At what speed should a dog run so that it doesn't hear the ringing of a frying pan tied to its tail? - briefly
To determine the speed at which a dog should run to avoid hearing the ringing of a frying pan tied to its tail, one must consider the Doppler effect. The dog will not hear the sound if it runs faster than the speed of sound relative to the observer (in this case, itself). This speed is approximately 343 meters per second at sea level and standard atmospheric conditions.
At what speed should a dog run so that it doesn't hear the ringing of a frying pan tied to its tail? - in detail
To determine at what speed a dog should run to avoid hearing the ringing of a frying pan tied to its tail, we need to delve into the principles of sound propagation and the Doppler effect.
Sound travels through air at approximately 343 meters per second (m/s) at room temperature. The frequency of the sound produced by the ringing frying pan is constant, but as the dog runs away from the source of the sound, the perceived frequency changes due to the Doppler effect. This effect occurs because the motion of the dog relative to the source causes a shift in the frequency of the sound waves.
When the dog is running towards the source, the perceived frequency increases, and when the dog is running away from the source, the perceived frequency decreases. The formula for the Doppler shift in frequency (f') can be expressed as:
[ f' = \frac{v + v_s}{v + v_d} f ]
Where:
- ( f ) is the original frequency of the sound produced by the ringing frying pan.
- ( v ) is the speed of sound in air.
- ( v_s ) is the speed of the source (in this case, the dog).
- ( v_d ) is the speed of the observer (also the dog).
For the dog to not hear the ringing of the frying pan, the perceived frequency must drop below the audible range for dogs. Dogs can typically hear frequencies between 40 Hz and 60 kHz, with the most sensitive range being around 8,000 Hz.
To find the speed at which the dog should run to make the sound inaudible, we set ( f' ) to a value below the lower limit of the dog's hearing range (40 Hz). Let's assume the original frequency (( f )) of the ringing pan is 1 kHz.
[ 40 = \frac{343 + v}{343} \times 1000 ]
Solving for ( v ):
[ 40 \times 343 = (343 + v) \times 1000 ]
[ 13720 = 343000 + 1000v ]
[ 1000v = 13720 - 343000 ]
[ 1000v = -329280 ]
[ v = \frac{-329280}{1000} ]
[ v = -329.28 \, m/s ]
Since speed cannot be negative, this calculation indicates that the dog would have to run faster than the speed of sound for the ringing frying pan to become inaudible. This is not physically possible under normal conditions. Therefore, the dog will always hear the ringing of the frying pan tied to its tail, regardless of the speed at which it runs.