You and your dog are standing on the side of the road. Someone approaches in a very fast car playing an eigth octave C on a piccolo (4186 Hz). How fast would they have to approach in m/s so that only your dog would hear the piccolo

In order to obtain a velocity that only the dog is be able to hear, you can consider that the perceived frequency is higher, at least, than 20,000 Hz.

This problem is about Doppler's effect. To calculate the speed of the source of the sound (a car playing a piccolo), you use the following formula for the case on which the source is getting closer to the observer:

[tex]f'=f(\frac{v+v_o}{v-v_s})[/tex] (1)

f: frequency of the source = 4186Hz

f': frequency perceived by the observer = 20,000Hz

v: speed of sound = 343 m/s

vo: speed of the observer = 0m/s

vs: speed of the source = ?

You solve the equation (1) for vs and replace the values of the other parameters:

[tex]f'=f(\frac{v}{v-v_s})\\\\(v-v_s)f'=fv\\\\-v_sf'=fv-f'v\\\\v_s=\frac{v(f'-f)}{f'}=\frac{(343m/s)(20000Hz-4186Hz)}{20000Hz}=271.21\frac{m}{s}[/tex]

The speed of the car must be vs > 271.21 m/s