Problem 7 - Algebra 2

Question

AVIATION When a jet travels at speeds greater than the speed of sound, a sonic boom is created by the sound waves foom ing a cone behind the jet. If \(\theta\) is the measure of the angle at the vertex of the cone, then the Mach number \(M\) can be determined using the formula \(\sin \frac{\theta}{2}=\frac{1}{M}\) . Find the Mach number of a jet if a sonic boom is created by a cone with a vertex angle of \(75^{\circ} .\)

Step-by-Step Solution

Answer

The Mach number is approximately 1.64.

Step 1: Understand the Problem

We need to determine the Mach number, M, for a jet creating a sonic boom with a cone vertex angle \(\theta = 75^{\circ}\). We will use the formula \(\sin \frac{\theta}{2} = \frac{1}{M}\), given that the Mach number \(M\) relates to the angle \(\theta\).

Step 2: Calculate \(\frac{\theta}{2}\)

To use the formula, determine \(\frac{\theta}{2}\). Given \(\theta = 75^{\circ}\), then \(\frac{\theta}{2} = \frac{75^{\circ}}{2} = 37.5^{\circ}\).

Step 3: Use Sine Function

Calculate \(\sin 37.5^{\circ}\) using a calculator or a trigonometric table. \(\sin 37.5^{\circ} \approx 0.6088\).

Step 4: Solve for Mach Number \(M\)

Using \(\sin \frac{\theta}{2} = \frac{1}{M}\) and \(\sin 37.5^{\circ} = 0.6088\), solve for \(M\) by rearranging the equation: \(M = \frac{1}{\sin 37.5^{\circ}}\).

Step 5: Calculate \(M\)

Calculate \(M = \frac{1}{0.6088} \approx 1.6439\). Thus, the Mach number is approximately 1.64.

Key Concepts

Mach number

The Mach number is a dimensionless unit that is used to express the speed of an object moving through air, typically in relation to the speed of sound. It plays a crucial role in aviation and aerodynamics. The Mach number is defined as the ratio of the speed of the object to the speed of sound in the surrounding medium:

• When Mach = 1, the object is moving at the speed of sound (called transonic).
• When Mach < 1, the object is moving slower than the speed of sound (subsonic).
• When Mach > 1, the object is moving faster than the speed of sound (supersonic).

For example, if a jet has a Mach number of 1.64, it is traveling 1.64 times the speed of sound. Understanding Mach numbers is essential for designing aircraft as it affects flight dynamics. At Mach 1, the properties of airflow such as density, pressure, and temperature change dramatically, creating shock waves.

Sonic boom

When an aircraft travels faster than the speed of sound, it generates a sonic boom. This phenomenon occurs due to the pressure waves or "shock waves" that build up and merge because the aircraft is disturbing the air at such high speeds. These shock waves form a cone shape around the aircraft.
Sonic booms can be incredibly loud and can cause the sound to reach the ground with a sudden increase in sound level, similar to an explosive noise. They are most commonly associated with

• Supersonic jet aircraft.
• Spaceships re-entering the Earth's atmosphere.

The intensity of a sonic boom is influenced by several factors such as the aircraft's altitude, weight, and the size of the sonic boom cone, which is related to the Mach number.

Sine function

The sine function is a fundamental concept in trigonometry, representing a periodic wave. In the context of airspeed and shock waves, the sine function helps relate the angle of the sonic boom cone to the Mach number. The formula given ewline\[\sin \left(\frac{\theta}{2}\right) = \frac{1}{M} \]shows how the sine of half the vertex angle of the cone can calculate the Mach number, a crucial factor in determining speed.
Sine values are numerical characteristics of these angles:

• For small angles, the sine value increases gradually.
• Sine of 0 degrees is 0, and sine of 90 degrees is 1.

Calculating sine values for specific angles is often done using calculators or trigonometric tables, making this function vital for practical applications in mathematics and science.