Equations from The Four Principal Kinetic States of Material Bodies. Copyright 2005 Joseph A. Rybczyk


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1 Equions fom he Fou Pinipl Kinei Ses of Meil Bodies Copyigh 005 Joseph A. Rybzyk Following is omplee lis of ll of he equions used in o deied in he Fou Pinipl Kinei Ses of Meil Bodies. Eh equion is idenified by he sme numbe used in he heoy nd gien only one in he ode of is fis use. Also inluded e he oiginl Appendixes A1, A nd B1 hough B6 onining he Mhd pogms used in he iled wok. 1. Reliisi fomuls fo ineil eleion, elie moion nd ime nsfomion Refeing o Figue 1, Speed Siony Fme u Disne in LY Siony Fme D D u FIGURE 1 Aeleion, Speed nd Disne Relionships Whee is he onsn e of eleion elie o he eleion fme (AF), is he insnneous speed ehed elie o he siony fme (SF) duing he nsiioned o unifom moion fme u (UFu) ime inel h oesponds o SF ime inel, nd is he speed of ligh in uum, (1) Consn Aeleion in AF () ime nsfomion in UF u (3) Relionship of Insnneous Speed o Unifom Moion Speed u 1
2 (4) ime nsfomion fo SF (5) Alene Fom of Reipol of Gmmnsfomion Fo (6) Alene Fom of Gmmnsfomion Fo Noe: Boh sides of Eq. (6) e simplified millennium heoy equilen fos fo he Loenz γ nsfomion fo gien in Eq. (7) below. 1 1 (7) Loenz nsfomion Fo, γ Whee D u is he SF disne eled SF unifom moion speed u duing SF ime inel h oesponds o nd n be lenely expessed s he sme disne eled UFu unifom moion speed U duing UFu ime inel, D u u (8) SF Disne fo Unifom Moion Obje eling SF Speed u D u U (9) SF Disne fo Unifom Moion Obje eling UF Speed U u U (10) Unifom Moion Speed u Obseed in SF U u (11) Coesponding Unifom Moion Speed U in ems of Relie ime Whee D is he SF disne eled while eleing o SF speed AF onsn e of eleion duing SF ime inel, o he oesponding SF elie e of eleion, D + (1) SF Disne fo Aeleion Consn Re Relie o AF D (13) SF Disne fo Aeleion Consn Re Relie o AF + + ( )
3 D (14) SF Disne fo Aeleion Relie Re Relie o SF + ( ) Whee is he SF elie e of eleion he insn SF speed is ehed SF ime inel, nd is he UF(u ) ime inel h oesponds wih SF ime inel, (15) Relie Re of Aeleion Obseed in SF (16) Relionship of SF Relie Aeleion Re o AF Consn Re (17) Relionship of AF Consn Aeleion Re o SF Relie Re. Spheil Refeene Fmes nd Unifom Moion Refeing o Figue, S 1  SF S  UF D u1 u 1 D u U D u u FIGURE Unifom Moion Relionships, SF s. UF Whee u nd u 1 e SF (Sphee S 1 ) unifom moion speeds nd U is unifom moion speed elie o UFu 1, (Sphee S ) u u + U 1 u 1 (18) Reliisi Veloiy Composiion 3
4 Whee D u is he UFu 1 disne eled unifom moion speed U elie o UFu 1 duing UFu 1 ime inel, nd u is he oesponding unifom moion speed elie o he SF duing SF ime inel fo he sme disne eled elie o he SF, D u U (19) SF Disne fo Unifom Moion Obje eling UF Speed U D u u U u (0) SF Disne fo Unifom Moion Obje eling SF Speed u U u (1) Unifom Moion Speed U Obseed in UF S () Coesponding Unifom Moion Speed u in ems of Consn ime 3. Newon s Seond nd Fis Lws of Ineil Mehnis Whee Σ F is he eo sum of he foe, m is he ineil mss nd is he e of eleion Newon s Seond Lw of Mehnis, Σ F m (3) Veo Sum of Foe ΣF (4) Re of Aeleion m Whee he eo sum of he foe, Σ F, is 0, Newon s Fis Lw of Mehnis n be expessed s, ΣF 0 (6) Fis Lw of Mehnis 4. Newonin Giionl Foe nd Aeleion Whee F is giionl foe, G is he uniesl giionl onsn, m is seondy mss of insignifin size, M is he pimy mss, nd R is he dius of he pimy mss, mm F G (5) Uniesl Lw of Giion R Noe: he boe fomul gies he foe he sufe of he pimy mss. If R is epled by h gies he ol eneoene () disne beween he wo msses, m n be ny size mss nd F gies he giionl foe beween he wo msses unde ll ondiions. Whee s nd p e he es of giionl eleion of he seondy mss m nd he pimy mss M espeiely, elie o n ineil fme he ene of mss beween he wo msses, 4
5 F GM s (7) Giionl Aeleion of Seondy Mss Relie o Ineil Fme m F Gm p (8) Giionl Aeleion of Pimy Mss Relie o Ineil Fme M Whee is he ombined e of eleion of he wo msses m nd M elie o eh ohe, F F G( M + m) + (9) Giionl Aeleion of Msses Relie o Eh Ohe m M Noe: Eq. (9) is millennium heoy fomul fom he Millennium heoy of Inei nd Giy. 5. Inegion of Millennium heoy Reliisi Lws of Aeleion nd Ineil Lws of Mehnis Whee is he onsn e of ineil eleion elie o he AF, F is ineil foe elie o he AF, m is he mss of he eleing obje o body, is he insnneous speed elie o he SF, nd is he ime inel elie o ineil fme UF(u ) duing whih he eleion kes ple, F (30) m Whee is he insnneous speed elie o he SF, is he UF(u ) ime inel duing whih speed ws hieed while eleing onsn e elie o he AF, F is he onsn ineil foe of eleion elie o he AF, F is he oesponding elie ineil foe of eleion elie o he SF, nd is he oesponding SF ime inel, F (31) Equliy of Consn Aeleion Rios m F (3) SF Insnneous Speed m m (33) UF ime Inel F F m (34) AF Consn Foe F m F (35) Consn Mss m 5
6 F (36) Equliy of Relie Aeleion Rios m Whee is he elie e of ineil eleion elie o he SF, is he onsn e of ineil eleion elie o he AF, is he insnneous speed ehed elie o he SF duing SF ime inel, F is he foe elie o he SF, F is he foe elie o he AF, nd is he oesponding UF(u ) ime inel, (37) Relionship of o (38) Alene Fomul fo F m (39) SF Relie Foe F F m (40) AF Consn Aeleion F m (41) AF Consn Foe F F F (4) Relionship of Consn Foe F o Relie Foe F F F (43) Relionship of Relie Foe F o Consn Foe F F (44) SF Insnneous Speed m m (45) SF ime Inel F F m (46) SF Relie Foe F m F (47) Consn Mss m 6
7 F F (48) Alene Fomul fo Relionship of F o F F F (49) Alene Fomul fo Relionship of F o F 6. Inegion of Newonin Ineil Lws of Mehnis wih he Lw of Giion fo he Newonin ime Bsed Pogm, nd Ohe Pogms whee Applible (Appendixes B1 hough B6) Noe: Equions (50) nd (51) show wo possible ondiions unde whih ineil eleion n emule giionl eleion in Newonin physis. Moe speifilly, beuse he eduion of disne duing giionl eleion uses he giionl foe F g, nd hus he e of giionl eleion g o inese duing he peiod of eleion, eihe he ineil mss m i mus deese in unison s shown in Eq. (50), o he ineil foe F i mus inese in unison s shown in Eq. (51). Sine he ineil hnges e diffeen fom he giionl hnges, howee, i is onluded h ue ineil/giionl equilene is no he se. Whee i is he ineil e of eleion, F i is he ineil foe, m i is he ineil mss of he eleing obje, g is he e of giionl eleion, F g is he giionl foe, m g is he giionl mss of he eleing obje, M g is he giionl mss of he pimy body, is he  disne beween he giionl msses, nd he up nd down ows show he dieion of hnge, F F GM (50) Consn Foe Ineil/Giionl Equilene i i g g mi m g i g g mi mg g F F GM (51) Consn Mss Ineil/Giionl Equilene i g Whee is he e of giionl eleion elie o n ineil fme beween he pimy giing mss nd he seondy mss, m 1 is he pimy mss, is he  disne beween msses, nd he ssignmen symbol hs simil mening o he sign, Gm1 (5) Re of Giionl Aeleion Noe: Aeleion e, insnneous speed, nd ime inel he diffeen definiions in Newonin physis fom hose of Millennium heoy physis nd e no o be onfused wih he Millennium heoy ibles of he sme ype. Whee is he e of eleion, nd is he insnneous speed hieed duing ime inel 1, 7
8 (53) Insnneous Speed fom Giionl Aeleion Sndd Newonin fomul fo onsn ineil eleion (54) Newonin Fomul fo Consn Aeleion Whee d is he disne eled duing he pesen ime inel, is he ol speed ehed duing he peious ime inel, nd is he inese in speed fom he eleion duing he pesen ime inel, d + (55) Disne eled Duing SF 1 Seond Inel Sndd Newonin fomul fo eleion disne 1 d (56) Newonin Fomul fo Consn Aeleion Disne Whee o he lef of he ssignmen ow is he ol insnneous speed fo he pesen ime inel, o he igh of he ssignmen ow is he ol insnneous speed fom he peious ime inel, nd is he inese in speed duing he pesen ime inel, (57) ol Insnneous Speed fom Giionl Aeleion + Whee d o he lef of he ssignmen ow is he ol disne fo he pesen ime inel, d o he igh of he ssignmen ow is he ol disne fom he peious ime inel, nd d is he inese in disne duing he pesen ime inel, d d d (58) ol Disne eled duing Giionl Aeleion + Whee is he emining  disne fo he pesen ime inel, i is he iniil  disne he s of he pogm, nd d is he ol el disne fo he pesen ime inel, (59) Remining CeneoCene Disne Obje o Eh i d Whee is he uen e of giionl eleion he end of eh ime inel, nd he e fo he finl emining disne he end of he pogm, nd m 1 is he pimy mss, Gm1 (60) Cuen Re of Aeleion 8
9 7. Inegion of Millennium heoy Reliisi Lws of Ineil Aeleion nd Mehnis wih he Lw of Giion fo he Reliisi ime nd Disne Bsed Pogms (Appendixes B3 hough B6) Whee is he elie e of giionl eleion of he eleing seondy mss elie o he SF (n ineil fme beween he wo msses), m 1 is he mss of he pimy giing body, is he ol insnneous speed ehed duing he peious ime inel 1, is he uen  disne beween he wo msses, G is he uniesl giionl onsn, nd is he speed of ligh in uum, Gm1 (61) Reliisi Re of Giionl Aeleion Obseed in SF Inlid ndide fomuls fo eleion e nd insnneous speed Gm1 (6) Cndide fo Reliisi Re of Aeleion Obseed in SF Gm1 (63) Cndide Reliisi Insnneous Speed fom Giionl Aeleion Whee is he insnneous speed ehed duing SF ime inel, nd he fionl em o he igh of ible is he equilen elie e of giionl eleion peiously defined in Eq. (61) boe, Gm1 (64) Insnneous Speed fom Giionl Aeleion Simplified fom of Eq. (64) boe fo ses whee n he ny lue, (65) Insnneous Speed fom Giionl Aeleion Simplified fom of Eq. (65) boe fo ses whee 1 (66) Insnneous Speed fom Giionl Aeleion fo 1 Whee d is he SF disne eled duing he uen SF ime inel 1, is he ol insnneous speed ehed elie o he SF duing he peious SF ime inel 1, nd is he inese in insnneous speed esuling fom giionl eleion duing he uen SF ime inel 1, d + (67) Disne eled Duing SF 1 Seond Inel + 9
10 Whee is he uen elie e of giionl eleion of he eleing seondy mss elie o he SF (n ineil fme beween he wo msses), m 1 is he mss of he pimy giing body, is he ol insnneous speed ehed duing he peious ime inel 1, nd is he uen  disne beween he wo msses, Gm1 (68) Cuen Re of Aeleion Exenl Mhd fomul fo oneing lue ssigned o ible K o disne in mees. (Shown in Appendix B1, bu used in ll pogms) 3 D : K 10 (69) Disne Conesion, Kilomees o Mees he following is n exenl Mhd funion fo epoing he esuls of Eq. (69) boe. he esuls fom Eq. (70) e hen opied nd psed ino he Mhd pogm nd e lso ilble fo ohe exenl ompuions. (Shown in Appendix B1, bu used in ll pogms) D (70) Compuionl Oupu fo Peeding Fomul 8. Inegion of Newonin Ineil Lws of Mehnis wih he Lw of Giion fo he Newonin, nd whee pplible, he Reliisi Disne Bsed Pogms (Appendixes B1, B3, nd B5) Whee is he  disne beween he wo msses, R 1 is he dius of he pimy mss, R is he dius of he seondy mss, nd D is he lene disne beween he wo msses, R + R + D 1 (71) CeneoCene Disne beween Bodies Whee is he ime inel o el 1 kilomee he pesen e of eleion, is he ol insnneous speed ehed duing he peious ime inel fo he peiously eled kilomee, x is onesion fo o one kilomees o mees, nd is he e of eleion fo he peiod, + x (7) SF ime Inel o el 1 km Pesen Re of Aeleion Whee is he ime inel o el disne x onsn e of eleion, (No used diely in pogm. Used long wih he fomul fo he unifom moion disne eled he insnneous speed ehed duing he peious ime inel o deie Eq. (7) boe. See iled wok fo deiled explnion.) x (73) ime Inel fo Disne x 10
11 Whee o he lef of he ssignmen ow is he ol ime inel o el ol disne d (See Eq. (76) below.), o he igh of he ssignmen ow is he peious ol ime inel, nd is ime inel fo he uen peiod duing whih he uen disne of 1 kilomee is eled, (74) ol ime duing Giionl Aeleion + Whee is he inese in insnneous speed fo he peiod, esuling fom eleing eleion e duing he uen ime inel, (75) Insnneous Speed fom Aeleion Whee d o he lef of he ssignmen ow is he ol disne eled he end of he pesen peiod, d o he igh of he ssignmen ow is he peious ol disne eled, nd x is he (1 km) disne eled duing he pesen peiod, d d x (76) ol Disne eled duing Giionl Aeleion + Whee o he lef of he ssignmen ow is he uen emining  disne beween he wo msses, o he igh of he ssignmen ow is he peious emining  disne, nd x is he (1 km) disne eled duing he pesen peiod, x (77) Remining CeneoCene Disne Obje o Eh 9. Inegion of Millennium heoy Reliisi Lws of Ineil Aeleion nd Mehnis wih he Lw of Giion fo he Reliisi Disne Bsed Pogms (Appendixes B3, nd B5) Whee is he SF ime inel fo n obje eleing elie e of eleion elie o he SF o el disne x (1 km), (Like Eq. (73) no used diely in pogm. Used long wih he fomul fo he unifom moion disne eled he insnneous speed ehed duing he peious ime inel o deie Eq. (79) h follows. See iled wok fo deiled explnion.) x ( x ) (78) SF ime Inel fo Disne x Whee is he SF ime inel o el SF disne x (1 km) duing he pesen peiod of eleion whee is he SF elie e of eleion fo he peiod, nd is he ol insnneous speed ehed elie o he SF he end of he peious peiod, x + + x x (79) SF ime Inel o el 1 km Re ( + ) 11
12 Whee is he Giionl Aeleion Fme (GAF) ime inel fo he pesen peiod, is he SF ime inel fo he sme peiod, nd is he ol insnneous speed ehed elie o he SF he end of he peious peiod, (See Appendixes A1 nd A fo onsn ineil eleion ppliion.) (80) Peiod ime Inel in he GAF Whee o he lef of he ssignmen ow is he ol GAF ime inel fo ll peiods of eleion, o he igh of he ssignmen ow is he ol GAF ime inel fo ll peious peiods of eleion, nd is he GAF ime inel fo he pesen peiod of eleion, + (81) ol ime Inel in he GAF Appendixes he following Appendixes e inluded s p of his ppe: Appendix A1 Reliisi Consn Ineil Aeleion g o.5 he Fou Pinipl Kinei Ses Appendix A Reliisi Consn Ineil Aeleion g o.75 he Fou Pinipl Kinei Ses Appendix B1 Eh Newonin Disne Pogm he Fou Pinipl Kinei Ses Appendix B Eh Newonin ime Pogm he Fou Pinipl Kinei Ses Appendix B3 Eh Reliisi Disne Pogm he Fou Pinipl Kinei Ses Appendix B4 Eh Reliisi ime Pogm he Fou Pinipl Kinei Ses Appendix B5 Supemssie Blk Hole Disne Pogm he Fou Pinipl Kinei Ses Appendix B6 Supemssie Blk Hole ime Pogm he Fou Pinipl Kinei Ses 1
13 Appendix A1 Reliisi Consn Ineil Aeleion g o.5 he Fou Pinipl Kinei Ses Copyigh 005 Joseph A. Rybzyk Veifying fomul o be used below in nge ible fomul. : 1.. ime inel fo nge ible fomul below. U 3 ( ) : Rnge ible fomul. Rnge of speeds (fom 1 s o s) ( ) + Unifom moion fme, u is ime e, u is inel in s, u /Y is inel in yes. U u : u : u u u u Y ( ) ( ) Addendum B  Reliisi Moion Pespeie ime, Re nd Inel Anlysis Consn Aeleion Fme s Siony nd Unifom Moion Fmes Copyigh 003 Joseph A. Rybzyk Aeleion fme, is nge of ime es, is inel in s, /Y is inel in yes. U 3 : : ( ) Jn 3, 003 Joseph A. Rybzyk Pespeie Rnge Vible1.md : g : he speed of ligh nd he e of giionl leion g in m/s. : g1 U :.5 he e of onsn leion in ems of g, nd he hieed speed U in ems of. Y : Defining Y fo le use in oneing inels fom seonds o yes. U : Fomul fo deemining he siony fme ime inel duing U whih speed U is hieed. is in s, /Y is in yes Y U U : ( ) Y u u 13
14 Appendix A Reliisi Consn Ineil Aeleion g o.75 he Fou Pinipl Kinei Ses Copyigh 005 Joseph A. Rybzyk Veifying fomul o be used below in nge ible fomul. : 1.. ime inel fo nge ible fomul below. U 3 ( ) : Rnge ible fomul. Rnge of speeds (fom 1 s o s) ( ) + Unifom moion fme, u is ime e, u is inel in s, u /Y is inel in yes. U u : u : u u u u Y ( ) ( ) Addendum B  Reliisi Moion Pespeie ime, Re nd Inel Anlysis Consn Aeleion Fme s Siony nd Unifom Moion Fmes Copyigh 003 Joseph A. Rybzyk Aeleion fme, is nge of ime es, is inel in s, /Y is inel in yes. U 3 : : ( ) Jn 3, 003 Joseph A. Rybzyk Pespeie Rnge Vible1.md : g : he speed of ligh nd he e of giionl leion g in m/s. : g1 U :.75 he e of onsn leion in ems of g, nd he hieed speed U in ems of. Y : Defining Y fo le use in oneing inels fom seonds o yes. U : Fomul fo deemining he siony fme ime inel duing U whih speed U is hieed. is in s, /Y is in yes Y U U : ( ) Y u u 14
15 Appendix B1 Eh Newonin Disne Pogm he Fou Pinipl Kinei Ses Copyigh 005 Joseph A. Rybzyk Newonin Disne Giionl Aeleion D&  EhA.md June 0, 005 Copyigh 005 J. Rybzyk K : 1 : G : Speed of ligh in m/s, Giionl Consn G, K in km m : m : 1 D K 10 3 : D Pim & Se msses, m 1 & m in kg, Clene D in m R : R : 1 : R 1 + R + D m 1 & m Rdii, R 1 & R nd  disne in m Noe: he boe lues e independen of he lues used in pogms Nd(K) Nds(K) N() Ns() e. hey e used below he pogms fo ompison puposes only. Mke sue ll lues gee wih hose used in pogms. Nd( K) G : Nds( K) G : g : Gm 1 ( R 1 + R ) m R R 1 D R 1 + R + D x d 0 fo i 1.. K Gm 1 + d + d d + x x Gm 1 + x D K 10 3 : D SF Disne in km K Nd( K) SF Disne Inpu in km K : Inse K lue. Copy D o ll Pogms N 1 : Nds( K) Re of Aeleion his peiod Z ime fo his peiod in s Y ol ime in s 3. Copy o Righ Loop Speed fo his Peiod m/s X ol el Disne Afe s Remining  Disne Afe s g Cuen Re of Aeleion ol Speed Imp m R R 1 D R 1 + R + D x fo i 1.. K Gm 1 x + + x Newonin Speed in ems of N X Y Z 15
16 Appendix B Eh Newonin ime Pogm he Fou Pinipl Kinei Ses Copyigh 005 Joseph A. Rybzyk Newonin ime Giionl Aeleion D&  EhA.md My 1, 005 Copyigh 005 J. Rybzyk Insuions: Se lue of K uppe igh fo lene disne in km. Copy lue fom D ino ll pogm loops. Copy ol ime fom (lef) disne dependen pogm ino (igh) ime dependen pogm. (Whole numbes only.) All ommon lues fom boh pogms should hen losely gee. Noe: x in lef pogm uenly se fo 1000 m s in 1 km. Vlue n be hnged s ppopie. Cuenly eh loop ieion of lef pogm 1 km nd eh loop ieion of he igh pogm 1 se. N ( ) G : m R R 1 D i R 1 + R + D i 0 0 d 0 0 d 0 0 fo i 1.. G m 1 d + + d d + d SF ime Inel Inpu in s : 6065 SF ime Inel in s Vlue opied fom 3. Ns ( ) G : m R R 1 D i R 1 + R + D i 0 0 d 0 0 d 0 fo i 1.. G m 1 d + d d + d i d + d d i d G m 1 R 1 + R g N ( ) N : Ns ( ) Newonin Speed in ems of N Re of Aeleion his peiod Disne eled his ime Uni ol Speed Afe s ol el Disne Afe s Remining  Disne Afe s Cuen Re of Aeleion 16
17 Appendix B3 Eh Reliisi Disne Pogm he Fou Pinipl Kinei Ses Copyigh 005 Joseph A. Rybzyk Reliisi Disne Giionl Aeleion Reliisi D& EhC.md June 4, 005 J. Rybzyk Rd( K) : Rds ( K) : G m R R 1 G m R R 1 D R : R : 1 Gm 1 g : R 1 + R ( R 1 + R ) R 1 + R + D x d 0 0 fo i 1.. K Gm 1 x + + d d + x x + x x + + G m 1 Noe: he below lues e independen of he lues used in pogms Rd(K) Rds(K) R() Rs() e. hey e used fo ompison puposes only. Mke sue ll lues d gee wih hose used in pogms. G : m : m : 1 g SF Disne Inpu in km 1. Inse K lue K : D K 10 3 :. Copy D o ll Pogs D SF Disne in km Rd( K) D R 1 + R + D x fo i G m 1 x x 1.. K + + x x + + Rds ( K) : Reliisi Speed in ems of Rds ( K) K R e1 : R e Re of Aeleion his Peiod ime fo his Peiod in s ol ime in s 3. Copy o igh Speed fo his Peiod m/s X X ol Speed Imp ol el Disne Afe s Remining  Disne Afe s Cuen Re of Aeleion Y Z Y Z 17
18 Appendix B4 Eh Reliisi ime Pogm he Fou Pinipl Kinei Ses Copyigh 005 Joseph A. Rybzyk Reliisi ime Giionl Aeleion Reliisi D& EhC.md June 4, 005 J. Rybzyk R ( ) : G m R R 1 D i R 1 + R + D i 0 0 d 0 0 d 0 0 fo i 1.. G m 1 d d d + d 4. Copy whole numbe lue fo fom Disne Pogm lowe lef : 6065 SF ime Inel in s 6065 Rs ( ) : G m R R 1 D i R 1 + R + D i 0 0 d 0 0 d 0 fo i 1.. G m 1 d + + d d + d i d + i d G m 1 d d R e : Rs ( ) Rs ( ) Reliisi Speed in ems of R e R 1 + R R ( ) Re of Aeleion his Peiod Disne eled his ime Uni ol Speed Afe s ol el Disne Afe s Remining  Disne Afe s Cuen Re of Aeleion 18
19 Appendix B5 Supemssie Blk Hole Disne Pogm he Fou Pinipl Kinei Ses Copyigh 005 Joseph A. Rybzyk Disne  Giionl Aeleion Reliisi D& Supemssie Blk Hole.md  July 10, J. Rybzyk Rd ( K) : G m R R 1 D R 1 + R + D x d 0 0 fo i 1.. K G m 1 x d d + x + x x + + Supemssie Blk Hole SF Disne Inpu in km 1. Inse K lue K : D mx K 10 3 :. Pogm D No o Exeed D mx D mx SF Disne in km K Rds ( K) : G m R R 1 D R 1 + R + D x fo : R e1 : i 1.. K G m 1 x x + Rds ( K) + x x + + Rds ( K) Reliisi Speed in ems of R e x G m 1 d Rd ( K) Re of Aeleion his Peiod SF ime fo his Peiod in s Y SF ol ime in s 3. Copy o igh AF ime fo his Peiod in s X Z AF ol ime in s Y Speed fo his Peiod m/s X ol Speed fo ol Disne ol el Disne Afe s Remining  Disne Afe s Cuen Re of Aeleion Z 19
20 Appendix B6 Supemssie Blk Hole ime Pogm he Fou Pinipl Kinei Ses Copyigh 005 Joseph A. Rybzyk ime  Giionl Aeleion Reliisi D& Supemssie Blk Hole.md  July 10, J. Rybzyk R ( ) : G m R R 1 D i R 1 + R + D i 0 0 d 0 0 d 0 0 fo i 1.. G m 1 d d d + d Supemssie Blk Hole 4. Copy whole numbe lue fo fom Disne Pogm lowe lef : 116 SF ime Inel in s 116 Rs ( ) : G m R R 1 D i R 1 + R + D i 0 0 d 0 0 d 0 fo i 1.. G m 1 d + + d d + d i d + i d G m 1 d d R e : Rs ( ) Rs ( ) Reliisi Speed in ems of R e R ( ) Re of Aeleion his Peiod Disne eled his ime Uni ol Speed Afe s ol el Disne Afe s Remining  Disne Afe s Cuen Re of Aeleion Noe: If you eneed his pge diely duing seh, you n isi he Millennium Reliiy sie by liking on he Home link below: Home 0
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