The Velocity Addition Formula

Velocity Addition Formula

Updated: February 5, 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 )

#### 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.

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 & 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.

Giancoli, Physics, Sixth Edition, Appendix E.  It doesn’t have the quotient shown here.  But the temporal transform with plus-sign local time rather than minus-sign local time is shown, page A-25.  A brief algebraic derivation of that version is given.  It is bizarre.

A. P. French, Special Relativity, Chapter 5 Mechanics, page 126.  Equations 5-1 are exactly the additive transforms given here.

#### Symbols

The v-a-f, Section §5:

V = ( v + w ) / ( 1 + vw/c² )

A simple Galilean addition of speeds is shown in 1920-Chapter VI.  No reduction factor.

Galileo: W = v + w

Car-v goes down the highway at speed-v.  Car-w begins when and where car-v begins, x = 0 and t = 0. Though motion-w of car-w is motion relative to car-v, not relative to the pavement starting line, car-w may be seen as motion relative to the starting line, and is then denoted with left-hand-side symbol-W, not symbol-V.

Lets use symbol-W rather than symbol-V of Section §5. The car representing a speed sum is car-w not car-v.

In any case, the equation of three speed variables, v, w and V is only two cars, not three cars.

When Galilean-W is revised to relativistic v-a-f output we’ll use a different symbol: speed-Wsrt, subscript srt, short for “spezielle Relativitätstheorie.”

Wsrt = ( v + w ) / ( 1 + vw/c² )

Reduction factor ( 1 + vw/c² ) being a dimensionless number greater than 1, we have relativistic-Wsrt less than Galilean-W

Wsrt < W

Section §5 calls speed sum ( v + w ) a “composition of velocities.” 

#### Speed-v, Section §4

Let’s do just one car, car-v of speed-v.  Further, let’s use the version of speed-v which Section §4 uses to do time dilation.  Car-v crosses highway milepost [ x = 0 ] at time [ t = 0 ].  Car-v proceeding at speed-v passes a succession of stationary mileposts, post-1, post-2, post-3 etc, ending up at stationary post-x, a location on the highway.  Each stationary post has an attached stationary clock reading time-t, and then the speed of car-v at any milepost is the quotient of post location divided by post time. 

Milepost speed:  v = x / t.

Footnote: Section §4 has x = vt.  It is not an equation of motion, not point-x of speed-v.  It is a selection, at time-t, of a stationary milepost.

Whenever car-v passes milepost-x, post-x looks in the window of the passing car and sees the dashboard clock.  The moving clock is seen keeping time-t′ prime.

Dashboard clock:  t′ = t / gamma

Face time-t′ on the moving dashboard clock is less than face time-t on the stationary clock at post-x, and quote “time marked by the moving clock ( viewed in the stationary system ) is slow.”

The dashboard clock of car-v can be viewed both by car driver and ( momentarily ) by a stationary milepost.

####  One-car questions

Can car-v have a dashboard odometer?  If so, how does that distance number relate to milepost number?

Odometer distance can’t be output-x′ of the spatial-LT which primed variable in a roundabout manner is the size of the car, not its location.

If somehow odometer distance can be contrived, can car-v have a speedometer which is odometer distance divided by dashboard clock?

Given a dashboard speedometer, how does that speed relate to the milepost quotient?  Can the dashboard speedometer be viewed by a stationary milepost?
 
What is one-car Galilean speed-v?  A dashboard quotient or a milepost quotient?  Section §4 simply doesn’t deal with it. 

#### Two cars

W = v + w

Change to subtraction:

w = W – v

Car-v crosses the starting line at speed-v.  So also, then and there, car-W crosses the starting line at speed-W greater than speed-v.  Subtractive output-w is both an algebraic value and an observable fact, though we don’t have any car at speed-w.  It is not necessary to figure out how a milepost quotient or a dashboard quotient might render speed-w.

Go back to Galilean addition:

W = v + w

Symbols v and W as car speeds are reasonably milepost quotients.  In Section §4 speed-v is explicitly a milepost quotient.  That works here in the addition model, and output speed-W can run off the same mileposts as speed-v.

Input speed-w must also be a car speed.  It can’t simply be an algebraic value.  What instrument renders car speed-w?   It can’t be a quotient on a milepost staked out with respect to highway post zero.  It can’t be a dashboard speedometer on that same basic highway.  It can be a speedometer on a new highway fixed to front bumper of car-v.

After a fashion, Section §4 has a highway attached to the front of car-v.  It is the Michelson-Morley device. But Section §4 doesn’t have car-w on the new highway and therefore has no need to figure out how one-way speed-w might be rendered.

#### Two car relativity

While speed-w is still undefined, the v-a-f asks another obvious question:

The left-hand-sides of Halliday & Resnick LT’s have unprimed-x and unprimed-t. Since output-x is supposedly a sum of distances, quotient x / t is a speed.

Wsrt = x / t

Is speed-Wsrt a dashboard quotient or a milepost quotient?  As stated, in Galilean mode, speed-W is a milepost quotient just like speed-v. As stated, speed-Wsrt is less than speed-W. 

Wsrt < W

Car-Wsrt cannot arrive at a milepost, side-by-side with Galilean car-W, and have its milepost quotient less than the quotient for car-W.

However, car-Wsrt can arrive double-file with car-W and have a dashboard speedometer reading less than the milepost quotient for car-W.   That will be the case if it’s dashboard clock reads more time than proper time ( whereas Section §4 has a dashboard clock reading less time than proper time. )

What’s going on here?