A spacecraft approaching the earth launches an exploration vehicle. After the launch, an observer on earth sees the spacecraft approaching at a speed of 0.50 c and the exploration vehicle approaching at a speed of 0.70 c . What is the speed of the exploration vehicle relative to the spaceship?

We start by using the relativistic velocity addition formula. If the spacecraft (moving at \( v \)) sees the exploration vehicle (moving at \( u \) with respect to Earth), then the speed of the exploration vehicle relative to the spacecraft is given by \[ u' = \frac{u-v}{1-\frac{uv}{c^2}}. \] Here, we have: - \( u = 0.70c \) (speed of the exploration vehicle relative to Earth), - \( v = 0.50c \) (speed of the spacecraft relative to Earth). Substitute these into the formula: \[ u' = \frac{0.70c - 0.50c}{1 - \frac{(0.70c)(0.50c)}{c^2}}. \] Simplify the numerator: \[ 0.70c - 0.50c = 0.20c. \] For the denominator, note that \[ \frac{(0.70c)(0.50c)}{c^2} = 0.35, \] so the denominator becomes \[ 1 - 0.35 = 0.65. \] Thus, \[ u' = \frac{0.20c}{0.65} \approx 0.3077c. \] Rounding to two significant figures, the speed of the exploration vehicle relative to the spaceship is approximately \[ u' \approx 0.31c. \]