If an observer on earth sees two space ships approaching each other and she measures the speed of ship 1 wrt (with respect to) herself to be \( 0.8 c \) to the right and ship 2 to be 0.7 c to the left. An observer in ship 1 also sees the approach of ship 2. What velocity does he see ship 2 moving at wrt his ship 1?

To find the velocity of ship 2 as observed from ship 1, we can use the relativistic velocity addition formula: \[ v' = \frac{u + v}{1 + \frac{uv}{c^2}} \] where: - \( v' \) is the velocity of ship 2 relative to ship 1, - \( u = 0.7c \) (velocity of ship 2 relative to earth, to the left, we'll take left as negative, so \( u = -0.7c \)), - \( v = 0.8c \) (velocity of ship 1 relative to earth, to the right). Plugging these values into the formula: \[ v' = \frac{0.8c + (-0.7c)}{1 + \frac{(0.8c)(-0.7c)}{c^2}} \] Calculating the numerator: \[ 0.8c - 0.7c = 0.1c \] Calculating the denominator: \[ 1 + \frac{(0.8)(-0.7)c^2}{c^2} = 1 - 0.56 = 0.44 \] Now plugging these into the equation: \[ v' = \frac{0.1c}{0.44} \approx 0.2273c \] Therefore, the observer in ship 1 sees ship 2 moving towards him at a speed of approximately \( 0.2273c \).