gen-define-length-contraction-and-give-one-short-worked-example-dffa

Define length contraction and give one short worked example.

# Length Contraction **Definition:** Length contraction is the relativistic effect where an object's length appears shorter when measured in a reference frame in which it is moving, compared to its length measured in its rest frame. The contracted length is given by: $$L = L_0\sqrt{1 - \frac{v^2}{c^2}} = \frac{L_0}{\gamma}$$ where: - $L$ = contracted length (measured in the frame where object is moving) - $L_0$ = proper length (rest length) - $v$ = relative velocity - $c$ = speed of light - $\gamma$ = Lorentz factor ## Worked Example **Problem:** A spacecraft has a proper length of 100 m. It flies past Earth at 0.8c. What length do observers on Earth measure? **Solution:** First, calculate the Lorentz factor: $$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1 - 0.8^2}} = \frac{1}{\sqrt{0.36}} = \frac{1}{0.6} = 1.67$$ Then find the contracted length: $$L = \frac{L_0}{\gamma} = \frac{100 \text{ m}}{1.67} = 60 \text{ m}$$ **Answer:** Earth observers measure the spacecraft to be only 60 m long, contracted by 40% in the direction of motion.