Final answer:
Using the Doppler effect equation, the maximum speed of a vibrating fetal heart with a Doppler frequency shift of 100 Hz and a source frequency of 3.6 MHz in the human body where the speed of sound is 1,540 m/s is calculated to be approximately 4.28 cm/s.
Step-by-step explanation:
To calculate the maximum speed of a vibrating fetal heart based on a Doppler frequency shift, we can use the Doppler effect equation for ultrasound in medical diagnostics:
f' = f (v + v_s) / (v),
where:
- f' is the observed frequency after the Doppler shift,
- f is the original frequency of the ultrasound,
- v is the speed of ultrasound in the medium, and
- v_s is the speed of the source relative to the medium.
In this scenario, the observed frequency shift (Δf) is 100 Hz when the heart is moving towards the ultrasound source:
Δf = f' - f,
thus,
f' = f + Δf.
Now, solving for v_s, which is the speed of the fetal heart:
v_s = Δf × v / f.
Given the values:
- f = 3.6 MHz (which is 3.6 × 106 Hz),
- Δf = 100 Hz, and
- v = 1,540 m/s,
we find:
v_s = (100 Hz × 1,540 m/s) / (3.6 × 106 Hz),
which gives:
v_s = 0.042778 m/s.
Converting this to cm/s (by multiplying by 100), we get:
v_s = 4.2778 cm/s.
Therefore, the maximum speed of the vibrating fetal heart is approximately 4.28 cm/s.