Final answer:
To find the minimum speed producing a perceptible Doppler shift of 0.300%, we use the formula vs = v((f'/f) - 1) with a frequency increase of 1.003, and the given speed of sound 331 m/s. Thus, the minimum speed vs is 0.993 m/s, which is rounded up to 1 m/s as the perceptible threshold exceeds the options given.
Step-by-step explanation:
The question asks to determine the minimum speed at which a source must move toward an observer for the frequency shift, due to the Doppler effect, to be perceptible at a minimum level of 0.300%. The speed of sound is given as 331 m/s. For the Doppler shift in frequency to be noticeable, the change in frequency relative to the original frequency must be at least 0.300%.
The formula for the Doppler effect, when the source is moving towards a stationary observer, is:
f' = f(v + v
s
) / v
where:
- f' is the frequency observed.
- f is the emitted frequency.
- v is the speed of sound in the medium.
- vs is the speed of the source towards the observer.
To find the required minimum speed of the source vs, we can rearrange the formula to:
vs = v((f'/f) - 1)
Since we are looking for a minimum speed that causes a 0.300% increase in frequency, we have (f'/f) = 1.003. Plugging in the values, we get:
vs = 331 m/s ((1.003) - 1) = 331 m/s (0.003) = 0.993 m/s
Since the actual minimum speed must be an integer value in meters per second, and it must meet or exceed the threshold for a perceptible Doppler shift, we can round up the value to the nearest whole number to get the minimum perceptible speed. This gives us:
vs > 0.993 m/s, which we can approximate to 1 m/s as the minimum perceptible speed.
Examining the provided answer choices, none of the provided answer options (a) through (d) match the correct calculation. The closest match is option a) which is significantly higher than the calculated value.