Question

A policeman with a very good ear and a good understanding of the Doppler effect stands on the shoulder of a freeway assisting a crew in a 40 -mph work zone. He notices a car approaching that is honking its horn. As the car gets closer, the policeman hears the sound of the horn as a distinct \(\mathrm{B} 4\) tone \((494 \mathrm{~Hz}) .\) The instant the car passes by, he hears the sound as a distinct \(\mathrm{A} 4\) tone \((440 \mathrm{~Hz})\). He immediately jumps on his motorcycle, stops the car, and gives the motorist a speeding ticket. Explain his reasoning.

Short answer

Answer: Yes, the car was speeding. The calculated speed of the car was approximately 59.7 mph, which is above the 40-mph speed limit in the work zone.

Step-by-step solution

Doppler effect formula for sound

We will use the Doppler effect formula for sound, which is given as: \[ f' = \frac{f(v \pm v_o)}{v \pm v_s}, \] where \(f'\) is the observed frequency, \(f\) is the source frequency, \(v\) is the speed of sound in the medium (air), \(v_o\) is the speed of the observer (policeman) and \(v_s\) is the speed of the source (car honking the horn). The positive sign is used when the observer and the source are moving towards each other, while the negative sign is used when they are moving apart. Since we are given the observed frequencies (B4 tone and A4 tone), the speed of the policeman, and we can assume the speed of sound in the air, we can solve for the speed of the car to find out if it was speeding.

Determine the observer's speed

The policeman is stationary while listening to the car approaching. Hence, the speed of the observer (policeman) \(v_o\) is 0 mph.

Speed of sound

We can assume the speed of sound in air \(v\) to be approximately 767 mph (about 1235 km/h).

Solving for the car's speed

When the car is approaching the policeman: \[ 494 \text{ Hz (observed frequency)} = \frac{440 \text{ Hz (source frequency)}(767 + 0 \text{ mph})}{767 \text{ mph} - v_s \text{ mph}} .\] We can solve for the speed of the car (\(v_s\)) as it approaches the policeman: \[ v_s = 767 - \frac{440(767)}{494} .\] Calculating the value, we get: \[ v_s \approx 59.7 \text{ mph}. \] The car was speeding as it approached the policeman, as 59.7 mph is above the 40-mph speed limit in the work zone. Hence, the policeman's reasoning for issuing a speeding ticket is justified based on the change in frequencies from the Doppler effect.

Key concepts

Doppler Effect Formula

The Doppler effect is a commonly observed phenomenon where the frequency of a wave, such as sound, changes based on the relative motion between the source and the observer. It's essential in many fields, including astronomy, radar technology, and even in everyday situations like hearing an ambulance siren change in pitch as it passes by.

The fundamental formula for the Doppler effect in acoustics is given by: \[ f' = \frac{f(v \pm v_o)}{v \pm v_s}, \]where \(f'\) represents the observed frequency, \(f\) is the frequency of the sound the source emits, \(v\) is the speed of sound in the medium, \(v_o\) is the velocity of the observer, and \(v_s\) is the velocity of the source. The correct sign is chosen based on the relative direction of motion, with '+' used when closing the distance and '−' when increasing it. This formula allows for the quantitative analysis of how the perceived pitch of a sound changes as the distance between the source and observer changes.

Speed of Sound

The speed of sound is not a constant; it depends on the medium through which the sound is traveling and the conditions of that medium, such as temperature, humidity, and air pressure. In air, at sea level, under standard conditions of temperature (15 degrees Celsius) and pressure, the speed of sound is roughly 1235 km/h or 767 mph.

However, these conditions can fluctuate, causing variations in sound speed. For example, sound travels faster in warm air and slower in cold air. Understanding the speed of sound is pivotal when applying the Doppler effect formula, as it defines how quickly the sound waves are reaching the observer after being emitted by the source.

Frequency Change

In the context of the Doppler effect, the frequency change is the difference between the observed frequency \(f'\) and the emitted frequency \(f\). This shift is what a listener discerns as a change in pitch. For a sound source approaching an observer, the waves are compressed, leading to a higher frequency or pitch; conversely, for a receding source, the waves are stretched out, resulting in a lower frequency or pitch.

Understanding this frequency change is crucial for various applications, such as in radar gun technology used by law enforcement to determine the speed of a vehicle. By measuring the change in frequency of a radar wave reflected off a moving object, one can derive the speed of that object. In our textbook problem, the officer applies this principle to sound waves to estimate the speed of a speeding car.

Acoustic Phenomena

Acoustic phenomena encompass all events related to sound and its propagation through different mediums. The Doppler effect is just one example of such phenomena. Others include the resonance of sound in musical instruments, echolocation used by animals like bats and dolphins, and the sonic boom created when an object moves faster than the speed of sound.

The study of acoustic phenomena often involves exploring how sound waves interact with environments, influencing everything from the design of concert halls to noise reduction in automobiles. Understanding these principles not only helps in problem-solving, like with the policeman's deduction, but also in enhancing technology and life experiences where sound plays a significant role.