Explanation:
The heart rate of the astronaut is 78.5 beats per minute, which means that the time between heart beats is 0.0127 min. This will be the time t measured by the moving observer. The time t' measured by the stationary Earth-based observer is given by
[tex]t' = \dfrac{t}{\sqrt{1 - \left(\dfrac{v^2}{c^2}\right)}}[/tex]
a) If the astronaut is moving at 0.480c, the time t' is
[tex]t' = \dfrac{0.0127\:\text{min}}{\sqrt{1 - \left(\dfrac{0.2304c^2}{c^2}\right)}}[/tex]
[tex]\:\:\:\:=0.0145\:\text{min}[/tex]
This means that time between his heart beats as measured by Earth-based observer is 0.0145 min, which is equivalent to 69.1 beats per minute.
b) At v = 0.940c, the time t' is
[tex]t' = \dfrac{0.0127\:\text{min}}{\sqrt{1 - \left(\dfrac{0.8836c^2}{c^2}\right)}}[/tex]
[tex]\:\:\:\:=0.0372\:\text{min}[/tex]
So at this speed, the astronaut's heart rate is 1/(0.0372 min) or 26.9 beats per minute.
