7. A spaceship traveling past Earth with a speed of 0.87 c , relative to Earth, is measured to be 48.0 m long by observers on Earth. What is the proper length of the spaceship?

To determine the **proper length** of the spaceship, we'll use the concept of **length contraction** from Einstein's theory of Special Relativity. ### **Length Contraction Formula** The length observed by a stationary observer (**contracted length**, \( L \)) is related to the proper length (\( L_0 \)) by the formula: \[ L = L_0 \times \sqrt{1 - \left(\frac{v}{c}\right)^2} \] Where: - \( v \) = relative velocity of the spaceship = \( 0.87c \) - \( c \) = speed of light ### **Solving for Proper Length (\( L_0 \))** Rearranging the formula to solve for \( L_0 \): \[ L_0 = \frac{L}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \] ### **Plugging in the Values** 1. **Calculate \( \frac{v}{c} \)**: \[ \frac{v}{c} = 0.87 \] 2. **Calculate \( \left(\frac{v}{c}\right)^2 \)**: \[ \left(0.87\right)^2 = 0.7569 \] 3. **Calculate \( 1 - \left(\frac{v}{c}\right)^2 \)**: \[ 1 - 0.7569 = 0.2431 \] 4. **Take the square root**: \[ \sqrt{0.2431} \approx 0.493 \] 5. **Finally, calculate \( L_0 \)**: \[ L_0 = \frac{48.0\, \text{m}}{0.493} \approx 97.3\, \text{m} \] ### **Conclusion** The **proper length** of the spaceship (its length in its own rest frame) is approximately **97 meters**. **Answer:** The spaceship’s proper length is approximately 97  meters.