In mechanics, one often uses the model of a perfectly rigid body to model and determine the motion of physical objects (see, for example, Chapter 10 on rotation). Explain how this model contradicts Einstein's special theory of relativity.

Albert Einstein's groundbreaking paper on the special theory of relativity, introduced in 1905, forever changed our understanding of space, time, and motion. At its core, this theory put forth two postulates which seem deceptively simple: the laws of physics are the same for all non-accelerating observers, and the speed of light in a vacuum is the same for all observers, no matter their relative speed.

What unfolds from these two postulates is a framework that blends space and time into a single continuum known as space-time. Events that occur are not absolute but are relative to the motion of observers. This leads to mind-bending predictions, such as time dilation and length contraction, which have been confirmed through numerous experiments. For instance, time dilation implies that time passes at a slower rate for objects moving at high speeds, while length contraction implies that objects appear shorter in the direction of motion when moving at high speeds, as perceived by a stationary observer.

In contrast to Relativity, Classical Mechanics often involves the concept of a perfectly rigid body. This model is a theoretical construct where the distances between any parts of the object do not change. In essence, it's an object that doesn't compress, stretch, or deform, no matter the amount of stress or force exerted on it.

This idealized model simplifies calculations and is a useful approximation for many engineering problems where speeds are not relativistic, which means they are much slower than the speed of light. For all intents and purposes, under ordinary conditions, an iron beam or a wooden plank could be thought of as 'almost' perfectly rigid. However, this notion becomes problematic when speeds approach the realm of relativistic effects.

When we look at Classical Mechanics, which encompasses the physics developed by Sir Isaac Newton, it is based on principles and laws that describe the motion of objects at everyday speeds and sizes. The cornerstone of classical mechanics, Newton's laws, accurately predicts how objects move under the influence of forces. For many practical applications in our daily life, from constructing buildings to playing sports, classical mechanics serves us extremely well.

However, it becomes evident that Classical Mechanics does not consider the effects of traveling at high velocities, where it's integral to account for relativistic phenomena. It's at these high velocities, nearing the speed of light, that classical presumptions, such as time being a constant and space being absolute, begin to crumble, revealing a realm where Einstein's theory takes precedence.

One of the intriguing outcomes of Einstein's theory is the concept of length contraction. This occurs because, according to relativity, as an object moves faster, space itself seems to contract along the direction of motion from the perspective of a stationary observer. The formula for length contraction is given by \( L = L_0 \sqrt{1-\frac{v^2}{c^2}} \) where \( L \) is the contracted length, \( L_0 \) is the rest length (the length of the object at rest), \( v \) is the velocity of the object, and \( c \) is the speed of light.

Closely related to length contraction is time dilation, a phenomenon where time passes at a slower rate for an object in motion relative to a stationary observer. This effect becomes appreciable only at speeds comparable to the speed of light. Mathematically, the time dilation effect is described by \( t = \frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}} \) where \( t \) is the time interval measured by the stationary observer, \( t_0 \) is the proper time interval (the time measured by a clock moving with the object), \( v \) is the relative velocity, and \( c \) is the speed of light.

This phenomenon is not just a theoretical prediction; it has practical implications and has been experimentally verified, for example, in the operation of the Global Positioning System (GPS), where engineers must account for time dilation effects to ensure accuracy.