The Velocity Addition Formula

Velocity Addition Formula

Updated: May 10, 2026

On the Electrodynamics of Moving Bodies

Einstein-EDoMB-1905-Section §5

https://www.fourmilab.ch/etexts/einstein/specrel/www/

Einstein: Relativity, The Special and the General Theory, 1920, Chapter VI page 19, Chapter XIII page 45

https://www.google.com/books/edition/Relativity/n8QKAAAAIAAJ?hl=en&gbpv=1

Both sources show the velocity-addition-formula as it is today.

Shown: V = ( v + w ) / ( 1 + vw/c² )

#### Abbreviations and notation

SRT = spezielle Relativitätstheorie, Special Relativity

rf = reduction factor ( denominator ) in the v-a-f.

Let’s change a notation without changing anything on the ground.  Bring in an asterisk* for relativistic output velocity-V* of addition or subtraction.

Classical addition or subtraction:  V  ← ( v ± w )

SRT addition or subtraction: V*  ← ( v ± w ) / ( 1 ± vw/c² )

As in most of SRT, we use left ← to ← right equation logic ( an assignment operator so-called ) rather than high school left = to = right, an identity.

5 ← 2 + 3

2 + 3 = 5

In subtraction and addition set-ups that follow, speed-w is the special case of speed-c, the speed-of-light.  That simplifies the denominator reduction factor to:

rf = ( 1 ± v/c )

The sun out there in free-space fires off a photon:

Alter! When the Hills do —
Falter! When the Sun
Question if His Glory
Be the Perfect One —

#### Lorentz Transforms

Lorentz Transforms: minus-sign, shown in EDoMB-1905 Section §3 and used in Section §4 ( length contraction and time dilation. ) Shown in Halliday & Resnick as equations 37-2I:

x′ = gamma • ( x – vt )

t′ = gamma • ( t – vx / c² )

Lorentz Transforms: plus-sign, shown in H&R as equations 37-22. 

x = gamma • ( x′ + vt′ )

t = gamma • ( t′ + vx′ / c² )

This plus-sign pair is shown neither in Section §5 ( v-a-f ) nor in 1920-Chapter XIII ( v-a-f ):

#### Derive the v-a-f Section §5

Section §5 has a statement which invokes the minus-sign pair of Section §3.   Quote: “With the help of Section §3 equations we solve for x and t.”   The statement does not lead to a display of the plus-sign pair.  A single additive-spatial is shown. It is not an LT. Gamma is missing:

x = ( w′ + v ) • t / rf

It is this equations which leads to the v-a-f:

Shown:  V* = ( v + w ) / ( 1 + vw/c² )

#### Derive the v-a-f H&R Chapter 37

H&R Chapter 37-3 has a similar statement:  Quote: “Simply solve equations 37-21 for x.” The plus-sign pair is then shown but the algebra of “solve for” is not shown.  Chapter 37-4 then has derivation of the v-a-f from the plus-sign pair shown as Equation 37-29.

The general idea is simple.  Speed is distance divided by time.  The plus-sign LT pair with speed-w:

x = gamma • ( wt′ + vt′ )

t = gamma • ( t′ + vwt′ / c² )

V* = x / t

Gamma divides out.  Time-t′ divides out.

V* = ( v + w ) / ( 1 + vw/c² )

Special case: w = c

V* = ( v + c ) / ( 1 + vc/c² )

V* = ( v + c ) / ( 1 + v/c )

V* = c

#### Solve for gold

x′ = x – vt

paper money ← gold standard minus gold standard

x = vt + x′

gold standard ← gold standard plus paper money

Is it really the gold standard or have we been defrauded?

Footnote:  Comment-1 on the blog has a version of “solve for x” taken off a web page.  The algebra seems to have a blatant falsehood.

False: vt′ = vt

#### Subtract

Let’s do a velocity formula based on the minus-sign spatial of Section §3:

x′ = gamma • ( x – vt )

Point-x is a point of speed-v like point-vt, although not shown as such.  Let it have speed-w.  Rewrite: x = wt.

x′ = gamma • ( wt – vt )

We still have a distance equation, not a speed equation.  Get rid of gamma and we move over to a distance of classical mechanics.

x′ = wt – vt

What’s up? The right-hand-side is subtraction of changing distances on essential axis-X, unprimed, subtraction of changing locations.  Both distances are with respect to axis-zero.  Output-x′ on the left-hand-side is not a location, is not a distance with respect to axis-zero.  It is a segment on axis-X, changing in both location and size.  Use of primed notation is statement of that fact.  Here in Galilean mode, we don’t really have a new primed axis, “two systems of co-ordinates” as they say. 

The two distances ( locations ) wt and vt are what you might call parallel distances, not a “composition” of distances wherein distance-wt is with respect to location-vt.

x′ = ( w – v ) • t

Divide by t:

x′ / t = w – v

We now have explicitly a speed equation.  Let’s say speed-w is faster than speed-v.  The point-wt is out in front of point-vt, but that’s because it is going faster, not because it is a serial speed, aka serial resister-w attached to resister-v.  Output speed x′ / t is speed of separation, not ordinary axis speed with respect to the starting line.

Let speed-w be speed-c, the speed-of-light:

x′ = ct – vt

x′ = ( c – v ) • t

x′ / t = c – v

Speed of separation of points ct and vt is speed x′ / t.  It is a good algebraic value.  Point-ct gets out in front of point-vt at speed-( c – v ).  But point-vt cannot view point-ct as moving ahead at speed-( c – v ).  Too slow.  Point-ct is a photon and a photon has speed-c regardless of who is looking at it.  Point-vt must view point-ct as moving ahead at speed-c.

Point-vt views point-ct as point-x′.  From that stance, the speed of point-x′ is speed judged by a clock moving at speed-v. The clock is adjusted by the factor: ( c – v ) / c < 1, less than 1. Less time for a projectile doing length-x′ is higher speed.  The adjustment factor allows point-vt to have for itself a photon on distance-x′ at speed-c.  Counter-intuitive because the clock announcing speed-v of point-vt is stationary on the pavement.  It is not adjusted.  

x′ = ( c – v ) • t

t′ = [ ( c – v ) / c ] • t

Divide distance by time to obtain speed:

V* = x′ / t′

V* = ( c – v ) • t / [ ( c – v ) / c ] • t

The velocity subtraction formula, v-s-f:

V* = ( c – v ) / ( 1 – v/c )

V* = c

#### Three distance additions

The plus-sign spatial of H&R 37-22, without gamma:

++ add:  x = x′ + vt′

Introductory spatial of Section §5, without reduction factor:

++ add:  x = ( w′ + v ) • t

Solve for x:

++ add:  x = x′ + vt

Three equations.  Which is it ?  Heaven only knows !

https://groups.google.com/g/alt.arts.poetry.comments/c/R5Aif2cXSdE

#### Critical

Actually those three distance sums have something in common:  all three are the sum of two serial distances not two parallel distances.   Evidently the “composition of velocities” must begin with composition of distances.

1920 Chapter VI, which lays out a “carriage and embankment,” makes roughly the same statement.

Let’s not observe subtraction or addition of distances.  Let’s observe subtraction and addition of speeds.

Subtraction:  A photon viewed by point-vt does distance-x′ and goes too slowly.  A clock associated with point-vt ( stationary with respect to point-vt but moving at speed-v on the pavement ) is adjusted, thereby restoring the photon to speed-c on distance-x′.  Adjustment of this moving clock has no effect on two stationary clocks: the clock at pavement point-x doing speed-c in x = ct, and the pavement clock momentarily associated with point-vt, doing speed-v.

Addition: A photon is going to fast because it is doing the sum of two distances: distance-ct when w = c, and distance-vt.  If you speed up pavement clock ( on axis-X ) at location x = ct + vt, you slow down something which has gone that distance in time-t.  More time slower speed.

Critical: Doctoring up the clock at pavement location x = ( c + v ) • t can slow down the speed of point-x from c + v to c.  But changing that pavement clock requires changing all pavement clocks, including the clock doing point-vt, in which case value-vt is no good.

Subtraction requires transformation of only one clock: clock c – v.  Addition requires transformation of two clocks: clock c + v and clock v.  A monumental crisis of institutions.