The Velocity Addition Formula

Velocity Addition Formula

Updated: November 27, 2025

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 )

#### Lorentz Transforms

Lorentz Transforms – subtraction, per 1905 Sections §3 and §4:

x′ = gamma • ( x – vt )

t′ = gamma • ( t – vx / c² )

Lorentz Transforms – addition, from modern textbooks:

x = gamma • ( x′ + vt′ )

t = gamma • ( t′ + vx′ / c² )

#### Textbook derivation of the v-a-f

Halliday & Resnick, 10th Edition, Chapter 37-4.

Precursor to the numerator ( v + w ) in the v-a-f is the plus-sign LT spatial, an addition of distances.

Let x′ be of speed-w:  x′ = wt′

Spatial:  x = gamma • ( wt′ + vt′ )

Temporal:  t = gamma • ( t′ + vwt′ / c² )

Speed is distance divided by time:

V = x / t = gamma • ( w + v ) • t′ / gamma • ( 1 + vw/c² ) • t′

gamma divides out:

V = x / t = ( w + v ) • t′ / ( 1 + vw/c² ) • t′

Time-t′ divides out:

V = ( v + w ) / ( 1 + vw/c² )

EDoMB-1905 Section §5 and Relativity-1920 Chapter XIII invoke the Lorentz Transforms but neither document provides an LT derivation.   In particular, the plus-sign algebra of Halliday and Resnick Chapter 37 is not shown.

#### See also

Paul A Tipler, Physics Fourth Edition, Chapter 39
Equation 39-1b: “inverse”
Equation 39-18a: v-a-f by means of quotient

####  2 + 2 = 3

Question: How do you add two plus two and get three?

Answer A: 2 = 1.5 and 2 = 1.5 and 1.5 + 1.5 = 3

Answer B: 2 + 2 = 4 and 4 = 3

#### Verbalize

How do we verbalize?  Let’s have an abbreviation for Special Relativity: spezielle Relativitätstheorie, SRT.

For starters, speed variables v and w can be projectile names, projectile-v and projectile-w. 

Then use subscripts to distinguish Galilean speed-V from relativistic speed-V. 

Galilean sum of speeds v and w is speed-Vg subscript-g:

Vg = v + w

Relativistic sum of speeds v and w is speed-Vsrt, subscript-srt:

Vsrt = ( v + w ) / ( 1 + vw/c² )

Misleading.  If you revert to Galilean mode, and get rid of the reduction factor, you have with speed-Vg on the left-hand-side, and right-hand-side speed-w is simply speed-w.  What happens in SRT mode?  Speed-V becomes speed-Vsrt.  Do speeds v and w become speeds vsrt and wsrt?

#### Transformation

The v-a-f equation is written backwards.  SRT equations are usually written that way.  A standard equation written left → to → right is an “identity”:

Identity: 2 + 2 → 4  is simply 2 + 2 = 4

An SRT equation written left ← to ← right is an “assignment” operator.

Assignment: 3 ← 2 + 2

But it’s more than that: it’s “transformation.”  Speeds v and w are both equal to 2 miles per hour.  Speed sum 2 mph + 2 mph equal to 3 mph is not merely addition, it is also transformation.

#### The distributive law

How do you transform 2 mph + 2 mph into 3 mph? You divide by the dimensionless reduction factor.

Reduction factor ( 1 + vw/c² ) = 4/3 > 1

3 mph ← ( 2 mph + 2 mph ) / ( 4/3 )

What is division by ( 4/3 ) doing?

The distributive law:

2 / ( 4/3 ) + 2 / ( 4/3 ) = 4 / ( 4/3 )

Side-A: 2 / ( 4/3 ) + 2 / ( 4/3 )

Side-B: 4 / ( 4/3 )

Algebraically equal, sides A and B are different physical situations.

Side-A:  You reduce the speeds of projectiles v and w from 2 mph to 1.5 mph and add:

3 ←  1.5 + 1.5  ← 2 / ( 4/3 ) + 2 / ( 4/3 )

Side-B: You add projectile speeds v = 2 and w = 2 with resultant sum 4 mph, and then reduce 4 mph by divisor ( 4/3 ) with resultant 3 mph.

3 ← 4 / ( 4/3 ) ← ( 2 + 2 ) / ( 4/3 )

#### Which is it?

Which side, A or B, is inherent in the v-a-f? 

Vsrt = ( 2 + 2 ) / ( 4/3 )

Halliday & Resnick is Side-B:  Spatial and temporal LT’s do a speed quotient.  The numerator adds speeds v and w unchanged, and then the denominator factors up terminal time of projectile-Vg by 4/3 > 1, thereby factoring down speed-Vg = 4 mph by 3/4, and speed-Vg = 4 becomes speed-Vsrt = 3 mph.

The reduction factor rf = ( 1 + vw/c² ) is greater than 1, which factor increases time on the terminal pavement clock doing speed-V, so that speed-V is speed-Vsrt, and speed-Vsrt ( requiring more time ) is then less than speed Vg. 

Section §5 is Side-A:  Quote, “It is worthy of remark that v and w enter into the expression for resultant velocity in a symmetrical manner.”  Unquote.  Seemingly you factor down speeds v and w ( not clocks ) by 3/4 and then add.  3/4 of 2 mph plus 3/4 of 2 mph is 1.5 mph plus 1.5 mph equal to 3 mph.

#### Another A-B dichotomy

The Galilean equation:

Vg = v + w

As before, subscript-g is Galilean.  Per 1920 Chapter VI, speed-v is the “carriage” on the “embankment,” speed-w is the walker on the carriage, aka the “man” on the carriage, and speed-Vg is walker-w on the embankment.   We use the special case where nothing is equal to speed-c.  We can easily contrive another Galilean walker directly on the embankment, walker Vge, going side-by-side, double file, with walker-Vg ( who is walker-w ):

Vge = Vg

What happens when you impose SRT on walker-Vg, while walker-Vge remains Galilean?  Speed-Vg becomes speed-Vsrt:

Vsrt = ( v + w ) / ( 1 + vw/c² )

How does walker-Vsrt relate to walker-Vge?  You have two scenarios:

Scenario-A: Walker-Vsrt, going slower than walker-Vge, falls behind and arrives at a ground destination later than walker-Vge.

Scenario-B: Walker-Vsrt and walker-Vge are side-by-side, double file. But, arriving at one ground destination at one moment, walker-Vsrt has a terminal clock reading more time than the Galilean terminal clock of walker-Vge.  Taking more time to get some place you are going slower.

Roughly, Scenario-A is Section §5 of “symmetric” while Scenario-B is college algebra where the denominator in the LT quotient is the terminal clock.

Which is it?