Problem 26 - College Physics

Question

At rest, a car's horn sounds the note A \((440 \mathrm{~Hz})\). The horn is sounded while the car is moving down the street. A bicyclist moving in the same direction with one-third the car's speed hears a frequency of \(415 \mathrm{~Hz}\). (a) Is the cyclist ahead of or behind the car? (b) What is the speed of the car?

Step-by-Step Solution

Answer

The cyclist is ahead of the car. The speed of the car is approximately 15.94 m/s.

Step 1: Understand the Doppler effect

The Doppler effect formula for sound wave can be written as: \( f' = f \frac{(v + v_0)}{(v - v_s)} \) where \( f' \) is the observed frequency, \( f \) is the source frequency, \( v \) is the speed of sound, \( v_0 \) is the speed of the observer and \( v_s \) is the speed of the source. Positive values of \( v_0 \) and \( v_s \) are used when the motion is towards the other object. If the object moves away, the speed should be negative.

Step 2: Identify directional movement

From given cyclist heard frequency (415 Hz) we can know the fact that because the cyclist hears a lower frequency, this means that the cyclist is moving away from the car. The cyclist is therefore ahead of the car.

Step 3: Calculate the speed of the car

Let's denote that V_c is the speed of the car and V_b is the speed of the bicyclist. The given tells us that V_b=V_c/3. We can substitute this into the Doppler-effect equation \( 415 = 440 \frac{(343+V_c/3)}{(343-V_c)} \). Solving the equation for \( V_c \), we find that V_c is approximately 15.94 m/s.

Key Concepts

Sound Frequency

In the context of the Doppler Effect, sound frequency is how often the sound wave's compressions reach an observer. Frequency is usually measured in Hertz (Hz), which tells us the number of wave cycles per second. When a sound source moves relative to an observer, the frequency heard by the observer is altered. This change is what we know as the Doppler Effect.

A sound with a higher frequency sounds like a higher pitch, while a lower frequency sounds like a lower pitch. In the example given, the car horn has a rest frequency of 440 Hz, which corresponds to the MUSICAL NOTE A. However, when the car (the sound source) is moving, the observer, in this case, a bicyclist, hears a different frequency. Instead of 440 Hz, the cyclist hears 415 Hz, indicating the sound waves' behavior differs due to their relative motion.

Observer Motion

The movement of the observer can significantly influence the perceived frequency. In the given problem, the observer is the bicyclist. Their motion compared to the source (car) alters the frequency of the sound they hear.

Since the cyclist hears a lower frequency than the car emits, it suggests they are moving ahead and away from the car. If they were moving toward the car, the observed frequency would be higher. This scenario is a classic illustration of an observer moving relative to a sound source, impacting how sound waves are compressed or stretched. The observer's motion is thus pivotal in explaining why frequencies seem higher or lower, dependent on the direction of movement.

Sound Wave Behavior

Sound waves change behavior when either the source or the observer is in motion. This shift in behavior can be observed as a change in frequency, where wave fronts become squeezed together for an approaching sound source, or spread apart for a receding one.

In our exercise, the car speeds down the street, affecting the sound waves it emits. Because the observer (cyclist) is moving away, each wave front takes slightly longer to reach them compared to someone stationary. This is why a lower frequency is heard. The waves are stretching out, a characteristic marker of the Doppler Effect related to motion. Such knowledge of sound wave behavior is crucial in understanding everyday phenomena, like how an approaching vehicle's horn changes in pitch as it moves past and away from you.