If two objects move in opposite directions, both traveling at half the speed of light relative to a stationary observer, the combined speed of the objects will not exceed the speed of light. According to the theory of special relativity, the velocity addition formula is used to calculate the relative velocity between two objects in different inertial frames of reference.
The velocity addition formula states that if two objects are moving with velocities u and v relative to a stationary observer, then their relative velocity, w, is given by:
w = (u + v) / (1 + (uv / c^2))
In this scenario, if u = v = c/2 (where c is the speed of light), the formula becomes:
w = (c/2 + c/2) / (1 + (c/2 * c/2) / c^2) = (c) / (1 + (1/4)) = (c) / (5/4) = 4c/5
So, the relative velocity between the two objects moving in opposite directions at half the speed of light would be 4/5 times the speed of light, which is less than the speed of light.
This result is consistent with the theory of special relativity, which prohibits objects with mass from attaining or exceeding the speed of light. The principle of relativity states that the laws of physics should be the same in all inertial reference frames, and the theory of special relativity provides the mathematical framework to describe the behavior of objects moving at relativistic speeds.