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. The formula is given by: \[ u' = \frac{u + v}{1 + \frac{uv}{c^2}} \] where: - \( u' \) is the velocity of ship 2 as observed from ship 1, - \( u \) is the velocity of ship 2 as observed from Earth (which is \( -0.7c \) since it is moving to the left), - \( v \) is the velocity of ship 1 as observed from Earth (which is \( 0.8c \)), - \( c \) is the speed of light. Now, substituting the values into the formula: \[ u' = \frac{-0.7c + 0.8c}{1 + \frac{(-0.7c)(0.8c)}{c^2}} \] Let's simplify this step by step. 1. Calculate the numerator: \[ -0.7c + 0.8c = 0.1c \] 2. Calculate the denominator: \[ 1 + \frac{(-0.7c)(0.8c)}{c^2} = 1 - 0.56 = 0.44 \] Now we can substitute these results back into the equation for \( u' \): \[ u' = \frac{0.1c}{0.44} \] Now, let's calculate this value. Calculate the value by following steps: - step0: Calculate: \(\frac{0.1}{0.44}\) - step1: Convert the expressions: \(\frac{\frac{1}{10}}{\frac{11}{25}}\) - step2: Multiply by the reciprocal: \(\frac{1}{10}\times \frac{25}{11}\) - step3: Reduce the numbers: \(\frac{1}{2}\times \frac{5}{11}\) - step4: Multiply the fractions: \(\frac{5}{2\times 11}\) - step5: Multiply: \(\frac{5}{22}\) The calculation gives us: \[ u' = \frac{0.1}{0.44} \approx 0.22727c \] Thus, the observer in ship 1 sees ship 2 moving at approximately \( 0.227c \) towards the right.