Millennium Theory Equations Original Copyright 2002 Joseph A. Rybczyk Updated Copyright 2003 Joseph A. Rybczyk Updated March 16, 2006

Size: px

Start display at page:

Berenice McBride
5 years ago
Views:

1 Millennim heoy Eqaions Oiginal Copyigh 00 Joseph A. Rybzyk Updaed Copyigh 003 Joseph A. Rybzyk Updaed Mah 6, 006 Following is a omplee lis o he Millennim heoy o Relaiviy eqaions: Fo easy eeene, all eqaions ae lised nde he heoy in whih hey wee deived and ae ideniied by he same nmbe oiginally assigned in ha heoy. Fo onsiseny, all vaiables ha oiginally sed pime maks ae eplaed wih sbsip vaiables.. Millennim heoy o Relaiviy. ime ansomaion Whee is he ime ineval in he saionay ame, is he ime ineval in he niom moion ame, v is he elaive speed, and is he speed o ligh, Eq. () ime ineval in he moving ame. Eq. () ime ineval in he saionay ame.. Disane ansomaion Whee d is disane in he saionay ame, d is he oesponding disane in he niom moion ame, d is he adis o he saionay sphee, and d is he adis o he moving sphee, d d Eq. (3) Disane in he niom moion ame. d Eq. (4) Radis o saionay sphee. d Eq. (5) Radis o moving sphee. d d Eq. (6) Disane in he saionay ame.

2 d d d d Eq. (7) Eqaliy o disane aios..3 Wavelengh and Feqeny ansomaion (See also: Eqaions om Relaivisi ansvee Dopple Ee) Whee is eqeny, λ is wavelengh, and λ ae he emied eqeny and wavelengh espeively, and x and λx ae he ansomed eqeny and wavelengh espeively, λ Eq. (8) Feqeny o wavelengh elaionship. λ Eq. (9) Wavelengh o eqeny elaionship. λ λ x Eq. (0) Wavelengh ansomaion, MF o SF. λ λ x Eq. () Wavelengh ansomaion, SF o MF. x Eq. () Feqeny ansomaion, MF o SF. x Eq. (3) Feqeny ansomaion, SF o MF..4 Dopple Ee (See also: Eqaions om Relaivisi ansvee Dopple Ee) Whee o and λ o ae he obseved eqeny and wavelengh espeively,

3 + v λo λx Eq. (4) Dopple ee, Obseved wavelengh. (v is + o eession o appoah) λx λo + v Eq. (5) Dopple ee, ansomed wavelengh. (v is + o eession o appoah) o Eq. (6) Dopple ee, Obseved eqeny. x + v (v is + o eession - o appoah) x o + v Eq. (7) Dopple ee, ansomed eqeny (v is + o eession - o appoah).5 Combined ansomaion and Dopple Ees (See also: Eqaions om Relaivisi ansvee Dopple Ee) ± v λo λ o λ λ o ± v Eq. (8) Obseved wavelengh. (v + o eession o appoah) o ± v o o ± v Eq. (9) Obseved eqeny. (v + o eession o appoah) 3

4 λ λo ± v o λ λ o ± v Eq. (0) Emied wavelengh. (v + o eession o appoah) o ± v o o ± v Eq. () Emied eqeny. (v + o eession o appoah).6 Dopple Ee, ansvee Moion Iniial Wavelenghs and Feqenies (Fo Reeene Only See Eqaions om Relaivisi ansvee Dopple Ee o pdaed eamen.) Whee A is he angle o obsevaion, ± v(os A) + λ λ o + v (os A) Eq. () Obseved wavelengh ansvee moion (v + o eession o appoah) o ± v(os A) + + v (os A) Eq. (3) Obseved eqeny ansvee moion (v + o eession o appoah) * (* De o some so o mahemaial, o possible ompe o sowae abeaion, he above eqeny eqaion loses aay when sed o appoah. A simila expeiene in he pas poved o be a ompe/sowae appliaion poblem. When dealing wih vey small nmbe sh as hose deining wavelenghs, he aay o olde deskop ompe and sowae an someimes ase poblems. In he pevios ase nmbe aived a ding ompaion someimes onained a i signiian digi ha ell oside o he ieen-digi limi o he sowae appliaion. his allowed division by zeo wiho lagging i as an eo. his in n led o eoneos answe simila o hose enoneed hee. Needless o say, aion ms be sed ding hese ypes o ompaions.) 4

5 vsin(80 A) sin A asin λ o λ sin(80 A) Eq. (4) Obseved wavelengh ansvee moion Alenae eqaion sin( 80 A) vsin(80 A) sin A asin o Eq. (5) Obseved eqeny ansvee moion * Alenae eqaion (* Use wih aion. See pevios ommens.). he Laws o Aeleaion. he Law o Consan Aeleaion Whee a is he ae o onsan aeleaion expeiened in he aeleaion ame and v is he speed o he niom moion ame ansiioned o ding niom moion ame ime ineval. v a (). he Law o Relaive Aeleaion Whee a is he elaive ae o aeleaion expeiened in he saionay ame and v is he speed o he niom moion ame ansiioned o ding saionay ame ime ineval. v a (4).3 Relaionship o Consan Aeleaion o Relaive Aeleaion 5

6 a a (7) a a (8) 3. ime and Enegy, Ineia and Gaviy 3. Kinei Enegy (Noe: hese omlas ae elaivisi veions o he Newonian omla, k mv. Fo elaivisi veions o he alenae Newonian omla, see Seion 5..) k mad Whee p is momenm, d is he disane aveled in he niom moion ame a a onsan ae o aeleaion ding niom moion ame ime ineval, m is mass, and k is kinei enegy, k pd (5) Kinei Enegy Whee p is momenm, D is he disane aveled in he saionay ame a a onsan ae o aeleaion ding saionay ame ime ineval, m is mass, and k is kinei enegy, k pd Kinei Enegy (See, Relaivisi Moion Pepeive, Seion 9.3) k m v (3) Kinei Enegy + 3. oal Enegy Whee E is oal enegy, 6

7 m v E + + m (4) oal Enegy 3.3 Momenm Whee p is momenm, mv p (7) Momenm 3.4 Disane aveled ding Consan Aeleaion (Noe: hese omlas ae o disanes aveled in he niom moion ame. Fo disanes aveled in he saionay ame, see Seion 5.) Whee d is he niom moion ame disane aveled by an obje nde onsan aeleaion, and a is he ae o onsan aeleaion, o elaive ime (niom moion ame ime ineval), d (9) + v Disane based on v and d a Disane based on (30) + ( a ) a and o onsan ime (saionay ame ime ineval), 7

8 d v (3) + Disane based on v and 3.5 Veloiy and Consan Aeleaion Using elaive ime (niom moion ame ime ineval), v a () and, v a () Consan Aeleaion Using onsan ime (saionay ame ime ineval), a v (33) + ( a ) Insananeos Veloiy based on a and a v (34) Consan Aeleaion based on v and 3.6 Veloiy and Relaive Aeleaion v a (3) and, v a (4) Relaive Aeleaion 3.7 Foe Whee F is oe, 8

9 v F ma (36) and, F m (37) Consan Foe 3.8 Ineia and Gaviy Whee I is es ineia, I is oal ineia, g is gaviy, G is he gaviaional onsan , and is he disane o he ene o gaviy, I m (38) Res Ineia I I + p (39) oal Ineia v I m + (43) oal Ineia v I g G (50) v + I v + (5) v g G Gaviy and Ineia 3.9 Aeleaion Based ime ansomaion Fomlas Whee a is he onsan ae o aeleaion, is he saionay ame ime ineval, and is he ansiioned o niom moion ame ime ineval. 9

10 (57) and, + ( a ) (58) ( a ) Aeleaion Based ime ansomaion 3.0 Aeleaion Based ansomaion Fao Whee a is he onsan ae o aeleaion, is he saionay ame ime ineval, and is he ansiioned o niom moion ame ime ineval. + ( a ) (59) and, ( a) (60) Saionay Fame o Moving Fame ansomaions ( a ) (6) and, + ( a) (6) Moving Fame o Saionay Fame ansomaions 4. Millennim Relaiviy Veloiy Composiion 4. Veloiy Composiion Fomla In he owad dieion, whee is he veloiy o he moving ame elaive o he saionay ame, is he veloiy o an obje (o anohe ame) moving elaive o, and in he same dieion as he moving ame, and v is he oal veloiy, v + (9) Millennim Veloiy Composiion Fomla In he evee dieion, whee is he veloiy o he moving ame elaive o he saionay ame, 3 is he veloiy o an obje (o anohe ame) moving elaive o, and in he same dieion as he moving ame, and v is he oal veloiy, 0

11 v + 3 () Millennim Veloiy Composiion Fomla (Revee Dieion) 4. Complemenay Veloiies he elaionship beween omplemenay veloiies and is, () Relaionship beween omplemenay veloiies, and. he elaionship beween omplemenay veloiies and is, 3 3 (3) Relaionship beween omplemenay veloiies, 3 and. 4.3 he Mass/Enegy ime-ansomaion Fomla m m + k (4) ime ansomaion sing mass and kinei enegy. giving, m m + k and is eipoal, m + k m as ohe oms o he elaivisi ansomaion ao. 5 Relaivisi Moion Pepeive Copyigh 003 Joseph A. Rybzyk 5. Disane aveled ding Consan Aeleaion

12 (Noe: hese omlas ae o disanes aveled in he saionay ame. Fo disanes aveled in he niom moion ame, see Seion 3.4) Whee D is he saionay ame disane aveled by an obje nde a onsan ae o aeleaion, a elaive o he aeleaion ame, o elaive ae o aeleaion, A elaive o he saionay ame, ding saionay ame ime ineval. D + v Eq. (6) SF disane sing ahieved speed v ding ineval. D + a + ( a ) Eq. (7) SF disane sing a and. D + A ( A ) Eq. (8) SF disane sing A and. 5. Kinei Enegy (Noe: hese omlas ae elaivisi veions o he alenae Newonian omla, k mad. Fo elaivisi veions o he Newonian omla, k mv see Seion 3.) Whee k is kinei enegy, m is mass, and D is he elaivisi disane aveled in he saionay ame nde a onsan ae o aeleaion, a expeiened in he aeleaion ame, and a elaive ae o aeleaion, A expeiened in he saionay ame ding he saionay ame ime ineval ha speed v is ahieved. (Noe: Sine a, A, v, and ae mally dependen vaiables, ae ms be exeised in he se o hese omlas.) k mad Eq. (0) he basi alenaive omla o kinei enegy. k + ma v Eq. () Deived sing Eq. (6) wih Eq. (0). k + ma + ( a ) Deived sing Eq. (7) wih Eq. (0).

13 k + ma A ( A ) Deived sing Eq. (8) wih Eq. (0). 5.3 ime Inevals Whee is he saionay ame ime ineval ding whih speed v is ahieved while aeleaing a he onsan ae, a expeiened in he aeleaion ame. a v Eq. (4) Saionay ame ime ineval. he elaionship beween saionay ame ime ineval, and aeleaion ame ime ineval (a ), and niom moion ame ime ineval, when he niom moion ame is ansiioned o om he saionay ame. ( a) Eq. (5) ime ineval elaionships. 5.4 ime Raes and Inevals om Addendm C Whee: SF is he saionay ame, UF is he niom moion ame, and he ollowing vaiables ae deined as shown: (Noe: sbsip m, sed in Addendm was eplaed by sbsip ) (SF ime ae as meased in he SF) (SF ime ae as meased in he UF) (SF ime ineval as meased in he SF) (SF ime ineval as meased in he UF) (UF ime ae as meased in he SF) (UF ime ineval as meased in he SF) 3

14 5.5 ime Rae and Inevals om Addendm E Whee, is he ime ineval in he saionay ame, and and ae he ime ae and ime ineval espeively, in he niom moion ame, Eq. () Uniom moion ame ineval, deined in ems o and. (Noe: om Addendm E eplaed by o agee wih he deiniions given above in Seion 5.4.) Whee, is he insananeos ime ae in he aeleaion ame and v is he insananeos speed ahieved by a onsan ae o aeleaion, a ding saionay ame ime ineval, v. a Eq. (3) Insananeos ime ae in aeleaion ame. Whee, is he niom moion ame ime ae o he niom moion ame ha has a speed eqal o he insananeos speed, v ahieved by a onsan ae o aeleaion, a ding saionay ame ime ineval,. a Eq. (4) Uniom moion ame ime ae o ansiioned ame. Noe: Eqaion (4) is simply a esaemen o eqaion (3) and may be edndan aep in a ase whee i migh be neessay o some eason o disingish beween he insananeos speed v, and he pesenly ansiioned o niom moion ame. 5.6 ansomaion Fao om Addendm E 4

15 Whee is he speed o he niom moion ame ansiioned o a insananeos speed v ahieved by an aeleaion ae, a ding saionay ame ime ineval,, he ollowing ansomaion ao ae deived, v a a Eq. (5) and, Eq. (6) UF o SF ansomaion ao. a a v Eq. (7) and, Eq. (8) SF o UF ansomaion ao. 5.7 ansomaion Fao om Addendm D Whee D is he SF disane aveled by a niom moion obje a speed U, and D a is he SF disane aveled by an obje aeleaing a he onsan ae, a ding he same SF ime ineval, o eah an insananeos speed V U, he ollowing disane elaionships povide anohe om o elaivisi ansomaion ao: D D D a a o, D D a UF o SF ansomaion ao. Da D D Da o, Da SF o UF ansomaion ao. 5.8 Aeleaion Fame ime Ineval he aahed opy o Addendm B om he Relaivisi Moion Pepeive model developed in Mahad, veion 4, an be sed o deemine he ime ineval, a in he onsan aeleaion ame elaing o he saionay ame ineval. 5

16 Addendm B - Relaivisi Moion Pepeive ime, Rae and Ineval Analysis Consan Aeleaion Fame vs Saionay and Uniom Moion Fames Copyigh 003 Joseph A. Rybzyk Jan 3, 003 Joseph A. Rybzyk Pepeive Range Vaiable.md g he speed o ligh and he ae o gaviaional aleaion g in m/s. a g. U.5. he ae o onsan aleaion a in ems o g, and he ahieved speed U in ems o. Y Deining Y o lae se in onveing inevals om seonds o yea. U. Fomla o deemining he saionay ame ime ineval ding whih a. U speed U is ahieved. is in s, /Y is in yea Y a.. U U 0.5 a. Veiying omla o be sed below in ange vaiable omla... ime ineval o ange vaiable omla below. a.. U 3 U. a Range vaiable omla. Range o speeds (om s o s) a.. Uniom moion ame, is ime ae, is ineval in s, /Y is ineval in yea Y Aeleaion ame, a is ange o ime aes, a is ineval in s, a /Y is ineval in yea. U 3 a a. a a a a Y a a Noe: I yo eneed his page diely ding a seah, yo an visi he Millennim Relaiviy sie by liking on he Home link below: Home nex > 6

Sections 3.1 and 3.4 Exponential Functions (Growth and Decay)

Sections 3.1 and 3.4 Exponential Functions (Growth and Decay) Secions 3.1 and 3.4 Eponenial Funcions (Gowh and Decay) Chape 3. Secions 1 and 4 Page 1 of 5 Wha Would You Rahe Have... $1million, o double you money evey day fo 31 days saing wih 1cen? Day Cens Day Cens