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Chapter 16: Problem 72

(II) An airplane travels at Mach 2.0 where the speed of sound is \(310 \mathrm{m} / \mathrm{s},(a)\) What is the angle the shock wave makes with the direction of the airplane's motion? \((b)\) If the plane is flying at a height of \(6500 \mathrm{m},\) how long after it is directly over- head will a person on the ground hear the shock wave?

Short Answer

Expert verified
(a) 30掳 (b) Approximately 21 seconds.

Step by step solution

01

Calculate the speed of the airplane

Mach number is defined as the ratio of the speed of an object to the speed of sound. Given, the airplane travels at Mach 2.0 and the speed of sound is 310 m/s. The speed of the airplane is calculated by multiplying the Mach number by the speed of sound: \[v_a = ext{Mach} \times v_{sound} = 2.0 \times 310 \, \text{m/s} = 620 \, \text{m/s}\]
02

Determine the angle of the shock wave

The angle \(\theta\) made by the shock wave with the direction of motion of the airplane is given by the sine of the angle, \[ \sin \theta = \frac{v_{sound}}{v_a} = \frac{310}{620} = 0.5. \] Using the inverse sine function, \[\theta = \sin^{-1}(0.5) = 30^\circ.\]
03

Calculate the time for shock wave to reach the ground

The time \(t\) taken for the shock wave to travel vertically to the ground is determined using the height of the airplane and the speed of sound. Given airplane height \(h = 6500 \, \text{m}\), and speed of sound \(v_{sound} = 310 \, \text{m/s}\),\[t = \frac{h}{v_{sound}} = \frac{6500}{310} \approx 20.97 \, \text{seconds}.\]

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Key Concepts

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

Mach Number
The Mach number is a crucial concept in the study of aerodynamics and is particularly relevant when discussing shock waves. It is a dimensionless quantity that represents the ratio between the speed of an object and the speed of sound in the surrounding medium. This means:
  • If an airplane travels at Mach 1, it is moving at the speed of sound.
  • If it travels at Mach 2, like in our exercise, it's moving twice the speed of sound.
In the given exercise, the airplane is flying at Mach 2.0 and the speed of sound is 310 meters per second, making the airplane's speed 620 meters per second. Understanding Mach numbers is essential when considering how different speeds impact the behavior of shock waves and how they interact with mediums like air.
Mach numbers help engineers and scientists predict airflow characteristics, noise, and aerodynamic heating around high-speed vehicles.
Speed of Sound
The speed of sound is the speed at which sound waves propagate through different materials. It varies depending on factors such as air temperature and altitude. Under standard conditions, the speed of sound in air is around 343 meters per second, but in this exercise, we have it as 310 m/s.
  • This lower value might reflect conditions at a higher altitude or a specific day.
  • The speed of sound is important because it helps define Mach numbers, which are used to label how fast objects travel through air.
The concept of speed of sound is linked to pressure, temperature, and medium density. In our exercise, knowing the speed of sound allows us to calculate the airplane's speed and then analyze the shock wave pattern produced by the plane traveling at Mach 2.
Inverse Sine Function
In mathematics, the inverse sine function, also known as arcsine, is used to determine angles from their sine values. In problems involving shock waves, this concept becomes essential. For instance, the angle (\(\theta\)) formed by the shock wave depends directly on the speed of the airplane and the speed of sound.
  • In the exercise, we use the inverse sine function to find the angle by setting \( \sin \theta = \frac{310}{620} \).
  • This calculation resulted in \( \theta = 30^\circ \), representing the cone angle of the shock wave relative to the airplane's path.
This angle, termed the "Mach angle," provides key insights into the shock wave pattern that occurs when an object moves faster than sound. The ability to use inverse functions to solve these types of physics problems is a powerful mathematical tool.

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Most popular questions from this chapter

For large concerts, loudspeakers are sometimes used to amplify a singer's sound. The human brain interprets sounds that arrive within \(50 \mathrm{~ms}\) of the original sound as if they came from the same source. Thus if the sound from a loudspeaker reaches a listener first, it would sound as if the loudspeaker is the source of the sound. Conversely, if the singer is heard first and the loudspeaker adds to the sound within \(50 \mathrm{~ms}\), the sound would seem to come from the singer, who would now seem to be singing louder. The second situation is desired. Because the signal to the loudspeaker travels at the speed of light \(\left(3 \times 10^{8} \mathrm{~m} / \mathrm{s}\right),\) which is much faster than the speed of sound, a delay is added to the signal sent to the loudspeaker. How much delay must be added if the loudspeaker is \(3.0 \mathrm{~m}\) behind the singer and we want its sound to arrive 30 ms after the singer's?

(I) A piano tuner hears one beat every 2.0 \(\mathrm{s}\) when trying to adjust two strings, one of which is sounding 370 \(\mathrm{Hz} .\) How far off in frequency is the other string?

A source emits sound of wavelengths \(2.64 \mathrm{~m}\) and \(2.72 \mathrm{~m}\) in air. (a) How many beats per second will be heard? (Assume \(T=20^{\circ} \mathrm{C} .\) ) \((b)\) How far apart in space are the regions of maximum intensity?

A uniform narrow tube \(1.80 \mathrm{~m}\) long is open at both ends. It resonates at two successive harmonics of frequencies \(275 \mathrm{~Hz}\) and \(330 \mathrm{~Hz}\). What is \((a)\) the fundamental frequency, and \((b)\) the speed of sound in the gas in the tube?

A tuning fork is set into vibration above a vertical open tube filled with water (Fig. 16-40). The water level is allowed to drop slowly. As it does so, the air in the tube above the water level is heard to resonate with the tuning fork when the distance from the tube opening to the water level is \(0.125 \mathrm{~m}\) and again at \(0.395 \mathrm{~m}\). What is the frequency of the tuning fork?
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