A commuter train passes a passenger platform at a constant speed of 40.0 m/s. The train horn is sounded at its characteristic frequency of 320 Hz. What overall change in frequency is detected by a person on the platform as the train moves from approaching to receding?
2 (V/a)*fo = (80/340)*320 Hz
assuming a = 340 m/s is the speed of sound.
There is a more accurate formula for moving sources and stationary observers, but this should be close enough
To calculate the overall change in frequency detected by a person on the platform as the train moves from approaching to receding, we can use the Doppler effect equation:
Δf = ((v + vo) / v) * f
Where:
Δf = the change in frequency detected
v = speed of sound in air (approximated as 343 m/s)
vo = speed of the observer (person on the platform)
f = frequency of the source (sound from the train horn)
In this case, the observer is on the platform, so their speed is 0 m/s (vo = 0). The speed of sound in air is approximately 343 m/s (v = 343 m/s), and the frequency of the source (train horn) is 320 Hz (f = 320 Hz).
Plugging in these values into the Doppler effect equation, we have:
Δf = ((v + vo) / v) * f
Δf = ((343 m/s + 0 m/s) / 343 m/s) * 320 Hz
Simplifying the equation:
Δf = (343 / 343) * 320 Hz
Δf = 1 * 320 Hz
Δf = 320 Hz
Therefore, the overall change in frequency detected by a person on the platform as the train moves from approaching to receding is 320 Hz.
To find the change in frequency detected by a person on the platform as the train moves from approaching to receding, we need to use the Doppler effect formula:
Δf/f₀ = (v/c) * (cosθ - 1)
Where:
Δf = Change in frequency detected
f₀ = Initial frequency (characteristic frequency of the train horn)
v = Speed of the train
c = Speed of sound in air
θ = Angle between the direction of motion of the train and the line connecting the source of the sound (train horn) and the observer (person on the platform)
In this case, since the train is moving from approaching to receding, the angle θ is 180°.
Now we can plug in the values:
f₀ = 320 Hz
v = 40.0 m/s
c = approximately 343 m/s (speed of sound in air)
θ = 180°
Δf/f₀ = (40.0/343) * (cos180° - 1)
Now we solve for Δf/f₀:
cos180° = -1, so:
Δf/f₀ = (40.0/343) * (-1 - 1)
Δf/f₀ = (40.0/343) * (-2)
Δf/f₀ ≈ -0.117
Therefore, the overall change in frequency detected by a person on the platform is approximately -0.117 times the initial frequency of 320 Hz.