆ ij, ˆ 4 4 ij, ˆ i j; 4 6 sin cos 4 sin cos 4 Ê ; :, É 4, 4 4, ; 6 sin cos, :, É9,,,. r( Š sec iš n j Ê x sec n y x y n Ê sec n ; Ê x y ; Š sec n iš sec j n Š sec n $ sec iš sec n j; r( i, ( j, ( i; % sec n sec $ % Ê ; 6 sec n 8 sec n sec n sec% :, É,,, Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

28 786 Chper Vecor-Vlue Funcions n Moion in Spce $Î $Î $Î $Î. r i j Ê i j Ê Ê mx mx. We wn o mximize : n Ê Ê. For, ; for, Ê occurs when Ê 4. r e cos ie sin j Ê e cos e sin ie sin e cos j Ê e cos e sin e sin e cos ie sin e cos e cos e sin j r e sin ie cos j. Le e he nle eween r n. hen cos Š e sin cos e sin cos Ée cos e sin Ée sin e cos e r cos cos Š cos for ll i j 5. i4 j n 5i5 j Ê 4 5 Ê 5; $ 5$ 5 Ê, y ww 6., e e Ê e e e e e w y $Î x x $Î, x x $Î x x &Î x x x x $Î x x &Î x x &Î x x x x &Î x e e e e e e c e e e e e ;, x x Ê e Ê e Ê x ln Ê x ln ln Ê y ; herefore is x mximum he poin Š ln ß, x y x 7. r xiy j Ê i j n i y Ê y. Since he pricle moes roun he uni circle x y y x x y x x y x y, x y Ê Ê (y x. Since y n x, we he y y yix j Ê (ß, jn he moion is clocwise. y y $ x x 8. 9y x Ê 9 x Ê x. If r xiy j, where x n y re ifferenile funcions of, x y x y x hen i j. Hence i 4 Ê 4 n j x ( (4 (ß. Also, x y y x i j n ˆ x ˆ ˆ x x x. Hence i Ê n y ((4 ( ( 6 he poin (x y (. j ß ß r r r r 9. orhoonl o r Ê r r r ( r r Ê r r K, consn. If r xiy j, where x n y re ifferenile funcions of, hen r r x y Ê x y K, which is he equion of circle cenere he oriin.. ( ( ( cos i( sin j Ê sin i cos j; ( n ( j; ( in ( j; ( n ( j; ( in ( j Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

29 Chper Prcice Exercises 787 (c Forwr spee he opmos poin is ( ( f/sec; since he circle mes reoluion per secon, he cener moes f prllel o he x-xis ech secon Ê he forwr spee of C is f/sec.. y y ( sin Ê y 6.5 (44 f/sec(sin 45 ( sec f/sec ( sec f Ê he sho pu is on he roun. ow, y Ê Ê. sec (he posiie roo Ê x (44 f/sec(cos 45 (. sec 66.7 f or ou 66 f, in. from he sopor ( sin [(8 f/sec(sin 45 ] ( f/sec. ymx y 7 f 57 f. x ( cos n y ( sin Ê n 9 y ( sin ( sin x ( cos cos sin cos n 9 Ê cos n 9 sin Ê, which is he ime when he olf ll his he upwr slope. A his ime x ( cos Š Š sin cos cos n 9. x sin cos cos n 9 cos 9 cos 9 ow OR Ê OR Š Š cos sin cos n 9 cos 9 cos 9 Š Š cos sin cos 9 cos sin 9 cos 9 Š Š cos Š cos 9 [sin ( 9]. he isnce OR is mximize when x is mximize: Š (cos sin n 9 x sin cos n Ê (cos sin n 9 Ê co n 9 Ê co n ( 9 Ê 9 Ê 5 4. ( x (cos 4 n y 6.5 (sin (sin 4 6 ; x 6 f n y f (cos 4 (cos 4 Ê 6 (cos 4 or n 6.5 (sin 4 6 Ê (cos 4 (.764 sec Ê.764 sec. herefore, (cos 4 (.764 sec Ê Ê 9 f/sec ( ymx y f ( sin (9(sin 4 (( 5. r ( cos i( sin j Ê ( sin i( cos j Ê ( sin ( cos ( Î4 Î% Ê Lenh ln ¹ ¹ É ln Š É r ( cos i( sin j Ê ( sin i( cos j Ê É( sin ( cos $Î Î Î 9 9 Ê Lenh ( 4 $Î $ 4 $Î 4 $Î Î Î r ( i ( j Ê ( i ( j Î Î Ê É ( Î ( Î ˆ Ê ( i ( j Î Î Ê ( i j ; ( i ( j Ê ( i j Ê ( i j Ê ( i j; B( ( ( 4 i j ; Î Î ( i ( j Ê ( i jn ( i j Ê ( ( Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

30 788 Chper Vecor-Vlue Funcions n Moion in Spce i j i j 4 Ê Š Ê ( ; 9 9 9, $ $ Š Þ $Î $Î Þ ˆ ˆ ( i ( j Ê ( i j Ê 7( 8. r e sin ie cos je Ê e sin e cos ie cos e sin je Ê Ée sin e cos e cos e sin e e Ê ˆ sin cos ˆ cos sin i j Ê ( i j ; ˆ cos sin iˆ sin cos j Ê ( i j Ê ( 5 i j ˆ 4 i j Ê ( i j; B( ( ( i j ; 5 Š e cos e sin ie cos 4e sin je Ê ( 4ij n ( ij i j Ê ( ( 8i4j Ê n ( Ê,( $ 9 ; Þ 4e cos 8e sin e sin 6e cos i e cos 6e sin 4e sin 8e cos j e Þ e cos e sin i e cos e sin j e Ê ( i j Ê 7( 9. r i e j Ê ie j Ê e 4 4 e Ê i j Ê (ln i j; e4 e e e 8 4 Ê (ln Ê (ln ; e4 $Î i e4 $Î j ˆ ˆ 7 i 7 7 j 7 i 7 j 7 i j 4 B(ln (ln (ln 7 7 ; e j Ê (ln 8 jn (ln i4j i j Þ Ê (ln (ln 4 8 Ê 8 n (ln 8 7 Ê,(ln ; 4e j Þ 6 Ê (ln 6 j Ê 7(ln. r ( cosh i( sinh j6 Ê (6 sinh i(6 cosh j6 Ê 6 sinh 6 cosh 6 6 cosh Ê Š nh i jš sech ˆ 8 ˆ 8 ˆ (ln Ê 7 i i Ê (ln i j ; Š sech iš sech nh Ê (ln Š Š Ê Š Š Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

31 Chper Prcice Exercises 789 i j Ê (ln i ; B(ln (ln (ln i j ; ( cosh i( sinh j Ê (ln ˆ 7 i ˆ j i j n i j (ln 6 ˆ 5 i6 ˆ j6 i j6 Ê (ln (ln $ 5 Š i5j7 Ê 5 5 n (ln 5 Ê,(ln ; Þ Þ (4 sinh i(4 cosh j Ê (ln 45i5 j Ê 7(ln. r i4 4 j(6 cos Ê ( 6 i(4 8 j(6 sin Ê ( 6 (4 8 (6 sin 5 6 sin Ê Î 5 6 sin ( 7 sin cos Ê ( ( ; 6i8 j(6 cos Ê 6 8 (6 cos 6 cos Ê ( 6 Ê É Ê ( 6. r ( i j Ê i( 4 j Ê ( 4 ( 8 Ê 8 Î (8 4 Ê ( ; 4j Ê 4 Ê É Ê Š Ê (. r (sin iš cos j(sin Ê (cos iš sin j(cos Ê Ê(cos Š sin (cos Ê Š cos i(sin jš cos ; Š sin (cos Š sin Ê i j ÊŠ sin ( cos Š sin i j ˆ Ê sin i(cos j sin ; B cos sin cos Š Š sin cos sin i j i ; ( sin i Š cos j (sin Ê cos sin cos sin cos sin Ê Þ i 4 Ê, ; ( cos iš sin j(cos $ Š $ cos sin cos sin cos sin Š Š Š cos sin cos (cos sin ( (cos 4 Ê 7 4. r i(5 cos j( sin Ê ( 5 sin j( cos Ê ( 5 cos j( sin Ê 5 sin cos 9 sin cos 6 sin cos ; Ê 6 sin cos Ê sin or cos Ê, or Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

32 79 Chper Vecor-Vlue Funcions n Moion in Spce 5. r iˆ 4 sin jˆ Ê r ( ij ( ˆ 4 sin ( Ê 4 sin Ê sin Ê Ê (for he firs ime $ 6. r( i j Ê ij Ê 4 9 % Ê ( Ê ( i j, which is norml o he norml plne Ê (x (y (z or x y z 6 is n equion of he norml plne. ex we clcule ( which is norml o he recifyin plne. ow, j6 Ê ( j6 Ê ( ( i j 6 6 Ê ( ( 76 9 s s i j 76 Ê,( $ ; ( Ê ¹ Š Î 4 i j Š Š 4 ˆ 9 7 i 7 j % $ s ¹, so ˆ s, Ê j6 4 Ê Ê (x (y (z or x 8y 9z is n equion of he recifyin plne. Finlly, B( ( ( i j 4 Š Š ˆ ( Ê (x (y (z or x y z i j is n equion of he osculin plne. 7. r e i(sin jln ( Ê e i(cos jˆ Ê ( ij; r( i Ê (ßß is on he line Ê x, y, n z re prmeric equions of he line 6 8. r Š cos iš sin j Ê Š sin iš cos j Ê ˆ Š sin Š i cos j ij is ecor nen o he helix when Ê he nen line is prllel o ˆ ; lso rˆ Š cos iš sin j Ê he poin ˆ ß ß is on he line Ê x, y, n z 4 re prmeric equions of he line 4 9. x cos n ˆ y sin Ê x ˆ y ÞÞÞ ÞÞÞ ÞÞÞ ÞÞÞ xxyy Þ Þ Þ Þ x y x y ÞÞ x ÞÞ y x Þ y Þ x Þ ÞÞ x xxyy Þ ÞÞ Þ ÞÞ y Þ ÞÞ y Þ ÞÞ Þ ÞÞ Þ ÞÞ Þ ÞÞ x y y x x x y y xy Þ ÞÞ yx Þ ÞÞ Þ Þ Þ Þ Þ Þ x y x y x y ÞÞ x ÞÞ y ÞÞ ÞÞÞ ÞÞÞ Þ Þ xyyx Þ Þ x y x y s $Î Þ Þ x y ÞÞ ÞÞ x y ÞÞ ÞÞÞ ÞÞÞ s xyyx, ÞÞ Þ Þ xxyy ÞÞ ÞÞ ÞÞ ÞÞ ÞÞ. s x y Ê x y s x y Ê Ê s s 9. s Ê Ê 9 Ê Ê, since s Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

33 DO O 68 O SO (? SO? OD Ê Ê Ê y Ê y 597 m; 68 ( VA x Ê Š x y y 68 Š y y 68 y 68 y c68y 597 &*( ( 6,95,469 m.69 m ; 6,95,469 m ( percene isile.% 4(68 m ( y Chper Aiionl n Ance Exercises 79 CHAPER ADDIIOAL AD ADVACED EXERCISES r r. ( r( ( cos i( sin j Ê [( sin i( cos j ] ; z Ê É z É Ê É 4 É Ê Î É Ê ; z Ê z ( É Ê É Î Ê É C; Ê Ê C r (c ( [( sin i( cos j ] [( sin i( cos j ] Š, from pr ( ( sin i ( cos j Ê ( Š ; r [( cos i ( sin j] ˆ [( sin i ( cos j ] [( cos i( sin j] [( sin i( cos j ] ( sin i ( cos j Š Š [( cos (sin ] i j Š (here is no componen in he irecion of. B Š Š r z r Î ˆ Î Î Î É u u c u u, where c. ( r( ( cos i( sin j Ê [( cos sin i( sin cos j ] ; Ê ( s u u u Ê s c u c ln ¹ u c u ¹ Š c c ln ¹ c ¹ c ln c ( er r ( er (e sin r ( er (e sin e cos ( e cos ( e cos r r. r Ê ; Ê Ê ( er (e sin Ê sin Ê or. oe h when sin n when sin. Since sin on ( er n sin on, r is minimum when n r( r e cos 4. ( f(x x sin x Ê f( n f( sin since sin Ÿ ; since f is coninuous on [ß ], he Inermeie Vlue heorem implies here is roo eween n ( Roo Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

34 79 Chper Vecor-Vlue Funcions n Moion in Spce ( x iy jn r ur r u r [(cos i(sin j] ˆ r [( sin i(cos j] Ê i x n i r cos r sin Ê x r cos r sin ; j y n j r sin r cos... Ê y r sin r cos.. ( ur (cos i(sin j Ê ur x cos y sin.... ˆ r cos r sin (cos ˆ r sin r cos (sin y pr (,.... Ê ur r ; herefore, r x cos y sin ;.. u (sin i(cos j Ê u xsin y cos..... ˆ r cos r sin ( sin ˆ r sin r cos (cos y pr ( Ê u r ;... herefore, r xsin y cos r w r ww 6. r f( Ê f ( Ê f ( ˆ w r f ( ; ur r u ˆ r cos r sin ˆ r sin r cos Ê ˆ r r ˆ w f f ˆ i j ; ÞÞÞ ÞÞÞ x r x y y x, where x r cos n y r sin. hen ( r sin (cos x ( sin r (r cos (r sin (cos r y ; (r cos (sin r ˆ y r r ( cos (r sin ˆ (r cos (sin. hen $ $ ww w much ˆ r r ˆ r ˆ Š Ê Ê (fer ler r r r f f f f Ê, ww w f f f f $Î w f f Î Î r r r r r ( ; r r r r ˆ r r ( 9 r 6 7. ( Le r n Ê n Ê. he hlfwy poin is (ß Ê ; u u Ê u u u u Ê u u ( I es he eele min o crwl o he oriin Ê he ro hs reole 6 rins Ê L É[f( ] cf w ( Ɉ ˆ É É 7 ( ln 6 Š in. ( 6 ( 6 ln 6 ( 6 8. ( x r cos Ê x cos r r sin ; y r sin Ê y sin r r cos ; hus x cos r r sin cos r r sin n y sin r r sin cos r r cos Ê s x y z r r z (c r e Ê r e ( ln 8 ln 8 e e e ln 8 e e ln 8 Ê L r r z 8 7 Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

35 i j 9. ( ur u cos sin Ê rih-hne frme of uni ecors sin cos u u Chper Aiionl n Ance Exercises 79 r ( ( sin i(cos j u n ( cos i(sin j ur Þ Þ Þ Þ ÞÞ Þ Þ Þ Þ ÞÞ Þ Þ ÞÞ (c From Eq. (7, rur r u z Ê rur r urˆ r u r u r u z ÞÞ Þ ÞÞ Þ Þ ÞÞ Š r r u ˆ r r u z r L. L( r( m ( Ê ˆ r m r L c Š r m Ê ( m ( r m r m ; F m Ê r L c c r r $ m Ê r m r Š r ( r r Ê L consn ecor $ $ r Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.

36 794 Chper Vecor-Vlue Funcions n Moion in Spce OES: Copyrih Person Eucion Inc. Pulishin s Aison-Wesley.