integrated relativistic velocity and acceleration compositioncv t da + − = and 2 2 2 1 1 c c a t (...

Equations from Integrated Relativistic Velocity and Acceleration Composition Copyright © 2004 Joseph A. Rybczyk

Following is a complete list of all of the equations used in and/or derived in the Integrated Relativistic Velocity and Acceleration Composition work. For ease of reference, all equations are identified by the same number originally assigned in the theory. Also included are examples using the same Figures including the originally assigned numbers from the referenced work. 1. Summary List of All Formulas Used in the Above Referenced Paper

NRP =

vc t

va = and 1

11

vc t

va = (1) Constant Acceleration Formulas

vctav = and (2) Instantaneous Speeds from Acceleration 111 vc tav =

cvcTtv

22 −= and

cvc

Ttv

21

2

1−

= (3) Time Transformation Formulas

22 )( TacTcav

c

c

+= and

21

21

1)( Tac

Tcavc

c

+= (4) Instantaneous Speed Formulas

cvc

Vvv u

21

2

21−

+= (5) Velocity Composition Formula (Vu2 intentionally shown as V2 in paper)

21

21

2)(

vc

vvcVu−

−= (6) Velocity Composition Formula for Vu2

TtVv v

uu1

22 = (7) Equivalent Speed Relative to SF

cvc

Uvu2

12

21−

+= (8) Velocity Composition Formula Modified

21

21

2)(

vc

vucU−

−= (9) Velocity Composition Formula for U2

TtUu v1

22 = (10) Uniform Motion Speed Relative to SF

1

cuc

Vuv u

21

2

21−


21

21

2)(

uc

uvcVu−


cuc

Ttu

21

2

1−

= (13) Time Transformation Formula for UF u1

TtVv u

uu1


cuc

Uuu2

12

21−

+= (15) Velocity Composition Formula, Standard Form

21

21

2)(

uc

uucU−

−= (16) Velocity Composition for U2

TtUu u1


22 vcccvTDa

−+= and

22

2

)( TaccTcaD

c

ca

++= (18) Acceleration Distance for v & ac

21

2

11

vcc

TcvDa−+

= and 2

12

21

1)( Tacc

TcaDc

ca

++= (19) Acceleration Distance for v1 & ac1

22

2

122

Vcc

tcVD va

−+= and

212

2

212

2)( vc

vca

tacctcaD

++= (20) Acceleration Distance for V2 & ac2

22

21

221

2

22

av

av

DtcDtcV+

= (21) Instantaneous Speed, V2 using Da2

12 aaa DDD −= (22) The Relationship of Da2 to Da and Da1

21

21

211

2

2 )()(2

aav

aav

DDtcDDtcV

−+−

= (23) Instantaneous Speed, V2 using Da and Da1

2

2

22

vc t

Va = (24) Acceleration Rate, ac2

cVc

tt vv

22

2

12−

= (25) UF v2 Time Transformation

2

22

vu

ucu t

Va = (26) Acceleration Rate, acu2

cVc

tt uvvu

22

2

12−

= (27) UF vu2 Time Transformation

21

22

1

221

2

2

2

)()( vc

vvvcT

cacu

+−−−

= (28) Acceleration Rate, acu2 based on v and v1 (New)

)(1 2

1222

2

42

vcTac

cV

cu

u

−+

= (29) Alternate Formula for Vu2 (New)

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−+−

−+=

21

21

222vcc

cvvcc

cvTDa (30) Formula for Da2

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

−+

+−

−+= 22

12

2222

)(u

ua V

vcc

Vvcvcc

cvTD (31) Formula for Da2 Based on Vu2 (New)

22

212

2

u

vuau

Vcc

tcVD−+

= and 2

122

212

2)( vcu

vcuau

tacctcaD

++= (32) Acceleration Distance for Vu2 & acu2

TtVv v1


uTDu = (34) Uniform Motion Distance for u

21

21

211

2

2 )()(2

auv

auv

DDtcDDtcV

−+−

= (35) Instantaneous Speed, V2 using Du and Da1

3

cucTtu

22 −= (36) Time Transformation for UF u

21

21

1

vca

cvTc −

= (37) SF Interval for Reaching Speed v1

221 uca

cuTc

i−

= (38) Intermediate SF Interval for Reaching Speed u

21

21

2)(

vc

vucVu−

−= (39) Modified Velocity Composition Formula for Vu2

cVc

tt uuvu

22

2

12−


TuDu 11 = (41) Uniform Motion Distance for u1

21

21

211

2

2 )()(2

auu

auu

DDtcDDtcV

−+−

= (42) Instantaneous Speed, V2 using Du1 and Da

TtVv u1


22

2

122

Vcc

tcVD ua

−+= and

212

2

212

2)( uc

uca

tacctcaD


22

212

2

u

uuau

Vcc

tcVD−+

= and 2

122

212

2)( ucu

ucuau

tacctcaD


cVc

tt uv

22

2

12−


21

21

2)(

uc

uucVu−

−= (47) Velocity Composition for Vu2

cuc

Vuu u

21

2

21−

+= (48) Velocity composition formula

4

21

21

211

2

2 )()(2

uuu

uuu

DDtcDDtcV

−+−

= (49) Instantaneous Speed, V2 using Du and Du1

212

212

2)( ucu

ucuu

tactcaV

+= (50) Instantaneous Speed, Vu2

212

212

2)( uc

uc

tactcaV

+= (51) Instantaneous Speed, V2

23

2

133

Vcc

tcVD ua

−+= and

213

2

213

3)( uc

uca

tacctcaD++

= (52) Acceleration Distance for V3 & ac3

21

22

21

2

32

uucuccu

V−+−

= (53) Instantaneous Speed, V3 using u and u1 (New)

3

33

vc t


cVc

tt uv

23

2

13−


213

213

3)( uc

uc

tactcaV


TtVv u1


22

22ucucv

+= (58) Instantaneous Speed, v using u

1utTvV = (59) Equivalent Speed Relative to UF u1

2. Examples of Formula Applications

5

2.1 Case 1 – Two Different Acceleration Rates

FIGURE 6 Case 1 – Part 2 – The ac2 Relationship to ac1 and ac

Da Da1

v & Vu2

v1

ac1

ac

Distance in LY’s (SF)

Spe

ed (

SF)

0

0.1

.75c

0

.25c

.5c

c

0.2 0.3 0.4 0.5

Spe

ed (

UF

v 1) .5c

.25c

0

V2 ac2

Da2

0.1 0.2 0.3 Distance in LY’s (UF v1)

Where, relative to the SF,

ac and ac1 are constant rates of acceleration, v and v1 are the respective speeds, Da and Da1 are the respective distances, and, T is the time interval,

while, relative to UF v1, ac2 is a constant rate of acceleration, V2 is the speed,

v2 is the corresponding speed in the SF, Da2 is the corresponding distance in the SF, and,

tv1 is the time interval corresponding to T, while, tv and tv2 are the time intervals in UFs v and v2 respectively, (not shown in the

illustration) Given: c, ac, ac1, and T

22 )( TacTcav

c

c

+= and

21

21

1)( Tac

Tcavc

c


6

cvcTtv

22 −= and

cvc

Ttv

21

2

1−


v

c tva = and

1

11

vc t

va = (1) Constant Acceleration Formulas

vctav = and (2) Instantaneous Speeds from Acceleration 111 vc tav =

22 vcccvTDa

−+= and

22

2

)( TaccTcaD

c

ca


21

2

11

vcc

TcvDa−+

= and 2

12

21

1)( Tacc

TcaDc

ca


12 aaa DDD −= (22) The Relationship of Da2 to Da and Da1

22

21

221

2

22

av

av

DtcDtcV+

= (21) Instantaneous Speed, V2 using Da2

21

21

211

2

2 )()(2

aav

aav

DDtcDDtcV

−+−

= (23) Instantaneous Speed, V2 using Da and Da1

Note, in the following, tv2 is the interval in UF v2 (not shown in the illustration)

corresponding to tv1

cVc

tt vv

22

2

12−


2

22

vc t


TtVv v1


22

2

122

Vcc

tcVD va

−+= and

212

2

212

2)( vc

vca

tacctcaD


7

FIGURE 7 Case 1 – Part 2 – The acu2 Relationship to ac1 and ac

Da Da1

v

v1

ac1

ac


Spe

ed (

SF)

0

0.1

.75c

0

.25c

.5c

c

0.2 0.3 0.4 0.5

Spe

ed (

UF

v 1) .5c

.25c

0

Vu2

acu2

Dau2


and also, relative to UF v1,

acu2 is a constant rate of acceleration, Vu2 is the speed,

vu2 is the corresponding speed in the SF, Dau2 is the corresponding distance in the SF, and,

whereas, tvu2 is the time interval in UF vu2 (not shown in illustration) corresponding to tv1,

cvc

Vvv u

21

2

21−

+= (5) Velocity Composition Formula (Vu2 intentionally shown as V2 in paper)

21

21

2)(

vc

vvcVu−


TtVv v

uu1


cVc

tt uvvu

22

2

12−


8

2

22

vu

ucu t


21

22

1

221

2

2

2

)()( vc

vvvcT

cacu

+−−−

= (28) Acceleration Rate, acu2 based on v and v1 (New)

)(1 2

1222

2

42

vcTac

cV

cu

u

−+

= (29) Alternate Formula for Vu2 (New)

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−+−

−+=

21

21

222vcc

cvvcc

cvTDa (30) Formula for Da2

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+

−+

+−

−+= 22

12

2222

)(u

ua V

vcc

Vvcvcc

cvTD (31) Formula for Da2 Based on Vu2 (New)

22

212

2

u

vuau

Vcc

tcVD−+

= and 2

122

212

2)( vcu

vcuau

tacctcaD


2.2 Case 2 – Low Instantaneous Speed vs. High Uniform Speed

Where, relative to the SF, ac1 is a constant rate of acceleration, v1 is the achieved speed, u is a uniform rate of speed ≥ v1Da1 and Du are the respective distances, and, T is the time interval,

while, relative to UF v1, acu2 and ac2 are constant rates of acceleration, Vu2 and V2 are the respective speeds, Dau2 and Da2 are the respective SF distances,

vu2 and v2 are the respective corresponding speeds in the SF, U2 is the speed corresponding to u, u2 is the corresponding speed for U2 in the SF,

tv1 is the time interval corresponding to T, whereas, tvu2, tv2 and tu are the time intervals in UFs vu2, v2, and u respectively (not shown

in the illustration)

Given: c, u, ac1, and T

9

21

21

1)( Tac

Tcavc

c

+= (4) Instantaneous Speed Formula

cvc

Ttv

21

2

1−

= (3) Time Transformation Formula

1

11

vc t

va = (1) Constant Acceleration Formula (not shown in paper)

111 vc tav = (2) Instantaneous Speed from Acceleration (not shown in paper)

FIGURE 8 Case 2 – Part 2 – The ac2 and acu2 Relationships to ac1 and u

Da1

v1

ac1


Spe

ed (

SF)

0

0.1

.75c

0

.25c

.5c

c

0.2 0.3 0.4 0.5

Spe

ed (

UF

v 1) .5c

.25c

0

u & U2

Du


ac2

Da2

c

.75c

Dau2

acu2

V2

Vu2

10

cvc

Uvu2

12

21−


21

21

2)(

vc

vucU−

−= (9) Velocity Composition Formula for U2

TtUu v1


Given, Vu2 = U2,

cVc

tt uvvu

22

2

12−


2

22

vu

ucu t


22

212

2

u

vuau

Vcc

tcVD−+

= and 2

122

212

2)( vcu

vcuau

tacctcaD



21

2

11

vcc

TcvDa−+

= and 2

12

21

1)( Tacc

TcaDc

ca


21

21

211

2

2 )()(2

auv

auv

DDtcDDtcV

−+−


TtVv v1


cVc

tt vv

22

2

12−


2

22

vc t


22

2

122

Vcc

tcVD va

−+= and

212

2

212

2)( vc

vca

tacctcaD


11

cucTtu


2.3.1 Case 3 – The ac2 and acu2 Relationships to u and ac1

Da1 Du

Da2

FIGURE 9 Case 3 – Part 2 – The ac2 and acu2 Relationships to u and ac1


Spe

ed (

SF)

0

0.1

.75c

0

.25c

.5c

c

0.2 0.3 0.4 0.5 S

peed

(U

F u)

.5c

.25c

0 0.1 0.2

Distance in LY’s (UF u) ac1

u

ac2

V2

.75c

c

0.3 0.4

acu2

ac2

v1 Vu2


ac1 is a constant rate of acceleration, v1 is the achieved speed, u is a uniform rate of speed ≥ v1Da1 and Du are the respective distances, and, T is the time interval,

while, relative to UF v1, acu2 and ac2 are constant rates of acceleration, Vu2 and V2 are the respective speeds, Dau2 and Da2 are the respective SF distances,

12


tv1 is the time interval corresponding to T, whereas, tvu2, tv2 and tu are the time intervals in UFs vu2, v2, and u respectively (not shown


Given: c, u, ac1, and T

21

21

1)( Tac

Tcavc

c


21

21

1

vca

cvTc −

= (37) SF Interval for Reaching Speed v1

221 uca

cuTc

i−

= (38) Intermediate SF Interval for Reaching Speed u

cvc

Vvu u

21

2

21−

+= (60) Velocity Composition Formula Modified (not given in paper)

21

21

2)(

vc

vucVu−

−= (39) Modified Velocity Composition Formula for Vu2

cvc

Ttv

21

2

1−


TtVv v

uu1


cVc

tt uvvu

22

2

12−


2

22

vu

ucu t


22

212

2

u

vuau

Vcc

tcVD−+

= and 2

122

212

2)( vcu

vcuau

tacctcaD



13

21

2

11

vcc

TcvDa−+

= and 2

12

21

1)( Tacc

TcaDc

ca


21

21

211

2

2 )()(2

auv

auv

DDtcDDtcV

−+−


TtVv v1


cVc

tt vv

22

2

12−


2

22

vc t


22

2

122

Vcc

tcVD va

−+= and

212

2

212

2)( vc

vca

tacctcaD


cucTtu


2.3.2 Case 3 – The ac2 and acu2 Relationships to u1 and ac


ac is a constant rate of acceleration, v is the achieved speed, u1 is a uniform rate of speed ≤ v Da and Du1 are the respective distances, and, T is the time interval,

while, relative to UF u1, acu2 and ac2 are constant rates of acceleration, Vu2 and V2 are the respective speeds, Dau2 and Da2 are the respective SF distances,

vu2 and v2 are the respective corresponding speeds in the SF, tu1 is the time interval corresponding to T,

whereas, tvu2, tv2 and tv are the time intervals in UFs vu2, v2, and v respectively (not shown in the illustration)

Given: c, u1, ac, and T

14

FIGURE 10 Case 3 – Part 2 – The ac2 and acu2 Relationships to u1 and ac

Du1

Da

v

u1

ac


Spe

ed (

SF)

0

0.1

.75c

0

.25c

.5c

c

0.2 0.3 0.4 0.5

Spe

ed (

UF

u 1) .5c

.25c

0 0.1 0.2

Distance in LY’s (UF u1)

Da2 = 0 (Final)

acu2

ac2

- 0.1

Dau2

V2

Vu2

Da2 (Initial)

22 )( TacTcav

c

c


cuc

Vuv u

21

2

21−


21

21

2)(

uc

uvcVu−


cuc

Ttu

21

2

1−


TtVv u

uu1


cVc

tt uuvu

22

2

12−


15

2

22

vu

ucu t


22

212

2

u

uuau

Vcc

tcVD−+

= and 2

122

212

2)( ucu

ucuau

tacctcaD



22 vcccvTDa

−+= and

22

2

)( TaccTcaD

c

ca


21

21

211

2

2 )()(2

auu

auu

DDtcDDtcV

−+−

= (42) Instantaneous Speed, V2 using Du1 and Da

TtVv u1


cVc

tt uv

22

2

12−


2

22

vc t


22

2

122

Vcc

tcVD ua

−+= and

212

2

212

2)( uc

uca

tacctcaD


cvcTtv

22 −= (61) Time Transformation for UF v (not given in paper)

2.4. Case 4 – The ac2 and acu2 Relationships to u1 and u


u is a uniform rate of speed u1 is a uniform rate of speed < u Du and Du1 are the respective distances, and, T is the time interval,

while, relative to UF u1, acu2 and ac2 are constant rates of acceleration, Vu2 and V2 are the respective speeds, Dau2 and Da2 are the respective SF distances,

16


tu1 is the time interval corresponding to T, whereas, tvu2, tv2 and tu are the time intervals in UFs vu2, v2, and u respectively (not shown


Given c, u, u1 and T

FIGURE 13 Case 4 – Part 2 – Two Different Uniform Motion Speeds

u1


Spe

ed (

SF)

0

0.1

.75c

0

.25c

.5c

c

0.2 0.3 0.4 0.5 S

peed

(U

F u 1

) .5c

.25c

0

u & U2

Du

Du1

.75c

acu2

ac2

Vu2

V2

Da2

Dau2

cuc

Uuu2

12

21−

+= (15) Velocity Composition Formula, Standard Form

21

21

2)(

uc

uucU−

−= (16) Velocity Composition for U2

cuc

Ttu

21

2

1−


17

TtUu u1




cuc

Vuu u

21

2

21−

+= (48) Velocity composition formula

21

21

2)(

uc

uucVu−

−= (47) Velocity Composition for Vu2

TtVv u

uu1


cVc

tt uuvu

22

2

12−


2

22

vu

ucu t


22

212

2

u

uuau

Vcc

tcVD−+

= and 2

122

212

2)( ucu

ucuau

tacctcaD


21

21

211

2

2 )()(2

uuu

uuu

DDtcDDtcV

−+−

= (49) Instantaneous Speed, V2 using Du and Du1

TtVv u1


cVc

tt uv

22

2

12−


2

22

vc t


18

22

2

122

Vcc

tcVD ua

−+= and

212

2

212

2)( uc

uca

tacctcaD


212

212

2)( ucu

ucuu

tactcaV

+= (50) Instantaneous Speed, Vu2

212

212

2)( uc

uc

tactcaV


cucTtu


3. Extrapolation of Principles

v

V3

ac3

ac

FIGURE 14 Extrapolation of Principles

u1


Spee

d (S

F)

0

0.1

.75c

0

.25c

.5c

c

0.2 0.3 0.4 0.5

Spee

d (U

F u 1

)

.5c

.25c

0

u & U2

Du Da

Du1

.75c

acu2

ac2 Vu2

V2

Da2

Dau2

c

Du Da3

0.1 0.2 0.3 0.4


u is a uniform rate of speed

19

u1 is a uniform rate of speed < u Du and Du1 are the respective distances, ac is a constant rate of acceleration, v is the achieved speed, Da is the distance, and, T is the time interval,

while, relative to UF u1, acu2, ac2 and ac3 are constant rates of acceleration, Vu2, V2 and V3 are the respective speeds, Dau2, Da2 and Da3 are the respective SF distances,

vu2, v2 and v3 are the respective corresponding speeds in the SF, U2 is the speed corresponding to u, u2 is the corresponding speed for U2 in the SF,

tu1 is the time interval corresponding to T, whereas, tvu2, tv2, tv3 and tu are the time intervals in UFs vu2, v2, v3 and u respectively (not

shown in the illustration)

Given: c, u, u1 and T


21

22

21

2

32

uucuccu

V−+−

= (53) Instantaneous Speed, V3 using u and u1 (New)

cuc

Ttu

21

2

1−


cVc

tt uv

23

2

13−


3

33

vc t


23

2

133

Vcc

tcVD ua

−+= and

213

2

213

3)( uc

uca

tacctcaD++

= (52) Acceleration Distance for V3 & ac3

213

213

3)( uc

uc

tactcaV


TtVv u1


20

22

22ucucv

+= (58) Instantaneous Speed, v using u

cvcTtv

22 −= (3) Time Transformation Formula

vc t

va = (1) Constant Acceleration Formula

22 )( TacTcav

c

c


1utTvV = (59) Equivalent Speed Relative to UF u1

Equations from Integrated Relativistic Velocity and Acceleration Composition

Copyright © 2004 Joseph A. Rybczyk

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21

integrated relativistic velocity and acceleration compositioncv t da + − = and 2 2 2 1 1 c c a t (...

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