10.A spacecraft travels along a space station/platform at 0.65 c relative to the platform. An astronaut on the spacecraft determines the platform to be \( 3.0 \times 10^{2} \) m long. What is the length of the platform as measured by an observer on the platform?

To find the length of the platform as measured by an observer on the platform, we need to use the concept of length contraction from the theory of Special Relativity. According to this theory, the length of an object moving relative to an observer will appear shorter than its proper length (the length measured in its rest frame). The formula for length contraction is: \[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \] Where: - \( L_0 \) is the proper length (300 m, as measured by the astronaut), - \( v \) is the relative velocity (0.65c), - \( c \) is the speed of light. First, we calculate \( \frac{v^2}{c^2} \): \[ v^2 = (0.65c)^2 = 0.4225c^2 \] Now substituting within the square root: \[ 1 - \frac{v^2}{c^2} = 1 - 0.4225 = 0.5775 \] Now we calculate \( \sqrt{0.5775} \): \[ \sqrt{0.5775} \approx 0.76 \] Now use this to find \( L \): \[ L = 300 \text{ m} \cdot 0.76 \approx 228 \text{ m} \] Thus, the length of the platform as measured by an observer on the platform is approximately **228 meters**.