The Mach number M of a supersonic airplane is the ratio of...

The Mach number $M$ of a supersonic airplane is the ratio of its speed to the speed of sound. When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane. The Mach number is related to the apex angle $\theta$ of the cone by $\sin (\theta / 2)=1 / M$. (a) Use a half-angle formula to rewrite the equation in terms of cos $\theta$. (b) Find the angle $\theta$ that corresponds to a Mach number of 1. (c) Find the angle $\theta$ that corresponds to a Mach number of 4.5. (d) The speed of sound is about 760 miles per hour. Determine the speed of an object with the Mach numbers from parts (b) and (c).

The Mach number $M$ of a supersonic airplane is the ratio of its speed to the speed of sound. When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane. The Mach number is related to the apex angle $\theta$ of the cone by $\sin (\theta / 2)=1 / M$.
(a) Use a half-angle formula to rewrite the equation in terms of cos $\theta$.
(b) Find the angle $\theta$ that corresponds to a Mach number of 1.
(c) Find the angle $\theta$ that corresponds to a Mach number of 4.5.
(d) The speed of sound is about 760 miles per hour. Determine the speed of an object with the Mach numbers from parts (b) and (c).

Key Concepts

Mach Number

The Mach number is a dimensionless quantity in fluid dynamics that represents the ratio of an object's speed to the speed of sound in the surrounding medium. It is crucial in aerodynamics for classifying flow regimes, such as subsonic, transonic, and supersonic speeds, which dictate different physical phenomena like shock wave formation at supersonic speeds.

Trigonometric Half-Angle Formulas

Half-angle formulas in trigonometry allow expressions involving angles to be rewritten in terms of functions of double or half those angles. They are valuable in solving equations where the angle in the trigonometric function needs to be adjusted, as in converting a sine function of half an angle into an expression involving the cosine of the full angle. This technique simplifies the manipulation and solution of trigonometric equations.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to determine the angle that yields a given trigonometric value. In the context of solving equations that involve relationships between speed and angle measures, these functions provide a way to retrieve the physical angle from the computed trigonometric ratio. This is essential for interpreting and applying mathematical results to practical scenarios, such as determining the shock wave cone angle.

Aerodynamic Shock Waves

Supersonic objects generate shock waves when traveling faster than the speed of sound, resulting in a cone-shaped disturbance called the Mach cone. The apex angle of this cone is directly related to the Mach number. This concept is fundamental in aerodynamics and is used to analyze and predict the behavior of objects moving through air at high speeds, as well as to understand phenomena such as sonic booms.

Unit Conversion in Aerodynamics

In aerodynamics, unit conversion is necessary when relating the dimensionless Mach number to actual speeds. Since the Mach number is defined as the ratio of an object's speed to the speed of sound, determining the real-world speed involves multiplying the Mach number by the speed of sound, which is expressed in specific units such as miles per hour or meters per second. This conversion is critical for engineering calculations and practical applications in vehicle design and performance assessment.

Transcript

00:01 We're asked to answer a question about the mock number using multiple angle and product to some formulas.

00:08 We're told the mock number m of a supersonic airplane is the ratio of its speed to the speed of sound.

00:15 When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane.

00:22 The mock number is related to the apex angle theta of the cone by sine of theta over 2 equals 1 over in part a, we're asked to use a half -angle formula to rewrite the equation in terms of cosine of theta.

00:43 Well, let's use the half -angle identity on the left -hand side.

00:51 This is equal to plus or minus the square root of 1 minus the cosine of theta over 2 equals 1 over m.

01:04 So now we want to solve this for cosine of theta.

01:09 To do this, we have a radical equation, so i'll square both.

01:12 Sides, we get 1 minus the cosine of theta over 2 equals 1 over m squared.

01:20 Therefore, 1 minus cosine of theta equals 2 over m squared, and therefore, cosine of theta equals 1 minus 2 over m squared.

01:34 Which is equal to m squared minus 2 over m squared.

01:44 Then in part b, we're asked to find the angle theta that corresponds to a model.

01:49 Number of 1.

01:55 In other words, want to find the value of theta, such that cosine of theta equals 1 squared minus 2 over 1 squared.

02:05 So this is negative 1.

02:10 For a cosine of theta to be equal to negative 1, this implies that theta must be equal to pi.

02:25 In part c, we're asked to find the angle theta that corresponds to a mock number of 4 .5.

02:30 In other words, so the cosine of theta is equal to 4 .5 squared minus 2 over 4 .5 squared.

02:39 Well, 4 .5 is the same as 9 halves...

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