At what speed should a dog run so as not to hear the ringing of a can tied to it? - briefly
The question of determining the speed at which a dog should run to avoid hearing the ringing of a can tied to it is a classic thought experiment often attributed to the French physicist Jean-Baptiste Biot. This scenario is a simplified illustration of the Doppler effect, where the perceived frequency of a sound changes relative to the observer's motion.
The Doppler effect explains that as the source of sound moves away from the observer, the frequency of the sound waves decreases. For a dog to not hear the ringing of the can, it would need to run at a speed where the frequency of the sound waves shifts below the audible range for the dog's ears. However, this speed is impractically high for a dog to achieve, making the scenario more of a theoretical curiosity than a practical consideration.
To provide a brief answer: A dog would need to run at a speed where the frequency shift due to the Doppler effect moves the sound of the ringing can below the dog's audible range, which is theoretically very high and impractical.
At what speed should a dog run so as not to hear the ringing of a can tied to it? - in detail
The phenomenon of a dog running with a can tied to its tail is a classic example used to illustrate principles of sound and motion. To understand the speed at which a dog must run to avoid hearing the ringing of the can, several scientific concepts must be considered.
Firstly, sound waves travel through the air at a specific speed, approximately 343 meters per second at room temperature. This speed is constant and does not depend on the motion of the source of the sound. Therefore, the ringing of the can will always travel at this speed relative to the air, regardless of the dog's movement.
However, the dog's perception of the sound depends on its own motion. When the dog runs, it moves through the sound waves. The perceived frequency of the sound changes due to the Doppler effect. This effect causes the frequency of a wave to increase when the source and observer are moving towards each other and to decrease when they are moving apart.
For the dog to not hear the ringing, the perceived frequency of the sound must be reduced to a level below the dog's hearing threshold. Dogs can typically hear frequencies ranging from about 67 to 45,000 Hertz (Hz). To avoid hearing the ringing, the dog would need to run at a speed that shifts the frequency of the sound below 67 Hz.
The frequency shift due to the Doppler effect can be calculated using the formula:
[ f' = f \left( \frac{v \pm v_o}{v \pm v_s} \right) ]
where:
- ( f' ) is the perceived frequency,
- ( f ) is the original frequency of the sound,
- ( v ) is the speed of sound in air,
- ( v_o ) is the speed of the observer (the dog),
- ( v_s ) is the speed of the source (the can).
For the dog to not hear the ringing, ( f' ) must be less than 67 Hz. Assuming the can rings at a typical frequency, say 1,000 Hz, and considering the dog is moving away from the source, the formula simplifies to:
[ 67 = 1000 \left( \frac{343}{343 - v_o} \right) ]
Solving for ( v_o ):
[ 67 (343 - v_o) = 1000 \times 343 ] [ 22941 - 67v_o = 343000 ] [ -67v_o = 320059 ] [ v_o = \frac{320059}{67} ] [ v_o \approx 4777 \, \text{m/s} ]
This calculation shows that the dog would need to run at an extremely high speed, approximately 4,777 meters per second, to avoid hearing the ringing of the can. This speed is well beyond the physical capabilities of any dog, making it an impractical scenario. Therefore, in reality, a dog cannot run fast enough to avoid hearing the ringing of a can tied to its tail.