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 428 Hz and a passenger in the front hears it at 402 Hz. Find the train's speed and the horn's frequency. Take the speed of sound to be 343 m/s.
(a) the trains speed (in m/s)
(b) the horns frequency (in Hz)
b. Fg = (402+428)/2 = 415 Hz. = Generator(horn) freq.
a. Fr = (Vs+Vr)/(Vs+Vg) * Fg = 428.
(343+Vr)/(343+0) * 415 = 428,
(343+Vr) * 415 = 146,804,
343+Vr = 353.7,
Vr = 10.7 m/s. = Velocity of receiver(train).
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To solve this problem, we can use the Doppler effect formula for frequency:
f' = f * (v +/- vr) / (v +/- vs)
Where:
f' is the observed frequency
f is the source frequency
v is the speed of sound
vr is the speed of the receiver (passenger in the back)
vs is the speed of the source (train's speed)
Given data:
f1 = 402 Hz (frequency heard by the passenger in the front)
f2 = 428 Hz (frequency heard by the passenger in the back)
v = 343 m/s (speed of sound)
We need to find:
(a) The train's speed (vs)
(b) The horn's frequency (f)
Step 1: Calculate the relative velocity between the train and the passenger in the back of the train.
vr = 0 m/s (since the passenger in the back is stationary relative to the train)
Step 2: Rearrange the formula to solve for the source's velocity (train's speed).
f'1 = f1 * (v - vr) / (v - vs)
f'2 = f2 * (v - vr) / (v - vs)
f1 / f2 = (v - vr) / (v - vs)
Step 3: Solve the above equation for vs (train's speed).
f1 / f2 = (v - vr) / (v - vs)
402 Hz / 428 Hz = (343 m/s) / (343 m/s - vs)
402 / 428 = 1 / (1 - vs / 343)
(402/428) - 1 = vs / 343
(1 - 0.939) = vs / 343
0.061 = vs / 343
vs = 0.061 * 343 = 20.023 m/s
(a) The train's speed is approximately 20.023 m/s.
Step 4: Solve for f (horn's frequency) using the value of vs obtained in the previous step.
f'2 = f2 * (v - vr) / (v - vs)
428 Hz = f * (343 m/s) / (343 m/s - 20.023 m/s)
428 Hz * (343 m/s - 20.023 m/s) = f * (343 m/s)
f = (428 Hz * (343 m/s - 20.023 m/s)) / (343 m/s)
f ≈ (428 Hz * 322.977 m/s) / 343 m/s
f ≈ 402 Hz
(b) The horn's frequency is approximately 402 Hz.
To solve this problem, we'll use the Doppler effect equation that relates the observed frequency of a sound to the source frequency and the relative velocity between them:
f_observed = (speed of sound ± velocity of the observer) / (speed of sound ± velocity of the source) * f_source
In this case, the source is the car's horn, the observer in the back of the train, and the observer in the front of the train.
Let's calculate the train's speed and the horn's frequency:
(a) The train's speed (in m/s):
1. Rear Observer's Perception:
The observed frequency is 428 Hz, and the observer is in the back of the train. Therefore, the relative velocity between the observer and the source (horn) is the sum of their speeds. Let's assume the observer's speed is v1 and the train's speed is v.
428 Hz = (343 m/s + v1) / (343 m/s + v) * f_source - (Equation 1)
2. Front Observer's Perception:
The observed frequency is 402 Hz, and the observer is in the front of the train. Therefore, the relative velocity between the observer and the source (horn) is the difference between their speeds. Let's assume the observer's speed is v2 and the train's speed is v.
402 Hz = (343 m/s - v2) / (343 m/s + v) * f_source - (Equation 2)
We have two equations with two unknowns (v and f_source). We can solve this system of equations to find v.
(b) The horn's frequency (in Hz):
Once you find the value of v, you can substitute it back into either Equation 1 or Equation 2 and solve for f_source to find the horn's frequency.
Remember to consider the sign (plus or minus) when calculating the relative velocities based on the direction of the observer and the source.
By solving these equations simultaneously, you can determine the train's speed and the horn's frequency.
Google Doppler effect
f = fo (c +/- Vr ) / (c +/- Vs )fo = real frequency
c = speed of signal
Vr = 0 here ( receiver not moving)
Vs = + when toward, - when away
so
402 = fo (343)/ ( 343 + Vs)
428 = fo (343)/ ( 343 - Vs )