Problem 78 - Physics

Question

Consult Multiple-Concept Example 11 in order to review a model for solving this type of problem. A car is accelerating while its horn is blowing. Just after the car passes a stationary person, the person hears a frequency of \(966.0 \mathrm{~Hz}\). Fourteen seconds later, the frequency heard by the person has decreased to \(912.0 \mathrm{~Hz}\). When the car is stationary, its horn emits a sound whose frequency is \(1.00 \times 10^{3} \mathrm{~Hz}\). The speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). What is the acceleration of the car?

Step-by-Step Solution

Answer

The car's acceleration is the change in velocity over time, based on Doppler shift equations.

Step 1: Use the Doppler Effect Formula

The problem involves the Doppler effect, which causes a change in the frequency of a sound as heard by an observer moving relative to the source of the sound. The formula for the observed frequency \( f' \) when the source is moving away from the observer is: \[ f' = \frac{f}{1 + \frac{v_s}{v}} \] where \( f \) is the original frequency of the sound, \( v_s \) is the speed of the source, and \( v \) is the speed of sound.

Step 2: Set Up Equations for Initial and Later Frequencies

Initially, the frequency heard by the person is \( 966.0 \) Hz and fourteen seconds later, it is \( 912.0 \) Hz. We set up two equations using the Doppler effect using these frequencies.\[966.0 = \frac{1000}{1 + \frac{v_{s1}}{343}} \] for the initial frequency and \[912.0 = \frac{1000}{1 + \frac{v_{s2}}{343}} \] for the frequency after 14 seconds.

Step 3: Solve for Speeds of the Car

From the equations in Step 2, solve for the velocities \( v_{s1} \) and \( v_{s2} \) corresponding to the initial and later frequencies. \[\begin{align*}966.0 &= \frac{1000}{1 + \frac{v_{s1}}{343}} \v_{s1} &= 343 \left(\frac{1000}{966.0} - 1\right) \912.0 &= \frac{1000}{1 + \frac{v_{s2}}{343}} \v_{s2} &= 343 \left(\frac{1000}{912.0} - 1\right)\end{align*}\]

Step 4: Calculate the Acceleration of the Car

Use the change in velocities and the time interval to find the acceleration. The acceleration \( a \) is defined as the change in velocity \( \Delta v \) over the time \( \Delta t \): \[ a = \frac{v_{s2} - v_{s1}}{14} \] where 14 seconds is the time duration between the two frequency measurements.

Key Concepts

Understanding Sound Frequency in the Doppler Effect

The Doppler Effect is an interesting phenomenon that explains why sound waves seem to change pitch or frequency as the source of the sound moves relative to an observer. It comes into play in various situations like when a car horn fades as a vehicle recedes into the distance. In this exercise, the sound frequency changes from an original value to a lower frequency as the car accelerates away from the stationary observer. Let's break down the concept further.

• Original Frequency: This is the frequency of the sound when the source is stationary. In our case, it's initially set as 1000 Hz for a stationary car horn.
• Observed Frequency: As the car moves, formula adjustments predict that the frequency (or pitch) heard by the observer will change. Initially, this frequency is 966 Hz and after some time it decreases to 912 Hz, indicating the car is moving away.
• Frequency Shift: This shift helps determine the speed at which the car is moving and, subsequently, its acceleration.

Understanding these frequencies is crucial to solving physics problems involving the Doppler Effect and provides insight into velocity changes of objects such as speeding vehicles.

Acceleration and Its Role in Doppler Problems

Acceleration is essential in physics for understanding how the velocity of an object changes over time. In this example, the car is not moving at a consistent speed but is accelerating, meaning its velocity is increasing as it moves away from the observer.
The rate of acceleration can be determined by the change in speed divided by the time it occurs over. Here’s how it works in this problem context:

• Initially, as soon as the car passes the observer, a particular frequency of 966 Hz is noted.
• 14 seconds later, the frequency drops to 912 Hz.
• This frequency drop indicates a change in speed or velocity over these 14 seconds.

Understanding acceleration allows us to determine how quickly an object's speed changes, which is vital for problems involving moving sources and the Doppler Effect.

Velocity Calculation Using the Doppler Effect

Calculating velocity in Doppler Effect scenarios involves understanding how frequency shifts provide insight into the speed of a moving object. We use these shifts to find out the speed of the car at two different points.

• Equation Setup: Using the observed frequencies and the known speed of sound ( 343 m/s), equations are set up to solve for the velocities at both times, before and after the 14-second interval.
• Finding Initial and Final Speeds: For the initial frequency of 966 Hz, we find the car's initial speed, and similarly, for a later frequency of 912 Hz, we get the car's speed after 14 seconds.

Solving these equations with the Doppler formula lets us estimate these speeds, which are crucial for finding the car's acceleration.

Physics Problem Solving with the Doppler Effect

Physics problems like this one often utilize a systematic approach for solutions. To effectively solve a Doppler Effect problem, a step-by-step method is generally employed. Here is a streamlined approach to tackle such problems:

• Identify Known Values: Start by noting all the known values such as the initial frequency, observed frequencies, and the speed of sound.
• Apply the Doppler Effect Formula: Use the formulated equations for each frequency measurement to determine the respective velocities.
• Calculate Speed: Extract the speeds from these frequency changes using algebraic manipulation of the Doppler formula.
• Determine Acceleration: Use the velocity changes and time difference to calculate acceleration, offering insights into movement dynamics.

Through understanding and applying these structured steps, one can efficiently solve complex physics problems involving the Doppler Effect. This systematic approach not only aids in problem-solving but also deepens comprehension of underlying concepts.