A whistle you use to call your hunting dog has a frequency of 24 kHz, but your dog is ignoring it. You suspect the whistle may not be working, but you can't hear sounds above 23 kHz. To test it, you ask a friend to blow the whistle, then you hop on your bicycle.
At what minimum speed should you ride to know if the whistle is working?
Answer in m/s
16.3
To determine if the whistle is working, you need to consider the Doppler effect. The Doppler effect describes the change in frequency of a sound wave when there is relative motion between the source of the sound and the observer.
In this scenario, you are the observer on your bicycle, and the whistle is the source of the sound. The frequency of the whistle is 24 kHz, while your hearing range is limited to sounds below 23 kHz.
To hear the sound of the whistle, the frequency heard by you needs to be shifted from 24 kHz to below 23 kHz. This shift can be achieved by the relative motion between you and the whistle.
The Doppler effect formula for frequency observed (fo) is given by:
fo = fs × (v + vr) / (v + vs)
Where:
fo = observed frequency
fs = source frequency (24 kHz)
v = speed of sound in air (approximately 343 m/s)
vr = velocity of the observer (bicycling speed)
vs = velocity of the source (speed at which the whistle is blown)
To find the minimum speed at which you should ride your bicycle to determine if the whistle is working, we need to rearrange the formula to solve for vr:
vr = ((fo * (v + vs)) / fs) - v
Substituting the known values, we have:
vr = ((23 kHz * (343 m/s + 0 m/s)) / 24 kHz) - 343 m/s
Simplifying the equation, we have:
vr = (23 * 343 / 24) - 343 m/s
vr ≈ 324.229 m/s - 343 m/s
vr ≈ -18.771 m/s
Since the velocity of the observer (vr) is negative, it means that you need to move towards the whistle (opposite to the source) to hear the sound. In practical terms, this means you need to ride your bicycle at a speed of at least **18.771 m/s** in the opposite direction of the sound source (blowing whistle) to hear it.
Please note that riding a bicycle at such a high speed may not be safe. This calculation is for theoretical purposes and assumes ideal conditions. It's essential to prioritize safety while using this information.