(a) How fast would an athlete need to be running for a \(100-\mathrm{m}\) race to look 100 yd long? (b) Is the answer consistent with the fact that relativistic effects are difficult to observe in ordinary circumstances? Explain.

Relativity in physics is a fundamental theory introduced by Albert Einstein that revolutionized our understanding of space, time, and the interplay between them. It consists of two theories: Special Relativity and General Relativity. Special Relativity, developed in 1905, delves into frames of reference moving at a constant speed relative to one another and introduces groundbreaking concepts like time dilation and length contraction. General Relativity, established in 1915, expands the theory by including gravity and acceleration. At its core, relativity challenges the notion of absolute time and space, suggesting they are relative to the motion of the observer.

Relativity is known for its non-intuitive predictions, such as clocks moving at different speeds in different frames of reference, or the way masses curve spacetime, which has been verified by experiments such as the bending of light by gravity. Despite its complexity, relativity is essential in technologies like GPS, where accurate time measurements are crucial and must account for relativistic effects.

The speed of light, denoted by the symbol '(c)', is a constant that plays a vital role in physics, particularly in the theory of relativity. Measured at approximately 299,792,458 meters per second (in a vacuum), it's a universal speed limit according to Special Relativity. No object with mass can reach or exceed this velocity, as doing so would require an infinite amount of energy.

What makes the speed of light so intriguing is not just its magnitude but also its constancy. Regardless of the observer's motion or the source of the light, its speed remains the same. This constancy underpins the Lorentz transformations and leads to the fascinating relativistic phenomena that question our everyday perception of reality. Light speed also acts as a bridge between matter and energy, famously represented by Einstein's equation E=mc^2, showing that mass can be converted into energy.

The Lorentz Transformation is a set of mathematical equations fundamental to Special Relativity. Proposed by Hendrik Lorentz, they describe how, according to an observer, the measurements of space and time change for another observer in uniform motion relative to the first. These transformations allow us to calculate the time dilation and length contraction experienced by objects as they approach the speed of light.

For example, if an object is moving at a high velocity, it will appear to contract along the direction of motion from the view of a stationary observer. This effect, known as length contraction, means that the length of the object is measured to be shorter than its length at rest – which is precisely what the original exercise about the 100-meter race illustrates.By applying Lorentz transformations, we can understand how speeds close to the speed of light transform perceptions of space and time. This has practical applications in particle physics and cosmology, where objects commonly move at relativistic speeds.