The spaceship Enterprise 1 is moving directly away from earth at a velocity that an earth- based observer measures to be 0.65 c . A sister ship, Enterprise 2 , is ahead of Enterprise 1 and is also moving directly away from earth along the same line. The velocity of Enterprise 2 relative to Enterprise 1 is 0.31 c . What is the velocity of Enterprise 2 , as measured by the earth-based observer?

To find the velocity of Enterprise 2 as measured by the earth-based observer, we can use the concept of relative velocity in special relativity. Given: - Velocity of Enterprise 1 as measured by the earth-based observer = 0.65 c - Velocity of Enterprise 2 relative to Enterprise 1 = 0.31 c Let's denote the velocity of Enterprise 2 as measured by the earth-based observer as \(v_{2}\). Using the formula for relative velocity in special relativity: \[ v_{2} = \frac{v_{1} + v_{2,rel}}{1 + \frac{v_{1} \cdot v_{2,rel}}{c^2}} \] where: - \(v_{1}\) is the velocity of Enterprise 1 as measured by the earth-based observer, - \(v_{2,rel}\) is the velocity of Enterprise 2 relative to Enterprise 1. Substitute the given values into the formula: \[ v_{2} = \frac{0.65c + 0.31c}{1 + \frac{0.65c \cdot 0.31c}{c^2}} \] Now, we can calculate the velocity of Enterprise 2 as measured by the earth-based observer. Simplify the expression by following steps: - step0: Solution: \(\frac{\left(0.65c+0.31c\right)}{\left(1+\frac{\left(0.65c\times 0.31c\right)}{c^{2}}\right)}\) - step1: Remove the parentheses: \(\frac{0.65c+0.31c}{1+\frac{0.65c\times 0.31c}{c^{2}}}\) - step2: Reduce the fraction: \(\frac{0.65c+0.31c}{1+\frac{403}{2000}}\) - step3: Add the terms: \(\frac{0.96c}{1+\frac{403}{2000}}\) - step4: Add the numbers: \(\frac{0.96c}{\frac{2403}{2000}}\) - step5: Rewrite the expression: \(\frac{\frac{24}{25}c}{\frac{2403}{2000}}\) - step6: Rewrite the expression: \(\frac{\frac{24c}{25}}{\frac{2403}{2000}}\) - step7: Multiply by the reciprocal: \(\frac{24c}{25}\times \frac{2000}{2403}\) - step8: Reduce the fraction: \(8c\times \frac{80}{801}\) - step9: Multiply the terms: \(\frac{8c\times 80}{801}\) - step10: Multiply the terms: \(\frac{640c}{801}\) The velocity of Enterprise 2 as measured by the earth-based observer is \( \frac{640c}{801} \) or approximately 0.80 c.