Addition of Velocities

Let a mass point move at a velocity in frame . What is the velocity of this point in frame moving at a velocity relative to ? (Let .)

Relying on the definition of velocity we write:

and .

From Lorenz’s transformation we get:

and .

We substitute the obtained expressions for and in the right-hand side of the formula for ,


Fig. 2.5

This is the Law of addition of velocities in relativistic mechanics.

We can easily notice that if and the last formula becomes , which is the Law of addition of velocities in non-relativistic mechanics. If a body and frame move towards each Addition of Velocities other (fig. 2.5), the negative sign in the last formula changes to the positive:

Now we shall try to put an end to doubt formulated at the beginning of this chapter concerning the speed of light.

Let and . It means that the object concerned is the light moving at speed in frame . We need to find the speed of light in frame which in its turn moves at the speed of light relative to the frame . To find the speed of light in frame , substitute for and . We get:


the result corresponds to Einstein’s postulate on the equality Addition of Velocities of the speed of light in vacuum in any inertial frame of reference. We can say that the relativistic Law of addition of velocities confirms the existence of limit velocity - the velocity of light in vacuum.

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