A healthy astronaut's heart rate is 60 beats/min. Flight doctors on Earth can monitor an astronaut's vital signs remotely while in flight.
How fast would an astronaut be flying away from Earth if the doctor measured her having a heart rate of 28 beats/min?
To determine how fast the astronaut would be flying away from Earth based on her heart rate, we need to make a few assumptions and use the concept of relative velocity.
Let's assume that the change in heart rate is solely due to the Doppler effect because of the astronaut's motion away from Earth. The Doppler effect occurs when there is a relative velocity between the source of a wave (in this case, the astronaut) and the observer (the doctor).
The formula for the Doppler effect is as follows:
f' = f * (v + vr) / (v + vs)
where:
f' is the observed frequency (heart rate) by the doctor,
f is the emitted frequency (normal heart rate of 60 beats/min),
v is the speed of sound in the medium (assumed to be irrelevant for simplicity),
vr is the relative velocity between the astronaut and the doctor, and
vs is the velocity of sound waves emitted by the astronaut (assumed to be zero since sound can't propagate in a vacuum).
Since we desire to know the relative velocity (vr), we can rearrange the equation:
vr = (f/f' - 1) * vs
Substituting the given values, with the normal heart rate (f) of 60 beats/min and the observed heart rate (f') of 28 beats/min, the equation becomes:
vr = (60/28 - 1) * 0
Calculating this, you'll find that the relative velocity (vr) is zero. Therefore, according to this assumption, the astronaut would not be flying away from Earth since her relative velocity is zero. However, it's important to note that this assumption may not necessarily reflect the reality of spaceflight conditions.
Keep in mind that this calculation neglects factors such as acceleration, gravitational forces, and other potential causes of changes in heart rate. It is always crucial to consult actual flight doctors and medical professionals who specialize in monitoring astronauts during spaceflight to obtain accurate and reliable information.