According to the theory of special relativity, the principle of adding velocities does not hold true when dealing with speeds close to the speed of light. In other words, you cannot simply add the velocities of two objects moving in opposite directions to determine their combined speed.
Instead, special relativity introduces a formula known as the relativistic velocity addition formula to calculate the combined velocity. If two objects are moving in opposite directions with velocities v1 and v2 relative to an observer, the formula for the combined velocity (v) is given by:
v = (v1 + v2) / (1 + (v1 * v2) / c^2)
where c is the speed of light.
Using this formula, let's consider the scenario where two objects are moving in opposite directions, each with a velocity of half the speed of light (0.5c):
v1 = 0.5c v2 = -0.5c (since they are moving in opposite directions)
Plugging these values into the formula, we get:
v = (0.5c + (-0.5c)) / (1 + (0.5c * -0.5c) / c^2) = 0 / (1 - 0.25) = 0 / 0.75 = 0
The result indicates that the combined velocity of the two objects is zero. This means that, according to special relativity, the two objects would not be observed to be moving relative to each other. This is a consequence of the relativistic effects that occur at high speeds and is often referred to as the relativistic velocity addition "paradox."
It's important to note that as the velocities approach the speed of light, the relativistic effects become more pronounced, and the usual intuitions about adding velocities no longer hold true. The theory of special relativity provides a consistent framework for understanding the behavior of objects moving at high speeds, but it requires the use of relativistic formulas and concepts to accurately describe their motion.