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Chapter 12: Problem 14

A person hears the sound of a jet aeroplane after it has passed over his head. The angle of the jet plane with the horizontal when the sound appears to be coming vertically downwards is \(60^{\circ}\). If the velocity of sound is \(v\), then the velocity of the jet plane should be (a) \(2 v\) (b) \(v / \sqrt{3}\) (c) \(\sqrt{3} v\) (d) \(v\)

Short Answer

Expert verified
The velocity of the jet plane should be \(\sqrt{3} v\) (option c).

Step by step solution

01

Understand the Problem

The problem involves sound perception of a jet plane moving at an angle with the horizontal. The key point is the angle where the sound seems to come vertically downwards, and this angle is given as \(60^{\circ}\).
02

Flight Path and Sound Perception

When the sound seems to come vertically downwards, the line of sight to the sound source makes a \(90^{\circ}\) angle with the horizontal. We need to connect the actual path of the plane, the speed of sound, and the perceived sound direction.
03

Identifying Relevant Physics Concept

This scenario is related to the Doppler effect and the geometry of the plane's path. The sound from the jet reaches the observer from a direction when the combination of the jet's velocity and sound velocity forms a right triangle.
04

Form the Geometry Equation

The velocity components create a right triangle where the plane's velocity forms the hypotenuse, and the sound velocity is the opposite side forming a \(90^{\circ}\) angle with the horizontal.
05

Calculate the Velocity of the Jet Plane

For a \(60^{\circ}\) angle of the jet's path with the line of sight when sound is perceived vertically downward, use trigonometric identity: \( \sin(60^{\circ}) = \frac{v}{V}\), where \(V\) is the velocity of the jet plane, and \(v\) is the velocity of sound. Solving gives \( V = \frac{v}{\sin(60^{\circ})} = \frac{v}{\frac{\sqrt{3}}{2}} = \sqrt{3}v\).
06

Verify the Options

Mathematically, we calculated the jet's velocity as \(\sqrt{3} v\). Thus, the correct option is (c).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sound Velocity
Understanding sound velocity is crucial in analyzing exercises related to sound perception. Sound travels as a wave, and its velocity depends on the medium it travels through. In air, at standard temperature and pressure, sound velocity is approximately 343 meters per second (m/s). However, this can change with variations in temperature, humidity, and atmospheric pressure.
Key points to remember about sound velocity in these scenarios:
  • It is the speed at which sound travels through a medium like air.
  • Sound velocity can be influenced by environmental conditions.
  • When dealing with physics problems like this one, usually a constant velocity is assumed unless stated otherwise.
Facts about sound velocity help us understand how sound reaches an observer after some time and how it relates to the Doppler effect and other phenomena in motion.
Jet Plane Speed
In exercises concerning moving objects like a jet plane, its speed is a critical factor in understanding sound perception. The speed determines how quickly a plane covers a distance and indirectly affects the time it takes for sound to be perceived by an observer.
The relation between the plane's speed and sound velocity forms the basis of solving these problems. To comprehend this better:
  • The plane's speed (often much faster than sound) affects how sound waves stretch or compress in the direction of motion.
  • Sound perception by an observer will differ based on whether the plane is moving towards or away from them.
  • For an angle-based problem like this, the plane鈥檚 speed must be calculated relative to the sound velocity using trigonometric identities. In this particular exercise, the plane's velocity is determined to be \( \sqrt{3} v \).
The understanding of such relationships is key in physics, allowing us to predict and explain real-world phenomena.
Trigonometry in Physics
Trigonometry often plays an essential role in physics, especially when analyzing motion and angles. In this exercise, trigonometry is used to calculate the jet plane's velocity concerning the sound velocity.To apply trigonometry effectively:
  • You need to recognize relationships between angles and other physical quantities. For instance, the angle of sound direction relates to the velocity vectors.
  • Using the sine function, the problem describes how the sound velocity appears at a specific angle. \( \sin(60^{\circ}) = \frac{v}{V} \), where \ V \ is the jet plane speed.
  • From this, you can compute the necessary velocity of the plane for a given perceived angle, highlighting the function of trigonometry in solving spatial problems.
By bridging trigonometry with physics, students can efficiently address problems involving angles and velocities.
Angle of Sound Perception
The angle of sound perception is a pivotal concept when analyzing sounds from moving objects. It describes the angle relative to the observer at which the sound appears to be coming.Here, more details:
  • The given problem states that the sound seems to originate vertically downward when the plane is at an angle of \(60^{\circ}\) with the horizontal. This represents a clear relationship between the angle and perceived sound direction.
  • Sound perception angles are associated with how sound waves reach an observer, with Doppler effects influencing the depth and pitch.
  • Understanding these angles helps in figuring out the relative velocity components and calculating the actual speed of moving objects like planes.
The angle of perception doesn't just apply to physics problems but extends to understanding sound in everyday life, such as how we perceive direction and distance of moving sources.

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Most popular questions from this chapter

When two progressive waves of intensity \(I_{1}\) and \(I_{2}\) but slightly different frequencies superpose, the resultant intensity fluctuates between (a) \(\left(\sqrt{I_{1}}+\sqrt{I_{2}}\right)^{2}\) and \(\left(\sqrt{I_{1}}-\sqrt{I_{2}}\right)^{2}\) (b) \(\left(\sqrt{I_{1}}+\sqrt{I_{2}}\right)\) and \(\left(\sqrt{I_{1}}-\sqrt{I_{2}}\right)\) (c) \(\left(I_{1}+I_{2}\right)\) and \(\sqrt{I_{1}-I_{2}}\) (d) \(\frac{I_{1}}{I_{2}}\) and \(\frac{I_{2}}{I_{1}}\)

On sounding tuning fork \(A\) with another tuning fork \(B\) of frequency \(384 \mathrm{~Hz}, 6\) beats are produced per second. After loading the prongs of \(A\) with some wax and then sounding it again with \(B, 4\) beats are produced per second. What is the frequency of the tuning fork \(A\) ? (a) \(388 \mathrm{~Hz}\) (b) \(378 \mathrm{~Hz}\) (c) \(380 \mathrm{~Hz}\) (d) \(390 \mathrm{~Hz}\)

When stationary waves are set up, pick out the correct statement from the following (a) All the particles in the medium are in the same phase of vibration at all times and distances (b) The particles with an interval between two consecutive nodes are in phase, but the particles in two such consecutive intervals are of opposite phase (c) The phase lag along the path of the wave increases as the distance from the source increases (d) Only antinodes are in same phase

In stationary waves, antinodes are the points where there is (a) minimum displacement and minimum pressure change (b) minimum displacement and maximum pressure change (c) maximum displacement and maximum pressure change (d) maximum displacement and minimum pressure change

Two coherent sources must have the same (a) amplitude (b) phase difference only (c) frequency only (d) both (b) and (c)
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