At what speed must a dog run not to hear the ringing of a frying pan tied to its tail? - briefly
To avoid hearing the ringing of a frying pan tied to its tail, a dog must run at a speed where the sound waves emitted by the frying pan do not reach it. This occurs when the dog's speed exceeds the speed of sound in air, which is approximately 343 meters per second.
At what speed must a dog run not to hear the ringing of a frying pan tied to its tail? - in detail
To determine the speed at which a dog must run to avoid hearing the ringing of a frying pan tied to its tail, one must consider several key factors: the principles of sound propagation, the Doppler effect, and the physiological aspects of canine hearing.
Sound travels in waves, and the speed of these waves depends on the medium through which they are traveling. In air at standard temperature and pressure, the speed of sound is approximately 343 meters per second (m/s). When a dog runs, it creates a moving reference frame relative to the observer. This movement affects the perceived frequency of the sound waves due to the Doppler effect.
The Doppler effect states that the frequency of a wave changes for an observer who is moving relative to its source. If the dog and the frying pan are moving towards the observer, the frequency increases; if they are moving away from the observer, the frequency decreases. Mathematically, this can be expressed as: [ f' = \frac{f}{1 - \frac{v_s}{c}} ] where ( f' ) is the observed frequency, ( f ) is the emitted frequency, ( v_s ) is the speed of the source (dog), and ( c ) is the speed of sound.
For the dog to not hear the ringing of the frying pan, the perceived frequency must drop below its hearing threshold. The canine hearing range typically spans from 40 Hz to 60 kHz. To calculate the required speed, we need to know the frequency at which the frying pan rings when struck (this is a given or assumed value).
Assuming the frying pan rings at a frequency within the human audible range (let’s say 1000 Hz for simplicity), we can rearrange the Doppler equation to solve for ( v_s ): [ v_s = c \left( 1 - \frac{f}{f'} \right) ] Plugging in the values, if the dog's hearing threshold is 40 Hz and the frying pan rings at 1000 Hz: [ v_s = 343 \, \text{m/s} \left( 1 - \frac{1000}{40} \right) ] Clearly, this calculation yields a negative speed, which is not physically meaningful. Therefore, the dog must run at a speed greater than the speed of sound to avoid hearing the ringing of the frying pan tied to its tail. This means the dog would need to exceed approximately 343 m/s (or about 1235 km/h).
In practical terms, this is not feasible for a dog or any terrestrial animal due to physiological and mechanical constraints. Thus, under normal conditions, a dog will always hear the ringing of the frying pan tied to its tail, regardless of its running speed.