/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 53 A jet plane flics overhead at Ma... [FREE SOLUTION] | 91影视

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Chapter 16: Problem 53

A jet plane flics overhead at Mach 1.70 and at a constant altitude of 950 \(\mathrm{m}\) (a) What is the angle \(\alpha\) of the shock-wave cone? (b) How much time after the plane passes directly overhead do you hear the sonic boom? Neglect the variation of the speed of sound with altitude.

Short Answer

Expert verified
(a) \( \alpha \approx 36.87^\circ \). (b) Sonic boom heard after \( 3.69 \) seconds.

Step by step solution

01

Understanding the Mach Angle

The Mach number is the ratio of the speed of the object to the speed of sound in the medium. The angle of the shock-wave cone, known as the Mach angle \( \alpha \), can be determined using the sine function: \( \sin(\alpha) = \frac{1}{\text{Mach number}} \). Here, the Mach number is 1.70.
02

Calculating the Mach Angle

By substituting the given Mach number into the sine function, \( \sin(\alpha) = \frac{1}{1.70} \approx 0.5882 \). To find \( \alpha \), take the arcsine: \( \alpha = \arcsin(0.5882) \). Using a calculator, \( \alpha \approx 36.87 \) degrees.
03

Understanding Time for Sonic Boom

To find out when the sonic boom is heard, calculate the time it takes for the shock wave to reach the observer on the ground after the plane passes overhead. The horizontal distance of the plane from the observer when the boom is heard forms a right triangle with the altitude of the plane.
04

Calculating Horizontal Distance

In the right triangle formed, the tangent of angle \( \alpha \) is the ratio of the altitude to the horizontal distance \( d \). Thus, \( \tan(36.87^\circ) = \frac{950 \text{ m}}{d} \). Solving for \( d \), \( d = \frac{950 \text{ m}}{\tan(36.87^\circ)} \approx 1267 \text{ m} \).
05

Calculating Time for Sonic Boom

The time delay \( t \) for hearing the sonic boom after the plane passes directly overhead is the horizontal distance divided by the speed of sound. Assuming the speed of sound is approximately \( 343 \text{ m/s} \), \( t = \frac{1267 \text{ m}}{343 \text{ m/s}} \approx 3.69 \text{ seconds} \).

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Key Concepts

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

Shock Wave
Shock waves are powerful, rapid disturbances that occur when an object moves faster than the speed of sound. When a jet plane, for example, flies at such high speeds, it compresses air molecules together forming a high-pressure wave. This gathered energy travels outward as a shock wave, commonly visualized as a cone-shaped pattern surrounding the object. The angle of this shock wave, or Mach angle, is crucial in understanding these wave formations.
The shock wave's Mach angle becomes more acute (narrower) as the speed of the object increases. This is because the faster the object travels, the faster the pressure wave propagates outward.
Key aspects of shock waves include:
  • Formation at speeds exceeding the speed of sound, leading to a rapid pressure change and energy release.
  • Cone-shaped patterns, which form due to the rapidity of pressure buildup.
  • Effects such as turbulence, which is often dangerous for nearby structures due to the explosive nature.
Understanding shock waves helps engineers improve aircraft design and safety by reducing potential turbulence impacts.
Sonic Boom
A sonic boom is the thunder-like sound associated with shock waves when an object travels through the air faster than the speed of sound. It is a direct result of the shock waves created by the supersonic object, like an airplane.
As the object compresses air at high speeds, it forms a continuous series of pressure waves. When these waves merge, they create a sharp and loud sound, or a sonic boom.
Some key points about sonic booms include:
  • Sonic booms occur when the shock waves reach the ground, compressing air and creating a sudden change in pressure.
  • The intensity of a sonic boom depends on the distance from the object and its speed.
  • Efforts are often made to mitigate sonic booms in populated areas to avoid structural damage and noise pollution.
Understanding the impact of sonic booms is vital for minimizing their effects and ensuring safety regulations are adhered to in aerospace operations.
Speed of Sound
The speed of sound is the rate at which sound waves travel through a medium, such as air. It is a crucial factor in aerodynamics and acoustics because it determines how quickly sound signals are perceived by humans and animals alike.
On average, the speed of sound in air is approximately 343 meters per second (m/s) at standard temperature and pressure (20掳C and 1 atmosphere). However, this value can vary based on environmental conditions.
Some influencing factors include:
  • Temperature: Sound travels faster in warmer air because the molecules are more active, facilitating quicker wave propagation.
  • Density: Higher density mediums, like water, allow sound to travel faster compared to less dense mediums like gas.
  • Humidity: More humid air has a slightly higher speed of sound due to the presence of lighter water vapor molecules.
Understanding the speed of sound is essential in designing technology and predicting how sound is experienced in various atmospheric conditions.
Mach Number
The Mach number is a dimensionless unit used to quantify the speed of an object relative to the speed of sound in the surrounding medium. It serves as an important metric in aerodynamics to assess whether an object is traveling below, at, or above the speed of sound.
A Mach number is calculated as the ratio of the object's speed to the speed of sound, given by the formula:\[\text{Mach Number} = \frac{\text{Object Speed}}{\text{Speed of Sound}}\]This ratio aids in categorizing flight regimes, which include:
  • Subsonic (Mach <1): Speeds lower than the speed of sound.
  • Transonic (Mach ~1): Speeds approximately equal to the speed of sound, often exhibiting a combination of subsonic and supersonic flow patterns.
  • Supersonic (Mach >1): Speeds greater than the speed of sound, leading to the formation of shock waves and potential sonic booms.
  • Hypersonic (Mach >5): Extremely high speeds that can cause significant aerodynamic heating and complex fluid dynamics.
The understanding and application of Mach numbers are critical in designing air and space vehicles to ensure optimal performance and safety.

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Most popular questions from this chapter

Ultrasound in Medicine. \(\quad\) A \(2.00-\mathrm{MHz}\) sound wave travels through a pregnant woman's abdomen and is retiected from the fotal heart wall of her unborn baby. The heart wall is moving toward the sound receiver as the heart beats. The reflected sound is then mixed with the transmitted sound, and 85 beats per second are detected. The speed of sound in body tissue is 1500 \(\mathrm{m} / \mathrm{s}\) . Calculate the speed of the fetal heart wall at the instant this measurement is made.

Example 16.1 (Section 16.1\()\) showed that for sound waves in air with frequency 1000 \(\mathrm{Hz}\) , a displacement amplitude of\( 1.2 \times 10^{-8} \mathrm{m}\) produces a pressure amplitude of \(3.0 \times 10^{-2} \mathrm{Pa}\) . Water at \(20^{\circ} \mathrm{C}\) has a bulk modulus of \(2.2 \times 10^{9} \mathrm{Pa}\) , and the speed of sound in water at this temperature is 1480 \(\mathrm{m} / \mathrm{s}\) . For \(1000-\mathrm{Hz}\) sound waves in \(20^{\circ} \mathrm{C}\) water, what displacement amplitude is produced if the pressure amplitude is \(3.0 \times 10^{-2}\) Pa? Explain why your answer is much less than \(1.2 \times 10^{-8} \mathrm{Pa}\) .

The fundamental frequency of a pipe that is open at both ends is 594 Hz (a) How long is this pipe? If one end is now closed, find (b) the wavelength and (c) the frequency of the new fundamental.

The shock-wave cone created by the space shuttle at one instant during its reentry into the atmosphere makes an angle of \(58.0^{\circ}\) with its direction of motion. The speed of sound at this altitude is 331 \(\mathrm{m} / \mathrm{s}\) (a) What is the Mach number of the shuttle at this instant, and (b) how fast (in \(\mathrm{m} / \mathrm{s}\) and \(\mathrm{mi} / \mathrm{h} )\) is it traveling relative to the atmosphere? (c) What would be its Mach number and the angle of its shock-wave cone if it flew at the same speed but at low altitude where the speed of sound is 344 \(\mathrm{m} / \mathrm{s} ?\)

Two loudspeakers, \(A\) and \(B,\) are driven by the same amplifier and emit sinusoidal waves in phase. The frequency of the waves emitted by each speaker is 860 \(\mathrm{Hz}\) Point \(P\) is 12.0 \(\mathrm{m}\) from \(A\) and 13.4 \(\mathrm{m}\) from \(B .\) Is the interference at \(P\) constructive or destructive? Give the reasoning behind your answer.
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