Addition of velocities in special relativity

In special relativity, when one observer sees another observer moving with a certain velocity and that second observer sees an object moving with its own velocity, the formula for the resulting velocity (as seen by the first observer) ensures that all relative motions are accounted for correctly without exceeding the speed of light.
If two velocities $u$ and $v$ are being added, where $u$ is the velocity of an object in one frame and $v$ is the velocity of the moving frame relative to a third frame, the resultant velocity $w$ is given by:

w = \frac{u + v}{1 + \frac{u v}{c^{2}}}

Here:

$u$ is the velocity of the object relative to one frame.
$v$ is the velocity of the second frame relative to a third frame.
$w$ is the velocity of the object relative to the third frame.
$c$ is the speed of light in a vacuum, approximately $3 \times 10^{8} m/s$ .

This formula ensures that the resultant velocity $w$ does not exceed the speed of light $c$ .

Sometimes it is written as

β_{w} = \frac{β_{u} + β_{v}}{1 + β_{u} β_{v}} $ $, w h e r e $ β_{x} = x / c $, i . e ., t h e v e l o c i t y i n n a t u r a l u n i t s .