The Velocity Addition Formula

Velocity Addition Formula

Updated: November 13, 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

#### Length contraction, Section §4

Length contraction in Section §4 uses a “rigid” sphere.  A one-dimensional Michelson-Morley device of time dilation is easier than the three-dimensional sphere. The MMD has unnamed stationary size, call it length-L, analogous to sphere radius-R, lengths L and R being fixed. Both devices have speed-v.  Length contraction uses the subtractive spatial LT of Section §3:

x′ = gamma • ( x – vt )

Begin with “Galilean” subtraction, so-called no gamma.

x′ = x – vt

Point-x is moving ground location x = vt + L at time-t, which is the front of the MMD at time-t, and point-vt is the rear of the MMD at time-t.

x′ = ( vt + L ) – vt

x′ = L

As esoteric transformation, point-x moving on the ground is transformed into point x′ = L, stationary on a moving axis.  As a practical matter, the Galilean model does not need the transformation concept.   Nothing is needed except algebraic subtraction:

( vt + L ) – vt = L

Bring in gamma and you have transformation.

x′ = gamma • ( [ vt + L ] – vt )

This equation, by itself, does not transform stationary size-L of the MMD into left-hand-side size-x′ greater than L .  Gamma transforms a certain distance of unprimed metric on the right-hand-side into that same distance of primed metric on the left-hand-side. Transformation of length-L is a second step not shown in the equation.

Example numbers: The MMD is Galilean length L = 10 meters, stationary or moving.  10 meters is 11 yards.

Bring in SRT ( spezielle Relativitätstheorie ) and gamma.  Gamma = 1.1 .

Gamma = 1.1 transforms a 10 meter span on the right-hand-side ( unprimed ) into an 11 yard span on the left-hand-side ( primed ).

11 = 1.1 • 10

Step one: a span once enumerated as 10 meters is now enumerated as 11 yards, and the moving span of x = 10 meters is reconsidered as moving length of x′ = 11 yards, transformed but unchanged in length.

Step two: The MMD of stationary length x = 10 meters is then pasted into the newly calibrated span where it has length not of x′ = 11 yards ( same as 10 meters ) but of x′ = 10 yards ( a shorter length because x′ = 10 yards is 9.14 meters. )  Looked at from the ground, Section §4 says quote the MMD “appears shorter” than its 10 meter original, but no equation of the sort: 9.14 = 10 / 1.1 is shown.

#### Footnote: Nomenclature

Seen by the ground guy, the moving device “appears shorter.”  He can say it’s 10 yards, not 10 meters.  But someone associated with the moving device can’t say yards.  He has 10 meters, a moving meter not as long as a stationary meter.  Both viewers then have variable-x,  and variable-x means meters.  But unprimed-x means a stationary meter and primed-x′ is a moving meter, or better, a stationary meter on a moving axis.

Verbalizing a primed-unprimed equation is complete chaos, whereas you can write about yards and meters.

#### Footnote: Location zero

Section §4 does the rigid sphere of size-R at time-zero, at which moment the origin of moving radius-R ( center of the sphere ) is at ground-zero: vt = 0.

x′ = gamma • ( x – vt )

x′ = gamma • ( [ vt + R ] – vt )

x′ = gamma • ( [ 0 + R ] – 0 ]

x′ = gamma • R

Moving radius-R judged from the ground is R / gamma.

For distributive law models which follow, it’s best to have both x and vt as non-zero.  Length contraction cranks out the same R / gamma effect at any location, zero or non-zero.

#### Contraction distributive law:

gamma • x – gamma • vt = gamma • ( x – vt )

An equality.   Let’s say it can be verbalized as Side-A equals Side-B:

Side-A = Side-B

Which version does length contraction need in terms of phenomenon?  Side-A or Side-B?  It is a matter of no consequence upon which everything depends.

Algebra by itself leaves the question unanswered. But professor Long Hair can see clearly that length contraction needs Side-B.

#### v-a-f distributive law

The v-a-f: V = ( v + w ) / ( 1 + vw/c² )

Denominator factor ( 1 + vw/c² ) is the reduction factor, abbreviated as rf.   Like gamma, it is greater than 1.

V = ( v + w ) / rf

Distributive law:

v / rf + w / rf = ( v + w ) / rf

Just as with length contraction, we need to know which side is required, Side-A or Side-B.

Side-B:  ( v + w ) / rf

You transform neither input speed-v nor -w.  You transform Galilean output-Vg.  You do that by ramping up the terminal ground clock which reports speed-Vg as ground speed.  Terminal time speeded up by factor-rf slows down speed-Vg by 1 / rf.  Roughly, that’s college algebra with the temporal transform in the denominator writing ground time as t = rf • t′.

Denominator:  t = ( 1 + vw/c² ) • t′

Side-A:  v / rf + w / rf

Section §5 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 each of speed-v and speed-w with 1 / rf and then add.  That’s Side-A of the distributive law.

So which is it, Professor ?

Side-A or Side-B ?

Albert Einstein or Halliday & Resnick ?

#### A two lane highway

Different example numbers.  Car-v of speed v = 40 miles per hour is on lane-left and car-w of speed w = 30 mph is on lane-right. Beginning at mile-zero and time-zero, the cars go down the pike.  Add speeds:

Va = v + w

70 mph = 40 mph + 30 mph

Notation Va = 70 mph is V-algebraic, a good number, but not the phenomenon of any car in motion, not the phenomenon of Vg ( Galilean ) from before.

Put both cars on one lane, lane-left.  Again, both cars leave mile-zero and at one moment, time-zero.  However, car-w is now speed with respect to car-v, not with respect to the starting line.  We have Galilean addition of speeds from before, and car-w on lane-left is then blacktop speed Vg, Vg = v + w, just as car-v is blacktop speed.

Vg = v + w

70 mph = 40 mph + 30 mph

Output speed 70 mph is something that’s really there although it is not a new car.

Revisit two lanes and bring in a third car, car-u of speed-Ug on lane-right.   All cars, v, w and u, leave mile-zero at time-zero.  We give the right-lane car-u speed-Ug equal to speed-Vg, 70 mph in this case.  Speed-Ug is an ordinary blacktop or Galilean speed just like speed-Vg of car-w on lane-left.  Cars w and u go down the pike double file, side-by-side, at speed 70 mph.

Now we redo our left lane Galilean model of cars v and w as an SRT model, where addition of speeds v and w yields car-w of diminished speed-Vsrt ( v-a-f output ), slower than simple Galilean speed-Vg.

Slow wheels:  Let speed-Vsrt of car-w be an obvious highway phenomenon: car-w falls behind car-u. Old fashioned car-u is out in front.

A fast clock: Car-w and car-u go side-by-side, double file, with simultaneous arrival at the terminal.  But we have two clocks there, two terminal clocks reading different arrival times.  Clock-w, faster than clock-u, reads more time than clock-u. More travel time from point-A to point-B slows you down. 

Which is it?   A wheel model or a clock model?

####  1920 Chapter XIII: the Experiment of Fizeau

Chapter XIII has a photon of speed-c struggling with Fizeau fluids.  Yes, the v-a-f solves a speed-of-light problem, but the v-a-f is in essence ( Section §5 ) “a composition of velocities which are less than c.”

On Route 66 a ’57 Chevy races out in front of a squad car.  Car-w might be Galilean speed sum-Vg, or alternatively, v-a-f speed sum-Vsrt.  With both cars v and w at much less than the speed-of-light, the difference between Galilean speed and SRT speed is vanishingly small, but is still there and must be explained.