A spaceship passes you at the speed of 0.90 c . You measure its length to be 50.0 m . What is its length when at rest?

According to the effects of special relativity, an object moving at a significant fraction of the speed of light will appear contracted in the direction of motion. This contraction can be calculated using the formula for length contraction: \( L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \), where \( L_0 \) is the proper length (length at rest), \( L \) is the length measured while in motion, and \( v \) is the velocity of the object. Taking your example, where \( L = 50.0 \) m and \( v = 0.90c \): 1. Calculate \( v^2/c^2 = (0.90)^2 = 0.81 \). 2. Thus, \( 1 - v^2/c^2 = 1 - 0.81 = 0.19 \). 3. Now, calculate \( L_0 \): \( L_0 = \frac{L}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{50.0 m}{\sqrt{0.19}} \approx 50.0 m / 0.4359 \approx 114.75 m \). So, the length of the spaceship when at rest is approximately **114.75 meters**. Keep in mind that these relativistic effects become significant as the speed approaches the speed of light!