Question

Rocket ship Able travels at \(0.400 c\) relative to an Earth observer. According to the same observer, rocket ship Able overtakes a slower moving rocket ship Baker that moves in the same direction. The captain of Baker sees Able pass her ship at \(0.114 c .\) Determine the speed of Baker relative to the Earth observer.

Short answer

Answer: The velocity of rocket ship Baker relative to the Earth observer is approximately \(0.273c\).

Step-by-step solution

Write down the problem information

We have the following information: - Velocity of rocket ship Able relative to Earth Observer, \(v_A = 0.400c\) - Velocity of rocket ship Able relative to rocket ship Baker, \(v_{AB} = 0.114c\) We need to find the velocity of rocket ship Baker relative to Earth Observer, \(v_B\).

Write down the relativistic velocity addition formula

The relativistic velocity addition formula is given by \(v_{AB} = \cfrac{v_A - v_B}{1 - \cfrac{v_A v_B}{c^2}}\)

Rearrange the formula to isolate \(v_B\)

To isolate \(v_B\), we can first multiply both sides by the denominator: \(v_{AB}(1 - \cfrac{v_A v_B}{c^2}) = v_A - v_B\) Next, expand the left side and simplify: \((v_{AB} - \cfrac{v_{AB} v_A v_B}{c^2}) = v_A - v_B\) Now, move the terms with \(v_B\) to the left side and all other terms to the right side: \(v_B + \cfrac{v_{AB} v_A v_B}{c^2} = v_A - v_{AB}\) factor \(v_B\) from the left side: \(v_B(1 + \cfrac{v_{AB} v_A}{c^2}) = v_A - v_{AB}\) Finally, divide by \((1 + \cfrac{v_{AB} v_A}{c^2})\) to isolate \(v_B\): \(v_B = \cfrac{v_A - v_{AB}}{1 + \cfrac{v_{AB} v_A}{c^2}}\)

Substitute the given values and solve for \(v_B\)

Substitute \(v_A = 0.400c\) and \(v_{AB} = 0.114c\) into the formula for \(v_B\) and solve: \(v_B = \cfrac{0.400c - 0.114c}{1 + \cfrac{0.114c \cdot 0.400c}{c^2}}\) \(v_B = \cfrac{0.286c}{1 + \cfrac{0.0456c^2}{c^2}}\) \(v_B = \cfrac{0.286c}{1 + 0.0456}\) \(v_B = \cfrac{0.286c}{1.0456}\) \(v_B \approx 0.273c\)

Write down the final answer

The velocity of rocket ship Baker relative to the Earth observer is approximately \(0.273c\).