Einstein's Formula for Adding Velocities

According to the special theory of relativity, the composition of two collinear velocities and is given by a well-known formula derived by Einstein: , where is the speed of light, 2.9979× m/sec. For , this reduces to the simple Galilean formula . The result can be derived by successive application of collinear Lorentz boosts, but it can be shown more intuitively by an argument outlined in the details. In the Demonstration the speeds are scaled as (also known as ) to keep the two rocket ships within the graphic. Einstein's formula is analogous to the law of addition for hyperbolic tangents, , where corresponds to the rapidity, defined by .


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


Suppose a baseball team is traveling on a train moving at 60 mph. The star fastball pitcher needs to tune up his arm for the next day’s game. Fortunately, one of the railroad cars is free, and its full length is available. If his 90 mph pitches are in the same direction the train is moving, the ball will actually be moving at 150 mph relative to the ground. The law of addition of velocities in the same direction is relatively straightforward, . But according to Einstein’s special theory of relativity, this is only approximately true and requires that and be small fractions of the speed of light, ≈ 3 × m/sec (or 186,000 miles/sec). Expressed mathematically, we can write if According to special relativity, the speed of light, when viewed from any frame of reference, has the same constant value . Thus, if an atom moving at velocity emits a light photon at velocity , the photon will still be observed to move at velocity , not .
Our problem is to deduce the functional form of consistent with these facts. It is convenient to build in the known asymptotic behavior for by defining . When , we evidently have , so , and likewise . If both and equal , . A few moments' reflection should convince you that a function consistent with these properties is , which gives Einstein's velocity addition law .
Reference:S. M. Blinder, Guide to Essential Math, Amsterdam: Elsevier, 2008 pp. 5–6.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+