At what speed must a dog move to not hear the ringing of a frying pan tied to its tail? - briefly
To determine the speed at which a dog must move to avoid hearing the ringing of a frying pan tied to its tail, one needs to consider the principles of the Doppler effect. This effect states that the frequency of a sound wave changes as the source and observer move relative to each other. Specifically, the perceived frequency increases when the source and observer approach each other and decreases when they move apart. In this scenario, if the dog moves fast enough, the change in pitch caused by the Doppler effect could potentially shift the ringing of the frying pan out of its audible range, thus making it inaudible to the dog.
At what speed must a dog move to not hear the ringing of a frying pan tied to its tail? - in detail
To determine the speed at which a dog must move to no longer hear the ringing of a frying pan tied to its tail, we need to consider the principles of sound propagation and the Doppler effect.
Sound travels as a wave, and the frequency of this wave is what our ears perceive as pitch. When an object moves relative to a stationary observer, the perceived frequency of the sound it emits changes. This phenomenon is known as the Doppler effect. If the object is moving away from the observer, the perceived frequency decreases; if it is moving towards the observer, the perceived frequency increases.
In this scenario, the dog acts as both the source and the observer of the sound. The frying pan tied to its tail creates a noise when it moves, and this noise travels at the speed of sound in air (approximately 343 meters per second at sea level). As the dog increases its speed, the perceived frequency of the ringing changes according to the Doppler effect.
The formula for the Doppler effect is: [ f' = \frac{f_0}{1 + \frac{v}{c}} ] where ( f' ) is the perceived frequency, ( f_0 ) is the original frequency of the sound, ( v ) is the speed of the source (the dog), and ( c ) is the speed of sound in air.
For the dog to no longer hear the ringing, the perceived frequency ( f' ) must be zero or below the threshold of human hearing, which is typically around 20 Hz. This means that the equation becomes: [ 0 = \frac{f_0}{1 + \frac{v}{343}} ] or [ v = -343 ]
However, a negative speed is not physically meaningful in this context. Instead, we look for the condition where the perceived frequency equals zero. Solving for ( v ) when ( f' = 0 ): [ 0 = \frac{f_0}{1 + \frac{v}{343}} ] [ 1 + \frac{v}{343} = \infty ] [ \frac{v}{343} = -1 ] [ v = 343 ]
Therefore, the dog must move at or above the speed of sound in air (approximately 343 meters per second) to no longer hear the ringing of the frying pan tied to its tail. This is theoretically possible but practically challenging, as achieving such speeds would require exceptional conditions and possibly specialized equipment.