A spaceship is traveling away from Earth at \(0.87 c .\) The astronauts report home by radio every 12 h (by their own clocks). (a) At what interval are the reports sent to Earth, according to Earth clocks? (b) At what interval are the reports received by Earth observers, according to their own clocks?

In order to answer this question, follow the step-by-step solution provided: Step 1: Calculate the time dilation factor, which is given by the formula \(\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\). In this case, \(v = 0.87c\). Step 2: Determine the time intervals according to Earth's clocks (t_Earth) using the equation: \(t_{Earth} = \gamma t_{astronaut}\), where \(t_{astronaut} = 12\) hours. Step 3: Calculate the observed frequency by Earth observers (f_Earth) using the Doppler effect formula for light: \(f_{Earth} = f_{astronaut}\frac{1}{1 + \frac{v}{c}}\). Step 4: Calculate the time intervals between received reports by Earth observers (t_received) by taking the inverse of the observed frequency: \(t_{received} = \frac{1}{f_{Earth}}\). The interval at which the reports are sent according to Earth clocks is \(t_{Earth}\), and the interval at which the reports are received by Earth observers is \(t_{received}\).