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 407 Hz. Find the train's speed and the horn's frequency. Take the speed of sound to be 343 m/s.

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 407 Hz. Find the train's speed and the horn's frequency. Take the speed of sound to be 343 m/s.

Answer 1

See your Aug. 28th post.

Answer 2

To solve this problem, we can use the Doppler effect equation. The Doppler effect is the change in frequency (or pitch) of a sound wave caused by the relative motion between the source of the sound and the observer.

The equation for the Doppler effect is:

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

where:
f' is the observed frequency (in this case, the frequency heard by the passenger in the back),
f is the source frequency (in this case, the frequency produced by the horn),
v is the speed of sound,
vo is the velocity of the observer (in this case, the speed of the train), and
vs is the velocity of the source (in this case, the speed of the car waiting at the crossing).

We are given:
f' = 435 Hz (frequency heard by the passenger in the back)
f = ?
v = 343 m/s (speed of sound)

We need to find:
vo (speed of the train)

To find vo, we first need to find f (source frequency) using the given frequency heard by the passenger in the front (f'').

First, let's rearrange the Doppler effect equation to solve for f:

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

Now, let's substitute the values we know into this equation:

f = (343 + vo) / (343 + 0) * 407

Simplifying this equation,

f = (343 + vo) / 343 * 407
f = (343 * 407 + 407 * vo) / 343

Now, we'll use the given frequency heard by the passenger in the front (f''):

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

Substituting the values we know:

407 Hz = (343 + vo) / 343 * 435

Now, we can solve this equation for vo (speed of the train):

407 * 343 = 435 * (343 + vo)
139,601 = 149,505 + 435vo
435vo = -9,904
vo ≈ -22.7 m/s

Since the speed of the train should be positive, we take the absolute value:

vo ≈ 22.7 m/s

Now that we have the speed of the train, we can substitute vo into the first equation we derived to solve for f (source frequency):

f = (343 + vo) / 343 * 407
f = (343 + 22.7) / 343 * 407
f ≈ 419 Hz

Therefore, the train's speed is approximately 22.7 m/s, and the horn's frequency is approximately 419 Hz.