Question

A stationary source is emitting sound at a fixed frequency \(\mathrm{f}_{0}\), which is reflected by two cars approaching the source. The difference between the frequencies of sound reflected from the cars is \(1.2 \%\) of \(\mathrm{f}_{0} .\) What is the difference in the speeds of the cars (in \(\mathrm{km}\) per hour) to the nearest integer ? The cars are moving at constant speeds much smaller than the speed of sound which is \(330 \mathrm{~ms}^{-1}\).

Short answer

To determine the difference in the speeds of the cars in km/h, use the Doppler Effect formulas to derive an equation for the speed difference and approximate for a stationary source and moving observer. Then, solve the equation and convert the resulting speed difference from m/s to km/h.

Step-by-step solution

Understand the Doppler Effect

The Doppler Effect is a change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. It is important here because the frequency changes when sound waves are reflected off of moving objects, in this case, the cars.

Calculate the Frequency Change

According to the Doppler Effect, the frequency observed will change based on relative speeds. Since the difference in frequencies is 1.2% of the original frequency, we can express this as \(\Delta f = 0.012 f_0\).

Express Frequencies for Each Car

Let \(f_1\) and \(f_2\) be the frequencies of sound reflected from the two cars. Assume \(f_1 > f_2\) without loss of generality. The difference between these frequencies is the change \(\Delta f\), so we have \(f_1 - f_2 = \Delta f\).

Derive the Formula with Observed Frequencies

You can use the formula for the observed frequency due to the Doppler Effect when a source is stationary and the observer is moving: \(f' = f_0\left(\frac{c}{c \pm v} \right)\), where \(c\) is the speed of sound and \(v\) is the velocity of the car (positive if approaching, negative if receding).

Write the Equations for Both Cars

Using the formula, we write \(f_1 = f_0\left(\frac{c}{c - v_1}\right)\) and \(f_2 = f_0\left(\frac{c}{c - v_2}\right)\), where \(v_1\) and \(v_2\) are the speeds of the first and second car respectively.

Equate the Difference in Frequencies to the Difference in Speeds

Since we know \(f_1 - f_2 = \Delta f\), substitute the expressions for \(f_1\) and \(f_2\) to get an equation: \(f_0\left(\frac{c}{c - v_1}\right) - f_0\left(\frac{c}{c - v_2}\right) = 0.012 f_0\). Simplify this to find an expression for \(v_1 - v_2\) in terms of \(f_0\) and \(c\).

Simplification and Solving for Speed Difference

Simplify the equation and solve for \(v_1 - v_2\). Note that since \(v_1\) and \(v_2\) are much smaller than \(c\), we can approximate \(\frac{c}{c - v} \approx 1 + \frac{v}{c}\). Using this approximation makes it possible to solve for the difference in speeds.

Convert the Speed Difference to km/h

After solving for \(v_1 - v_2\) in \(m/s\), convert this speed difference into kilometers per hour by using the conversion factor \(1 \mathrm{m/s} = 3.6 \mathrm{km/h}\).

Key concepts

Acoustic Frequency Shift

The phenomenon where the frequency of sound waves changes due to the motion of the source or the observer is known as the acoustic frequency shift. For example, the pitch of a siren sounds higher as an ambulance approaches you and lower as it drives away. This effect is quantified using the Doppler Effect, which states that the observed frequency increases if the source and observer are moving closer together and decreases if they move apart.

When applied to our exercise problem involving two cars and a stationary sound source, we find that the frequency of the sound reflects differently from each car due to their unique speeds. This variability in speed results in a difference in frequencies observed, which is quantified as a percentage of the original frequency emitted by the stationary source. By calculating the frequency shift, we can dedetermine the difference in the speeds of the cars relative to the sound source.

Relative Motion in Waves

Understanding relative motion in waves is crucial when discussing wave phenomena such as the Doppler Effect. Relative motion tells us that the characteristics of wave interaction—like frequency or wavelength—depend on the movement of the source relative to the observer. In the context of sound waves, when a wave source and an observer are in motion relative to each other, the frequency of the waves detected by the observer will be different from those emitted by the source.

For our exercise, we can assume that one car might be moving faster towards the source than the other. This relative difference in motion causes each car to reflect sound waves with slightly altered frequencies, leading us to the conclusion that the cars are moving at different speeds based on the frequency shift observed.

Wave Propagation

Wave propagation refers to the manner in which waves travel through a medium. For sound waves, which are mechanical waves, they require a medium like air or water to travel. These waves move outward from the source in all directions, and their behavior changes when they encounter obstacles, boundaries, or other media. In our exercise problem, we consider sound waves propagating through the air, encountering, and reflecting off moving cars.Propagation comes into play as we explore how frequency shifts when waves reflect from a moving object. With a stationary sound source and two cars reflecting the sound back towards that source, we need to account for how these waves propagate through air and are perceived differently due to the relative motion of the cars.

Speed of Sound

The speed of sound is the velocity at which sound waves travel through a given medium and is a fundamental constant in problems involving acoustic phenomena. In air at room temperature, this speed is approximately 343 meters per second (m/s). However, the speed of sound can change based on factors such as temperature, humidity, and the medium itself. Knowing this constant is essential for solving Doppler Effect problems.

In our textbook exercise, we use the given value of 330 m/s as the speed of sound to solve for the difference in car speeds. This value allows us to determine how fast the sound waves reflect off the moving cars and find their relative velocities by combining the concept of the acoustic frequency shift with the known speed of sound. With the correct application of this constant, calculating the difference in frequencies becomes a straightforward process, ultimately leading to the difference in the speeds of the cars.