Problem 23 - College Physics

Question

A commuter train passes a passenger platiorm at a constant speed of \(40.0 \mathrm{~m} / \mathrm{s}\). The train horn is sounded at its characteristic frequency of \(320 \mathrm{~Hz}\). (a) What overall change in frequency is detected by a person on the platform as the train moves from approaching to receding? (b) What wavelength is detected by a person

Step-by-Step Solution

Answer

The overall change in frequency detected by the person on the platform as the train moves from approaching to receding is the subtraction of the frequency calculated in Step 2 from that in Step 1. The wavelength detected by the person can be calculated using the frequency obtained from the approaching train in Step 1.

Step 1: Calculating Frequency as the Train Approaches

The formula for frequency when the source is moving towards the observer is given by \( f' = f \cdot \frac{v+v_s}{v} \), where \( f' \) is the frequency heard by the observer, \( f \) is the source frequency, \( v \) is the speed of sound in air (343 m/s), and \( v_s \) is the velocity of the source (40 m/s). Substituting the given values, calculate \( f' \).

Step 2: Calculating Frequency as the Train Recedes

The formula for frequency when the source is moving away from the observer is given by \( f' = f \cdot \frac{v-v_s}{v} \). Using the same symbols as in the previous step, calculate \( f' \) for the receding train.

Step 3: Total Change in Frequency

Subtract the frequency calculated in Step 2 from the frequency calculated in Step 1 to find the total change in frequency as the train approaches and recedes from the observer.

Step 4: Calculate the Wavelength

The wavelength can be calculated using the formula \( \lambda = \frac{v}{f'} \). We use the frequency calculated in Step 1, as the observer would detect this frequency while the train is approaching.

Key Concepts

Frequency Change

The Doppler Effect is all about how waves, like sounds, change frequency when they come towards us or move away. Imagine hearing a train's horn as it rushes past you. First, the sound becomes higher, and then it gets lower. This change is due to the Doppler Effect.
When a train approaches, the sound waves are compressed, causing the frequencies to increase. We use the formula:

• \( f' = f \cdot \frac{v+v_s}{v} \)

Here, \( f' \) is the frequency heard by the observer. \( f \) is the actual frequency which is the source frequency known as 320 Hz. \( v \) is the speed of sound in air, usually 343 m/s, and \( v_s \) is the speed of the train coming at us, which is 40 m/s.
For an approaching train, this means the frequency increases.
Once the train starts moving away, those waves spread out, causing the frequencies to decrease. Here's the formula:

• \( f' = f \cdot \frac{v-v_s}{v} \)

Using these formulas, we can calculate the frequencies when the train moves towards and then away from us. The total frequency change is the difference between the two calculated frequencies.
The result is a noticeable difference in the pitch of the sound we hear.

Wavelength Calculation

Wavelengths are all about measuring distances between waves. Think of it as the length from one wave crest to the next. When using wavelengths in sound, we calculate based on how the sound is heard.
To figure out the wavelength as the train approaches, we use:

• \( \lambda = \frac{v}{f'} \)

"Lambda" \( \lambda \) represents the wavelength. \( v \) is the speed of sound, which is 343 m/s, and \( f' \) is the frequency heard by the person on the platform while the train approaches. When you plug in the frequency calculated using the Doppler Effect formula for an approaching source, you get the wavelength.
This helps us know how far apart the sound waves are as the observer perceives them while the train moves closer. It isn't just about the sounds we hear but how those sounds occupy and move through space.

Speed of Sound

The speed of sound is a constant that we often use in sound wave calculations. It's the speed at which sound waves travel through the air. In normal conditions, it is around 343 meters per second (m/s). This might vary slightly due to temperature and air pressure.
Understanding this speed is crucial for solving Doppler Effect problems because it serves as a reference point. Whether you're calculating the frequency change or the wavelength, the speed of sound is the denominator in the equations we use.

• It's important for predicting how sounds travel through different mediums.
• Knowing it helps us calculate both how fast sound reaches us and how it changes when it moves.

When combined with the velocity of a moving object, like a train in our exercise, it allows us to apply formulas and better understand how waves interact with motion.
Without understanding the speed of sound, we wouldn't accurately measure changes in frequency or wavelength, especially in real-time environments, such as trains passing by.