(a) At what relative velocity is \(\gamma=1.50 ?\) (b) At what relative velocity is \(\gamma=100 ?\)

The concept of special relativity, introduced by Albert Einstein in 1905, revolutionized the way we understand space, time, and motion. At its core, this theory postulates that the laws of physics are the same for all non-accelerating observers, and consequently, the speed of light in a vacuum is the same for all observers, regardless of their relative motion. This leads to fascinating and counterintuitive consequences, such as time dilation and length contraction, which can be quantified using the Lorentz factor, \(\gamma\).

Special relativity becomes significant at speeds approaching the speed of light, \(c\), and it's crucial for correcting the predictions of classical mechanics, which fail at such extreme velocities. The Lorentz factor, which appears in many relativistic equations, measures how much time, length, and relativistic mass change for an object moving relative to an observer's frame of reference. It's important to understand that these effects are not felt incrementally; instead, they become drastic as one nears light speed.

Relative velocity is a measure of how quickly one object is moving in relation to another object. It's a vector quantity, meaning it has both magnitude and direction. In classical mechanics, relative velocity is simply the difference in velocity between two bodies. However, as we move into the realm of special relativity, the concept of relative velocity becomes more nuanced. To reconcile the seemingly paradoxical idea that the speed of light is constant in all reference frames, relative velocities can no longer be added or subtracted linearly when they approach the speed of light.

In the context of special relativity, calculating relative velocity involves using the Lorentz transformation. For our exercise example, the Lorentz factor, which accounts for effects like time dilation and length contraction, was used to determine the required velocity for a given \(\gamma\) value. This provides a more accurate depiction of the speed at which two observers would see each other moving when one or both are traveling at a significant fraction of the speed of light.

The speed of light, commonly denoted as \(c\), is a universal physical constant that is fundamental to the structure of both space and time. In the vacuum of space, it is exactly 299,792,458 meters per second. It is the ultimate speed limit of the universe, meaning that no physical object or information can travel faster than this speed. This concept is a cornerstone of the theory of special relativity and influences how we understand the behavior of objects as they approach light-speed.

Why is the speed of light so important? It not only dictates how we observe astronomical objects at great distances but also plays a role in technologies like GPS and the functioning of particle accelerators. In the textbook example exercise, we use the known value of the speed of light to calculate the velocities corresponding to different Lorentz factors. This illustrates how theoretically, as an object's speed increases towards the speed of light, its relative velocity changes in ways that are quite distinct from our daily experiences governed by Newtonian physics.