Question

A vehicle with a horn of frequency \(\mathrm{n}\) is moving with a velocity of \(30 \mathrm{~m} / \mathrm{s}\) in a direction perpendicular to the straight line joining the observer and the vehicle. The observer perceives the sound to have a frequency \(\left(\mathrm{n}+\mathrm{n}_{1}\right) .\) If the sound velocity in air is \(300 \mathrm{~m} / \mathrm{s}\), then \(\ldots \ldots . .\) (A) \(\mathrm{n}_{1}=10 \mathrm{n}\) (B) \(\mathrm{n}_{1}=0\) (C) \(\mathrm{n}_{1}=0.1 \mathrm{n}\) (D) \(\mathrm{n}_{1}=-0 . \ln\)

Short answer

The correct answer is (C) \(\mathrm{n}_{1} = 0.1 \mathrm{n}\).

Step-by-step solution

Understanding the Doppler effect formula

The Doppler effect formula relates the observed frequency (f' ) to the source frequency (f), the speed of sound (v), the speed of the source (vs), and the speed of the observer (vo). The formula is given as follows: \(f'=f\left(\frac{v\pm vo}{v\mp vs}\right)\) Here, f' = observed frequency, f = source frequency, v = speed of sound, vo = speed of the observer, and vs = speed of the source. In this case, the source is the vehicle and the observer is stationary. Hence, vo = 0 m/s.

Apply the given values to the Doppler effect formula

We are given: n = source frequency, n + n1 = observed frequency, v = 300 m/s (speed of sound), and vs = 30 m/s (speed of the source, the vehicle). Since the vehicle is moving perpendicular to the observer and the observer is stationary, the Doppler effect formula becomes: \((\mathrm{n}+\mathrm{n}_{1})=n\left(\frac{v}{v-vs}\right)\)

Solve for n1

Now we have: \((\mathrm{n}+\mathrm{n}_{1})=n\left(\frac{300}{300-30}\right)\) Divide both sides by n: \(\frac{(\mathrm{n}+\mathrm{n}_{1})}{n} = \frac{300}{(300-30)}\) \((1+\frac{\mathrm{n}_{1}}{\mathrm{n}})=\frac{300}{270}\) \(\frac{\mathrm{n}_{1}}{\mathrm{n}}=\frac{300}{270}-1\) \(\frac{\mathrm{n}_{1}}{\mathrm{n}}=0.1\)

Compare with the given options

We are to find which of the given options is correct: (A) \(\mathrm{n}_{1} = 10 \mathrm{n}\) (B) \(\mathrm{n}_{1} = 0\) (C) \(\mathrm{n}_{1} = 0.1 \mathrm{n}\) (D) \(\mathrm{n}_{1} = -0 . \ln\) Comparing the answer, \(\frac{\mathrm{n}_{1}}{\mathrm{n}} = 0.1\), with the given options, we can see that the correct answer is (C) \(\mathrm{n}_{1} = 0.1 \mathrm{n}\).

Key concepts

Sound Frequency

Sound frequency is a crucial element in understanding the nature of sound waves. It refers to the number of vibrations or cycles per second, measured in Hertz (Hz). In the context of the Doppler Effect, the source frequency, often referred to as the intrinsic frequency of a sound-emitting object like a car horn, is key. This frequency remains constant from the source’s perspective.

However, when an observer is in a different relative motion compared to the sound source, they might perceive this frequency as different. This apparent frequency perceived is influenced by the movement of the source and the observer, leading to either a higher or lower perceived frequency. In the exercise, the horn of the vehicle had a source frequency denoted by \(n\). Understanding how this frequency interacts with other variables in the Doppler Effect helps in predicting what the observer will hear.

Speed of Sound

The speed of sound is the speed at which sound waves travel through a medium. In air, at room temperature and at sea level, the speed of sound is typically about 343 meters per second. However, a typical assumption for simplicity in many problems is 300 meters per second, as seen in this exercise.

• Temperature and medium affect the speed: Sound travels faster in warmer air and generally faster in water than in air.
• Constancy in calculations: For doppler effect problems, assuming a constant speed of sound simplifies the calculations.

In our scenario, the speed of sound was given as 300 m/s. This is a foundational element in the Doppler Effect equation which determines how the frequency is perceived by an observer.

Velocity of Source

The velocity of a sound source has a significant impact on the frequency observed by an observer. It refers to the speed at which the source emitting the sound, such as a car horn, is moving. In this exercise, the velocity of the source was 30 m/s in a direction perpendicular to the line between the observer and the source.

• The direction of movement: If the source moves towards the observer, the sound waves compress and the frequency increases.
• Perpendicular movement: In this case, as the vehicle moves perpendicularly, the Doppler effect is minimized or nullified as there is no effective change in distance along the line connecting the source and the observer.

Since the velocity of the source affects how sound waves reach the observer, it is a vital factor in calculating the observed frequency using the Doppler Effect formula.

Observed Frequency

Observed frequency is the frequency detected by an observer, which can differ from the actual frequency emitted by the source due to relative motion. It is the key element predicted by the Doppler Effect formula, which calculates how events like a moving vehicle can alter perception of sound.

• Observer’s standpoint: If the observer is stationary as in our problem, the frequency change depends solely on the source’s movement.
• Calculation via formula: The relation \( f' = f \left( \frac{v}{v - v_s} \right) \) was used to find the observed frequency in the provided solution.

In this problem, the observer perceived a frequency of \( n + n_1 \), indicating a slight change due to the movement of the vehicle. By accounting for the given speed values, the correct option revealed is \( n_1 = 0.1n \), meaning a 10% increase of the source frequency was heard.