Problem 10 - Applied Physics

Question

What frequency is heard by an observer who hears the \(45 \overline{0}\) - \(\mathrm{Hz}\) siren on a police car traveling at \(35 \mathrm{~m} / \mathrm{s}\) away from her? Assume that the velocity of sound in air is \(345 \mathrm{~m} / \mathrm{s}\)

Step-by-Step Solution

Answer

The observed frequency is approximately 409 Hz.

Step 1: Understand the Doppler Effect Formula

The frequency heard by the observer when there is relative motion between the source and the observer can be calculated using the Doppler effect formula:\[ f' = \frac{f \times (v + v_o)}{v + v_s} \]Where:- \( f' \) is the observed frequency.- \( f \) is the source frequency (450 Hz in this case).- \( v \) is the velocity of sound in air (345 m/s).- \( v_o \) is the velocity of the observer (0 m/s, since the observer is stationary).- \( v_s \) is the velocity of the source (35 m/s, and it's moving away from the observer).

Step 2: Identify Given Values

Extract the given values from the problem statement:- Source frequency, \( f = 450 \) Hz- Velocity of sound, \( v = 345 \) m/s- Velocity of observer, \( v_o = 0 \) m/s- Velocity of source, \( v_s = 35 \) m/s (since the source is moving away from the observer, this will be considered as positive in the denominator).

Step 3: Plug Values into the Doppler Effect Formula

Insert the given values into the Doppler effect formula:\[ f' = \frac{450 \times (345 + 0)}{345 + 35} \]

Step 4: Simplify the Expression

Calculate the expression:\[ f' = \frac{450 \times 345}{380} \]

Step 5: Calculate the Observed Frequency

Perform the arithmetic operation to find \( f' \):\[ f' = \frac{155250}{380} \approx 408.55 \text{ Hz} \]Rounding, the observed frequency is approximately \( 409 \) Hz.

Key Concepts

Frequency Calculation

When dealing with sound, the frequency refers to how many wave cycles occur in a second. It is measured in Hertz (Hz). Calculating the observed frequency involves determining what frequency a stationary observer hears when a source of sound is moving.
The frequency calculation for the Doppler effect can be simplified to:

• If the source and observer are moving closer, the observed frequency increases; this is known as a blue shift.
• If they move apart, the frequency decreases, signaling a red shift.

In our problem, the police car siren has a frequency of 450 Hz but is moving away from the observer at 35 m/s.
To calculate the new frequency heard, we use the Doppler effect formula: \[ f' = \frac{f \times (v + v_o)}{v + v_s} \] Where:- \( f' \) is the observed frequency - \( f \) is the source frequency - \( v \) is the velocity of sound - \( v_o \) is the observer's velocity (stationary in this case) - \( v_s \) is the source velocity (away from the observer, hence considered positive as per convention)
Using the provided parameters, the process becomes straightforward and results in the observer noting a lower frequency of approximately 409 Hz.

Velocity of Sound

The velocity of sound is a crucial factor in understanding how sound waves propagate through different mediums. It indicates how fast the sound waves travel and can be affected by the medium's temperature, density, and phase (solid, liquid, or gas).
For this exercise, the sound speed in air is given as 345 m/s, presumed under standard atmospheric conditions. Understanding how velocity influences frequency is important:

• A higher velocity of the sound medium would typically suggest quicker propagation, altering how quickly sound waves reach the observer.
• Conversely, lower velocities indicate slower wave travel, changing the timing and possibly the frequency name frequency an observer experiences.

The speed of 345 m/s helps us calculate the frequency changes when the sound source moves relative to the observer. It provides a base for the ratio calculations in the Doppler effect formula, making it essential to accurately plug into your calculations.

Observer Effect

The observer effect in the context of the Doppler effect can be understood as how motion affects sound perception. Here, the observer remains stationary, impacting wave interaction with the incoming sound.
The key impact happens when the observer changes position or speed, aligning with or diverging from the sound source. In this exercise, the observer does not move, but the source does. Thus, changes in the perceived frequency primarily come from the relative approach or separation by the sound source, like the police car moving away. Observers usually hear:

• A higher frequency if approaching or being approached by the sound source.
• A lower frequency if the distance increases, much like receding car sirens.

Variables like wind can also inadvertently act as observer motion corrections, subtly altering the perceived frequency with respect to the observer's position.

Source and Observer Motion

The interaction between the motion of the source and the observer significantly impacts the observed frequency. Understanding their motion types is essential:

• Source moving towards the observer results in higher frequencies (sounds approaching faster).
• Source retreating from the observer leads to lower frequencies (sounds moving away slower).

In our given problem, the police car is moving away from the observer, which means the frequency the observer hears is less than the actual emitted frequency.
By moving at 35 m/s away, it causes a noticeable drop recorded at roughly 409 Hz instead of the 450 Hz emitted frequency. The interaction of such motion scenarios is core to understanding Doppler effect applications across various scientific fields, including astronomy and radar monitoring.