Relativistic Addition of Velocities
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 .
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.