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Special relativity is a profound theory proposed by Albert Einstein that revolutionized our understanding of space and time. Unlike Newtonian mechanics, where time and space are considered absolute and unchanged regardless of the observer, special relativity introduces a relative framework, where the laws of physics remain consistent across different inertial frames of reference.

A central idea in special relativity is that the speed of light, denoted by \( c \) and approximately equal to \( 3 \times 10^8 \) m/s, is constant for all observers, no matter their motion. This constancy leads to fascinating effects such as time dilation, where time slows down for fast-moving objects, and length contraction, where objects appear shorter in the direction of motion when moving close to the speed of light.

Length contraction plays a pivotal role in our exercise scenario, where a spaceship's perceived length changes with respect to an observer depending on the spaceship's velocity. As the spaceship travels at significant speeds, its length contracts along its direction of travel, embodying the principle of special relativity.

Einstein's theory of special relativity, published in 1905, fundamentally changes our perception of time and space. It rests on two main postulates:

• The laws of physics are identical in all inertial frames of reference, meaning there is no preferred viewpoint in the universe for observing the laws of physics.
• The speed of light in a vacuum is constant, uninfluenced by the motion of the light source or the observer.

These postulates bring forth intriguing consequences such as the relativity of simultaneity, the equivalence of mass and energy as stated in the famous equation \( E = mc^2 \), and the observed phenomena of length contraction in the exercise. When observing the spaceship, an observer sees a contracted length due to its high speed, translating Einstein's abstract theory into a tangible effect.

The length contraction can be quantified using the formula \( L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \), where \( L_0 \) is the proper length and \( L \) is the contracted length. This equation shows how Einstein's theoretical ideas manifest in practical observations, such as determining the speed of the spaceship relative to an observer.

Understanding velocity calculation in the context of special relativity adds an intriguing layer of complexity compared to classical physics. In our exercise, the key is finding the velocity at which the spaceship travels relative to the observer to result in the observed contraction of its length.

Starting with the length contraction formula \( L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \), we rearrange the equation to solve for \( v \), the spaceship's velocity. By isolating \( v \), we initially have the expression \( \sqrt{1 - \frac{v^2}{c^2}} = \frac{L}{L_0} \). Rearranging gives \( v^2 = \left( \frac{L}{L_0} \right)^2 c^2 \), and finally, we find \( v \) by taking the square root:

\[ v = c \cdot \sqrt{\frac{9}{25}} \l = 3 \times 10^8 \times \frac{3}{5} = 1.8 \times 10^8 \text{ m/s} \]

This velocity represents a significant fraction of the speed of light, reflecting the relativity-driven effects on the spaceship's length observed from the reference frame of a stationary observer. This exercise demonstrates not only how velocity affects the perceived length of objects in motion but also how critical understanding special relativity is to solving such problems in modern physics.