A train is moving past a crossing where cars are waiting for it to pass. While waiting, the driver of the lead car becomes sleepy and rests his head on the steering wheel, unintentionally activating the car's horn. A passenger in the back of the train hears the horn's sound at a frequency of 435 Hz and a passenger in the front hears it at 413 Hz. Find the train's speed and the horn's frequency, assuming the sound travels along the tracks. Take the speed of sound to be 343 m/s.HINT(a)the train's speed (in m/s) m/s(b)the horn's frequency (in Hz) Hz
Question
A train is moving past a crossing where cars are waiting for it to pass. While waiting, the driver of the lead car becomes sleepy and rests his head on the steering wheel, unintentionally activating the car's horn. A passenger in the back of the train hears the horn's sound at a frequency of 435 Hz and a passenger in the front hears it at 413 Hz. Find the train's speed and the horn's frequency, assuming the sound travels along the tracks. Take the speed of sound to be 343 m/s.HINT(a)the train's speed (in m/s) m/s(b)the horn's frequency (in Hz) Hz
Solution
To solve this problem, we need to use the Doppler effect formula for sound. The Doppler effect describes the change in frequency of a wave in relation to an observer who is moving relative to the wave source.
The formula for the Doppler effect is:
f' = f * (v + vo) / (v + vs)
where: f' is the observed frequency, f is the source frequency, v is the speed of sound, vo is the speed of the observer, and vs is the speed of the source.
In this case, the car is the source and the train is the observer. When the train is moving towards the source (front of the train), the frequency increases. When the train is moving away from the source (back of the train), the frequency decreases.
Let's denote: f1 = 435 Hz (frequency heard at the back of the train), f2 = 413 Hz (frequency heard at the front of the train), v = 343 m/s (speed of sound), vs = ? (speed of the train, which we need to find), f = ? (frequency of the horn, which we need to find).
We have two equations:
f1 = f * (v + 0) / (v - vs) (1) (because the train is moving away from the source)
f2 = f * (v + 0) / (v + vs) (2) (because the train is moving towards the source)
We can solve these two equations simultaneously to find vs and f.
First, divide equation (1) by equation (2):
f1 / f2 = (v - vs) / (v + vs)
Solving this equation for vs gives:
vs = v * (f1 - f2) / (f1 + f2) = 343 m/s * (435 Hz - 413 Hz) / (435 Hz + 413 Hz) = 5.14 m/s
Substitute vs = 5.14 m/s into equation (1) or (2) to find f:
f = f1 * (v - vs) / v = 435 Hz * (343 m/s - 5.14 m/s) / 343 m/s = 424 Hz
So, the train's speed is 5.14 m/s and the horn's frequency is 424 Hz.
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