10.A spacecraft travels along a space station/platform at 0.65 c relative to the measue 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 can use the concept of length contraction from Einstein's theory of special relativity. The formula for length contraction is: \[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \] where \( L_0 \) is the proper length (the length of the object in its rest frame), \( v \) is the velocity of the object relative to the observer, and \( c \) is the speed of light. 1. In this case, \( L_0 = 3.0 \times 10^{2} \, m \) (the length of the platform as measured by the astronaut). 2. The velocity \( v = 0.65c \). 3. To find \( L \): \[ L = 3.0 \times 10^{2} \sqrt{1 - (0.65)^2} \] Calculating \( (0.65)^2 \): \[ (0.65)^2 = 0.4225 \] Now, substituting into the equation: \[ L = 3.0 \times 10^{2} \sqrt{1 - 0.4225} \] \[ L = 3.0 \times 10^{2} \sqrt{0.5775} \] \[ L \approx 3.0 \times 10^{2} \times 0.76 \approx 228.0 \, m \] Therefore, the length of the platform as measured by an observer on the platform is approximately **228.0 meters**.