Two spaceships \( A \) and B are exploring a new planet. Relative to this planet, spaceship A has a speed of 0.60 c , and spaceship B has a speed of 0.80 c . What is the ratio \( \mathrm{D} / \mathrm{Da}_{3} \) of the values for the planet's diameter that each spaceship measures in a direction that is parallel to its motion?
Solución de inteligencia artificial de Upstudy
Responder
Solución

¡Inicia sesión para desbloquear respuestas gratis!
Una plataforma de aprendizaje en la que confían millones de estudiantes y profesores reales.

Respondido por UpStudy AI y revisado por un tutor profesional
Beyond the Answer
To find the ratio \( D / D_A \) of the planet's diameter measured by the two spaceships \( A \) and \( B \) in a direction parallel to their motion, we will use the concept of length contraction from the theory of special relativity. The length contraction formula states that the length measured in a moving frame (denoted as \( L' \)) is related to the proper length (denoted as \( L \)) by: \[ L' = L \sqrt{1 - \frac{v^2}{c^2}} \] where \( v \) is the speed of the moving observer and \( c \) is the speed of light. 1. For spaceship \( A \) with speed \( v_A = 0.60c \): - The contracted diameter of the planet as observed by \( A \) is given by the formula: \[ D_A = D \sqrt{1 - \left(0.60\right)^2} \] \[ D_A = D \sqrt{1 - 0.36} = D \sqrt{0.64} = D \cdot 0.8 \] 2. For spaceship \( B \) with speed \( v_B = 0.80c \): - The contracted diameter of the planet as observed by \( B \) is given by the formula: \[ D_B = D \sqrt{1 - \left(0.80\right)^2} \] \[ D_B = D \sqrt{1 - 0.64} = D \sqrt{0.36} = D \cdot 0.6 \] 3. Now, we need to find the ratio \( \frac{D}{D_A} \) and \( \frac{D}{D_B} \): - From \( D_A = D \cdot 0.8 \): \[ \frac{D}{D_A} = \frac{D}{D \cdot 0.8} = \frac{1}{0.8} = 1.25 \] - From \( D_B = D \cdot 0.6 \): \[ \frac{D}{D_B} = \frac{D}{D \cdot 0.6} = \frac{1}{0.6} \approx 1.6667 \] 4. Finally, we are looking for the ratio \( \frac{D_A}{D_B} \): \[ \frac{D_A}{D_B} = \frac{0.8}{0.6} = \frac{4}{3} \approx 1.3333 \] Thus, the ratio of the values for the planet’s diameter that each spaceship measures in a direction that is parallel to its motion is \( \frac{D_A}{D_B} = \frac{4}{3} \).
