Explain Lorentz-Fitzgerald contraction with an example.

Lorentz-Fitzgerald Contraction (Length Contraction)
The Lorentz-Fitzgerald contraction states that the length of an object moving at a relativistic speed (close to the speed of light) appears shorter when measured by an observer at rest. This phenomenon is a direct consequence of Einstein's Special Theory of Relativity.
Mathematical Expression
If an object has a proper length (length measured in its own rest frame) L₀ and moves with velocity v relative to an observer, its observed length L is given by:
L=L01−v2c2L = L_0 \sqrt{1 - \frac{v^2}{c^2}}
where:
• L0L_0 = Proper length (length in rest frame)
• LL = Contracted length (observed by an external observer)
• vv = Velocity of the object
• cc = Speed of light in vacuum ( 3×1083 \times 10^8 m/s)
As the speed v approaches c, the factor 1−v2/c2\sqrt{1 - v^2/c^2} decreases, making LL smaller than L0L_0, showing that the object appears contracted.
Example of Length Contraction
Moving Spacecraft
Consider a spaceship traveling at 0.8c (80% of the speed of light). Suppose its proper length (when at rest) is 100 m. The length observed from Earth would be:
L=100×1−(0.8c)2c2L = 100 \times \sqrt{1 - \frac{(0.8c)^2}{c^2}} L=100×1−0.64L = 100 \times \sqrt{1 - 0.64} L=100×0.36L = 100 \times \sqrt{0.36} L=100×0.6L = 100 \times 0.6 L=60 mL = 60 \text{ m}
Thus, an observer on Earth would measure the spaceship’s length as 60 m instead of 100 m due to length contraction.
Key Observations
1. Length contraction only occurs in the direction of motion. The dimensions perpendicular to motion remain unchanged.
2. The contraction is negligible at everyday speeds but becomes significant as v approaches c.
3. It is a purely relativistic effect and has been experimentally confirmed using high-speed particle accelerators.
This phenomenon plays a crucial role in relativity and helps explain why fast-moving particles behave differently from everyday objects.