# Relativistic Addition of Velocities

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This Demonstration considers the composition of velocities in accordance with the special theory of relativity. Consider a system moving with velocity represented by the red arrow, with magnitude and direction , with respect to a stationary frame of reference. The red disk recapitulates this magnitude, which has an upper limit extending to the red circle, corresponding to the speed of light . The blue arrow represents a second velocity, which has a magnitude and direction , with respect to the moving frame of reference. The velocity with respect to the original stationary frame is then represented by . A compact formulation gives the components of parallel and perpendicular to : The gray arrow represents the vector .

Contributed by: S. M. Blinder (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: for , the Galilean result is a good approximation

Snapshot 2: if or , then

Snapshot 3: the collinear case reduces to Einstein's well-known formula

Snapshots 4, 5: velocity addition is not commutative;

Reference: J. D. Jackson, *Classical Electrodynamics*, 3rd ed., New York: John Wiley & Sons, 1998 p. 531.

## Permanent Citation