A physics student sits by the open window on a train moving at 25 m/sec towards the east. Her boyfriend is standing on the station platform, sadly watching her leave. When the train is 150 meters form the station, it emits a whistle at a frequency of 3000 Hz.
What is the frequency of does the boyfriend hear ?
A chemistry student (with nothing better to do than ride around on trains all day) is on a west bound train moving at a velocity of 25 m/sec towards the station. Before the two trains pass each other, what frequency will he hear from the train whistle from the physics student's train?
What frequency will the chemistry student hear after his train passes the physics student's train?
To find the frequency that the boyfriend hears, we need to apply the Doppler effect formula for sound waves.
The Doppler effect is the change in frequency or pitch that occurs when there is relative motion between the source of the sound and the observer. When the source and the observer are approaching each other, the frequency appears higher, and when they are moving apart, the frequency appears lower.
Given:
- Train velocity = 25 m/s towards the east
- Distance of the train from the station when the whistle is emitted = 150 meters
- Frequency of the whistle emitted by the physics student's train = 3000 Hz
To find the frequency heard by the boyfriend, we need to consider the relative velocity between the observer (the boyfriend) and the source of the sound (the train). In this case, the relative velocity is the sum of the velocity of the train and the velocity of sound waves.
Step 1: Find the relative velocity between the observer and the train.
Since the train is moving towards the east, and the observer is stationary on the station platform, the relative velocity between them is only the velocity of the train, which is 25 m/s.
Step 2: Use the Doppler effect formula to find the frequency heard by the boyfriend.
The general formula for the Doppler effect of sound waves is:
f' = (v + vo) / (v + vs) * f
Where:
- f' is the frequency heard by the observer
- f is the frequency emitted by the source
- v is the velocity of sound waves
- vo is the velocity of the observer (in this case, the boyfriend)
- vs is the velocity of the source (in this case, the train)
In this scenario, the velocity of the sound waves (v) is constant and can be approximated as 343 m/s (the speed of sound in air). The velocity of the observer (vo) is 0 m/s since the boyfriend is stationary. The velocity of the source (vs) is the velocity of the train, which is 25 m/s towards the east.
Plugging in the values into the formula:
f' = (343 + 0) / (343 + (-25)) * 3000
f' = 343 / 318 * 3000
f' ≈ 3241 Hz
Therefore, the boyfriend hears a frequency of approximately 3241 Hz.
Now, let's move on to the chemistry student:
Before the two trains pass each other, the chemistry student is on a westbound train moving towards the station. The physics student's train is moving towards the east. Since they are moving towards each other, the relative velocity between the chemistry student and the physics student's train is the sum of their velocities.
Step 1: Find the relative velocity between the chemistry student and the physics student's train.
The chemistry student's train's velocity is given as 25 m/s towards the west, and the physics student's train is moving towards the east at the same velocity.
Relative velocity = velocity of chemistry student's train + velocity of physics student's train
Relative velocity = (-25 m/s) + 25 m/s
Relative velocity = 0 m/s
Step 2: Use the Doppler effect formula to find the frequency heard by the chemistry student.
Since the relative velocity is 0 m/s, the observed frequency will be the same as the emitted frequency.
Therefore, the frequency heard by the chemistry student before the two trains pass each other is 3000 Hz.
After the chemistry student's train passes the physics student's train, the relative velocity changes.
Step 1: Find the new relative velocity between the chemistry student and the physics student's train.
After passing each other, the chemistry student's train moves away from the physics student's train. The relative velocity is now the difference between their velocities.
Relative velocity = velocity of chemistry student's train - velocity of physics student's train
Relative velocity = (-25 m/s) - 25 m/s
Relative velocity = -50 m/s (since the chemistry student's train is moving towards the west, relative to the reference frame of the physics student's train)
Step 2: Use the Doppler effect formula to find the frequency heard by the chemistry student.
Applying the same Doppler effect formula:
f' = (343 + 0) / (343 + (-50)) * 3000
f' ≈ 3564 Hz
Therefore, after the chemistry student's train passes the physics student's train, he hears a frequency of approximately 3564 Hz.