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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
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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 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 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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