Show that for any relative velocity \(v\) between two observers, a beam of light projected by one directly away from the other will move away at the speed of light (provided that \(v\) is less than \(c,\) of course).

At the heart of modern physics lies the theory of relativity, developed by Albert Einstein. This groundbreaking theory fundamentally altered our understanding of space, time, and gravity. It's composed of two parts: special relativity and general relativity.

Special relativity, which is pertinent to the discussion of relativistic velocity addition, addresses the physics of objects moving at constant speeds in straight lines, known as inertial frames of reference. One of the most astonishing revelations of special relativity is that the laws of physics are the same for all observers in uniform motion relative to one another, regardless of their individual velocities.

This has several counterintuitive implications, such as time dilation and length contraction. But most relevant here is that it introduced the concept that the speed of light in a vacuum is constant and is the maximum speed at which all energy, matter, and information in the universe can travel. It also introduced the idea that observers moving at different velocities may perceive time and distances differently, but all will measure the speed of light to be the same.

The speed of light in a vacuum, often denoted by the letter 'c', is a fundamental constant of nature valued at approximately 299,792 kilometers per second (or about 186,282 miles per second). It represents the fastest speed at which information or matter can travel through space.

According to Einstein's special relativity, the speed of light is the same for all observers, no matter the motion of the light source or observer. This constancy of 'c' leads to many interesting phenomena such as the need for relativistic velocity addition, which students often encounter in physics.

It's also worth noting that nothing can travel faster than the speed of light, and any attempt to add to the velocity of light must adhere to relativistic addition rules. These rules ensure that even when an observer moves toward or away from a light source, they still measure its speed as 'c'. This cornerstone principle maintains the structure of causality in the physical universe and ensures that the speed of light acts as a cosmic speed limit.

An inertial reference frame is a frame of reference wherein an object is either at rest or moving at a constant velocity, implying that it is not accelerating (no net force is acting on it). In the absence of external forces, an object in an inertial frame either remains stationary or continues to move at a uniform speed in a straight line, as stated by Newton's first law of motion.

Special relativity is particularly concerned with how events are measured in different inertial frames that are in relative motion to one another. When considering the speed of light and its invariance in different inertial frames, we encounter scenarios where the classical (Galilean) concept of velocity addition does not suffice. Instead, we use the relativistic velocity addition formula, which accounts for the effects of moving close to the speed of light.

For instance, if an observer in one inertial frame measures a beam of light moving at the speed of light, another observer moving relative to the first will also measure the beam's speed as 'c', consistent with our exercise. This invariant measurement across inertial frames underscores the non-intuitive aspects of relative motion at velocities approaching the speed of light.