A commuter train passes a passenger platform at a constant speed of 31 m/s. The train horn is sounded at its characteristic frequency of 330 Hz.Assume the speed of sound is 343 m/s.

(a) What is the frequency detected by a person on the platform as the train approaches? (b) What frequency is detected by a person on the platform as the train recedes? (c) What overall change in frequency is detected by a person on the plat-form as the train moves from approaching to receding?
(d) What wavelength is detected by a person on the platform as the train approaches?

fo=(1/(1±vs/v))fs

Where the plus sign is negative if the source is approaching the observer and positive if the source is moving away from the observer.

fo is frequency of the observer
fs is the frequency of the source
vs is the speed of the source
v is the speed of sound

a) So as the train approaches,
fs = (1 / (1 - 31/343))*330

b) As the train recedes,
fs = (1 / (1 + 31/343))*330

c) The overall change in frequency is the answer in part a minus the answer in part b

d) The wavelength lamda is given by

lamda = v/f
where v is the speed of sound, and f is the frequency.

To solve this problem, we can use the Doppler effect formula for sound waves:

f' = f * (v + v0) / (v + vs)

Where:
- f' is the observed frequency,
- f is the source frequency,
- v is the speed of sound,
- v0 is the velocity of the observer,
- vs is the velocity of the source.

Given:
- Source frequency (f): 330 Hz
- Speed of sound (v): 343 m/s
- Velocity of the observer on the platform (v0): 0 m/s (since the person is stationary)
- Velocity of the source (vs): 31 m/s (approaching train)

(a) What is the frequency detected by a person on the platform as the train approaches?

Using the formula, we can substitute the given values to find the observed frequency as the train approaches:

f' = 330 * (343 + 0) / (343 + 31)
f' = 330 * 343 / 374
f' = 302.196 Hz

So, the frequency detected by a person on the platform as the train approaches is approximately 302.2 Hz.

(b) What frequency is detected by a person on the platform as the train recedes?

In this case, the velocity of the source (vs) will be negative since the train is receding. Let's substitute the values into the formula:

f' = 330 * (343 + 0) / (343 - 31)
f' = 330 * 343 / 312
f' = 362.05 Hz

Therefore, the frequency detected by a person on the platform as the train recedes is approximately 362.1 Hz.

(c) What overall change in frequency is detected by a person on the platform as the train moves from approaching to receding?

To find the overall change in frequency, we can calculate the difference between the frequencies detected in parts (a) and (b):

Change in frequency = Frequency Detected (receding) - Frequency Detected (approaching)
Change in frequency = 362.1 Hz - 302.2 Hz
Change in frequency = 59.9 Hz

So, the overall change in frequency detected by a person on the platform as the train moves from approaching to receding is approximately 59.9 Hz.

(d) What wavelength is detected by a person on the platform as the train approaches?

The wavelength can be found using the formula:

λ = v / f'

Substituting the values:

λ = 343 / 302.196
λ = 1.135 m

Therefore, the wavelength detected by a person on the platform as the train approaches is approximately 1.135 meters.

To answer these questions, we need to apply the Doppler effect formula, which relates the frequency observed by a stationary observer (on the platform) to the frequency emitted by a moving source (train).

The formula for the observed frequency, also known as the apparent frequency (f'), is given by:

f' = f * (v + vo) / (v + vs)

Where:
f = frequency emitted by the source (known as the characteristic frequency in this case)
v = speed of sound
vo = velocity of the observer (stationary)
vs = velocity of the source (train)

Let's solve each part of the question:

(a) What is the frequency detected by a person on the platform as the train approaches?

In this case, the train is approaching the observer, so the velocity of the source (vs) is negative. We can substitute the known values into the formula:

f' = 330 Hz * (343 m/s + 0 m/s) / (343 m/s - 31 m/s)
f' = 330 Hz * (343 m/s) / (312 m/s)
f' = 362.77 Hz

So, the frequency detected by the person on the platform as the train approaches is approximately 362.77 Hz.

(b) What frequency is detected by a person on the platform as the train recedes?

Now, the train is moving away from the observer. The velocity of the source (vs) is positive. Let's use the formula:

f' = 330 Hz * (343 m/s + 0 m/s) / (343 m/s + 31 m/s)
f' = 330 Hz * (343 m/s) / (374 m/s)
f' = 301.07 Hz

Therefore, the frequency detected by the person on the platform as the train recedes is approximately 301.07 Hz.

(c) What overall change in frequency is detected by a person on the platform as the train moves from approaching to receding?

To find the overall change in frequency, we can subtract the frequency detected as the train recedes from the frequency detected as the train approaches:

Overall change in frequency = Frequency as train approaches - Frequency as train recedes
Overall change in frequency = 362.77 Hz - 301.07 Hz
Overall change in frequency = 61.7 Hz

So, the overall change in frequency detected by a person on the platform as the train moves from approaching to receding is 61.7 Hz.

(d) What wavelength is detected by a person on the platform as the train approaches?

The formula for the wavelength (λ) is given by:

λ = v / f'

Let's calculate the wavelength detected by the person on the platform as the train approaches:

λ = 343 m/s / 362.77 Hz
λ ≈ 0.9466 m

Therefore, the wavelength detected by a person on the platform as the train approaches is approximately 0.9466 m.