10.A spacecrat travels along a space station/platform at 0.65 c relative to the plattom. An astronaut on the spacecratt determines the plattorm to be \( 3 . \sigma \times 10^{2} \) m long. What is the length of the plattorm as measured by an observer on the plattorm?

To find the length of the platform as measured by an observer on the platform, we can use the principles of length contraction in special relativity. The length measured by the observer on the platform (proper length) is given by the formula: \[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \] Where: - \(L_0\) is the length of the platform as measured in the spacecraft, which is \(3. \sigma \times 10^{2}\) m. - \(v\) is the velocity of the spacecraft relative to the platform, which is \(0.65c\). Now, let's plug in the values: 1. Calculate \(v^2/c^2\): \[ \left(0.65\right)^2 = 0.4225 \] 2. Substitute it into the contraction formula: \[ L = 3. \sigma \times 10^{2} \cdot \sqrt{1 - 0.4225} \] 3. Calculate the square root: \[ \sqrt{1 - 0.4225} = \sqrt{0.5775} \approx 0.76 \] 4. Finally, calculate \(L\): \[ L \approx 3. \sigma \times 10^{2} \cdot 0.76 \approx 2.28 \times 10^{2} \, \text{m} \] Thus, the length of the platform as measured by an observer on the platform is approximately \(2.28 \times 10^{2}\) m.