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Chapter 16: Problem 52

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 \(\mathrm{m} / \mathrm{s} ?\)

Short Answer

Expert verified
Mach number at high altitude is 1.179; speed is 390.8 m/s or 874 mi/h. At low altitude, Mach number is 1.136, angle is 60.0掳.

Step by step solution

01

Understanding Mach Number and Shockwave Angle

The angle of the shock-wave cone, known as the Mach angle \( \theta \), is related to the Mach number \( M \) by the relation \( \sin(\theta) = \frac{1}{M} \). We are given \( \theta = 58.0^\circ \). We will use this to find the Mach number.
02

Calculating Mach Number at High Altitude

Using the relation \( \sin(58.0^\circ) = \frac{1}{M} \), calculate the Mach number \( M \). Thus, \( M = \frac{1}{\sin(58.0^\circ)} \). Compute the sine of the angle and find \( M \).
03

Formula Application

Compute \( \sin(58.0^\circ) \approx 0.848 \). Therefore, \( M \approx \frac{1}{0.848} \approx 1.179 \).
04

Calculating Shuttle Speed at High Altitude

The Mach number \( M \) is defined by \( M = \frac{v}{v_s} \), where \( v \) is the speed of the shuttle and \( v_s = 331 \ \mathrm{m/s} \) is the speed of sound at high altitude. Solve for \( v \): \( v = M \times v_s \approx 1.179 \times 331 \ \mathrm{m/s} \approx 390.8 \ \mathrm{m/s} \).
05

Converting Speed to Miles per Hour

Convert the speed from \( \mathrm{m/s} \) to \( \mathrm{mi/h} \) using the conversion factor \( 1 \ \mathrm{m/s} \approx 2.237 \ \mathrm{mi/h} \). Thus, \( v \approx 390.8 \times 2.237 \approx 874 \ \mathrm{mi/h} \).
06

Recomputing Mach Number at Low Altitude

At low altitude, where the speed of sound is \( 344 \ \mathrm{m/s} \), the Mach number is recalculated using \( M = \frac{v}{344} \). Since the speed \( v \approx 390.8 \ \mathrm{m/s} \), \( M \approx \frac{390.8}{344} \approx 1.136 \).
07

Calculating Shock-Wave Cone Angle at Low Altitude

Determine the new angle \( \theta \) at low altitude using \( \sin(\theta) = \frac{1}{M} \). Thus, \( \theta = \sin^{-1}\left( \frac{1}{1.136} \right) \approx 60.0^\circ \).
08

Review and Summary

At high altitude, the Mach number is approximately 1.179, and the speed is 390.8 m/s or 874 mi/h. At low altitude, the Mach number is approximately 1.136, and the shock-wave cone angle is about 60.0掳.

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Key Concepts

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

Shock-Wave Cone
When an object travels faster than the speed of sound, it generates a shock wave. This shock wave appears as a conical shape around the object and is known as the shock-wave cone. The angle of this cone can provide insights into the object's speed relative to the speed of sound. This angle, called the Mach angle, helps us understand how pressurized sound waves are displaced due to the object's high speed. The sharper the cone, the faster the object is traveling compared to the speed of sound. The angle is crucial in aerodynamics, providing engineers with necessary information regarding supersonic aircraft design.
Speed of Sound
The speed of sound represents how fast sound waves travel through a medium such as air. At standard sea level conditions, the speed of sound is approximately 344 meters per second (m/s). However, it can vary due to temperature, humidity, and pressure changes. At higher altitudes, the speed of sound may be lower due to the density and temperature differences in the atmosphere. Understanding the speed of sound is crucial, as it determines the transition point between subsonic and supersonic flight regimes. This transition has direct implications on an aircraft's design and structure.
Mach Angle
The Mach angle is a direct measure of the amount by which a shock wave created by an object in supersonic flight leads the object. It is calculated using the relationship \sin(\theta) = \frac{1}{M}\, where \( \theta \) is the Mach angle, and \( M \) is the Mach number of the object. This angle decreases as the Mach number increases, indicating a tighter shock-wave cone. For aerodynamics specialists, the Mach angle helps in understanding wave drag and other aerodynamic forces acting on a vehicle at high speeds. It is one of the key parameters in assessing the performance of high-speed aircraft and rockets.
Speed Conversion
When converting speed, especially in physics and engineering contexts, it is important to ensure accuracy for different measurement systems. For instance, converting the shuttle speed from meters per second (m/s) to miles per hour (mi/h) requires the use of a conversion factor. The factor commonly used is 1 m/s 鈮� 2.237 mi/h. Accurate conversion is critical in making calculations understandable and useful across various contexts, such as when reporting flight speeds to international audiences or aligning to regulatory standards from different countries.
Altitude Effects
Altitude has a significant impact on various properties of flight, particularly the speed of sound. At higher altitudes, air density and temperature decrease, which in turn affects the speed of sound. For instance, the speed of sound is 331 m/s at higher altitudes, compared to 344 m/s closer to sea level. This variance must be taken into account when determining the Mach number because the same actual speed in m/s can correspond to different Mach numbers depending on altitude. Understanding these effects allows pilots and engineers to optimize aircraft performance for efficiency and safety.
Supersonic Speed
Supersonic speed is achieved when an object travels faster than the speed of sound in the surrounding medium. This regime introduces various physical phenomena, such as sonic booms, which are explosive sounds created by the shock waves. Supersonic speeds significantly influence how aircraft are designed and operated. They require specialized materials and shapes to handle the intense aerodynamic forces. Additionally, going supersonic means considering aspects such as temperature changes, drag increase, and changes in lift forces. It is a fascinating area that challenges engineers to innovate continually in search of more efficient and safer high-speed travel options.

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Most popular questions from this chapter

The intensity due to a number of independent sound sources is the sum of the individual intensities. (a) When four quadruplets cry simultaneously, how many decibels greater is the sound intensity level than when a single one cries? (b) To increase the sound intensity level again by the same number of decibels as in part (a), how many more crying babies are required?

A long tube contains air at a pressure of 1.00 atm and a temperature of \(77.0^{\circ} \mathrm{C}\) . The tube is open at one end and closed at the other by a movable piston. A tuning fork near the open end is vibrating with a frequency of 500 \(\mathrm{Hz}\) . Resonance is produced when the piston is at distances \(18.0,55.5,\) and 93.0 \(\mathrm{cm}\) from the open end. (a) From these measurements, what is the speed of sound in alr at \(77.0^{\circ} \mathrm{C} ?\) (b) From the result of part (a), what is the value of \(\gamma ?\) (c) These data show that a displacement antinode is slightly outside of the open end of the tube. How far outside is it?

Find the fundamental frequency and the frequency of the first three overtones of a pipe 45.0 \(\mathrm{cm}\) long (a) if the pipe is open at both ends and (b) if the pipe is closed at one end. Use \(v=344 \mathrm{m} / \mathrm{s} .\) (c) For each of these cases, what is the number of the highest harmonic that may be heard by a person who can hear frequencies from 20 \(\mathrm{Hz}\) to \(20,000 \mathrm{Hz}\) ?

Moving Source vs. Moving Listener. (a) A sound source producing \(1.00-\mathrm{kHz}\) waves moves toward a stationary listener at one-half the speed of sound. What frequency will the listener hear? (b) Suppose instead that the source is stationary and the listener moves toward the source at one- half the speed of sound. What frequency does the listener hear? How does your answer compare to that in part (a)? Explain on physical grounds why the two answers differ.

A soprano and a bass are singing a duet. While the soprano \(\operatorname{sings}\) an \(A^{\prime \prime}\) at 932 . Hz, the bass sings an \(A^{\prime \prime}\) but three octaves lower. In this concert hall, the density of air is 1.20 \(\mathrm{kg} / \mathrm{m}^{3}\) and its bulk modulus is \(1.42 \times 10^{5} \mathrm{Pa}\) . In order for their notes to have the same Rnund intensity level, what must he (a) the ratio of the pressur amplitude of the bass to that of the soprano, and (b) the ratio of the displacement amplitude of the bass to that of the soprano? (c) What displacement amplitude (in \(\mathrm{m}\) and \(\mathrm{nm}\) ) does the soprano produce to sing her \(A^{\prime \prime}\) at 72.0 \(\mathrm{dB} ?\)
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