Problem 114 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Suppose an airplane flying faster than sound goes directly over you. Assume that the plane is flying at a constant altitude. At the instant you feel the sonic boom from the plane, the angle of elevation to the plane is $$\alpha=2 \arcsin \frac{1}{m}$$ where \(m\) is the Mach number of the plane's speed. (The Mach number is the ratio of the speed of the plane to the speed of sound.) Find \(\alpha\) to the nearest degree for each value of \(m\) (a) \(m=1.2\) (b) \(m=1.5\) (c) \(m=2\) (d) \(m=2.5\)

Step-by-Step Solution

Answer

(a) 113°, (b) 84°, (c) 60°, (d) 47°.

Step 1: Understand the Formula for \( \alpha \)

To find the angle of elevation \( \alpha \), we use the formula \( \alpha = 2 \arcsin \frac{1}{m} \), where \( m \) is the Mach number. This tells us the angle at which the sonic boom is heard when a plane moves faster than the speed of sound.

Step 2: Calculate \( \alpha \) for \( m = 1.2 \)

Substitute \( m = 1.2 \) into the formula: \( \alpha = 2 \arcsin \frac{1}{1.2} \). Calculate \( \frac{1}{1.2} = 0.8333 \). Find \( \arcsin(0.8333) \approx 56.44^\circ \). Therefore, \( \alpha = 2 \times 56.44^\circ \approx 112.88^\circ \). Round to the nearest degree: \( \alpha = 113^\circ \).

Step 3: Calculate \( \alpha \) for \( m = 1.5 \)

Substitute \( m = 1.5 \) into the formula: \( \alpha = 2 \arcsin \frac{1}{1.5} \). Calculate \( \frac{1}{1.5} = 0.6667 \). Find \( \arcsin(0.6667) \approx 41.81^\circ \). Therefore, \( \alpha = 2 \times 41.81^\circ \approx 83.62^\circ \). Round to the nearest degree: \( \alpha = 84^\circ \).

Step 4: Calculate \( \alpha \) for \( m = 2 \)

Substitute \( m = 2 \) into the formula: \( \alpha = 2 \arcsin \frac{1}{2} \). Calculate \( \frac{1}{2} = 0.5 \). Find \( \arcsin(0.5) = 30^\circ \). Therefore, \( \alpha = 2 \times 30^\circ = 60^\circ \). No rounding needed: \( \alpha = 60^\circ \).

Step 5: Calculate \( \alpha \) for \( m = 2.5 \)

Substitute \( m = 2.5 \) into the formula: \( \alpha = 2 \arcsin \frac{1}{2.5} \). Calculate \( \frac{1}{2.5} = 0.4 \). Find \( \arcsin(0.4) \approx 23.58^\circ \). Therefore, \( \alpha = 2 \times 23.58^\circ \approx 47.16^\circ \). Round to the nearest degree: \( \alpha = 47^\circ \).

Key Concepts

Mach Number

The Mach number is a dimensionless quantity representing the speed of an object relative to the speed of sound in the surrounding medium. It is named after the Austrian physicist Ernst Mach. Understanding the Mach number is crucial in aviation and aerospace engineering when dealing with high-speed travel such as supersonic and hypersonic flights. Here are some key aspects:

• A Mach number of 1 indicates the speed of sound.
• Supersonic speeds are greater than Mach 1.
• A Mach number less than 1 signifies the object is moving slower than sound.

The formula for the Mach number is usually given as:\[M = \frac{v}{c}\]where \(v\) is the velocity of the object and \(c\) is the speed of sound. Students working on trigonometry problems involving Mach numbers usually deal with angles of elevation using this concept. Beyond mathematics, this knowledge helps in understanding phenomena such as sonic booms.

Angle of Elevation

The angle of elevation is the angle between the horizontal line from the observer and the line of sight to an object above the horizontal. It is a common application in trigonometry used to determine heights and distances indirectly. Here’s how you can think about it:

• It is measured upwards from the horizontal.
• Commonly used in problems involving heights and distances.

When considering the context of a sonic boom and Mach numbers, the formula provides the angle at which the sound is first heard when an object travels at supersonic speed.Formula:\[\alpha = 2 \arcsin \frac{1}{m}\]This computes the real-world angle at which the observer senses the sonic boom as a plane travels faster than sound, offering a practical application of trigonometry in physics.

Sonic Boom

A sonic boom is a loud sound associated with the shock waves created when an object exceeds the speed of sound. This phenomenon is why you sometimes hear a sudden explosive noise when jets fly overhead at high speeds. Here’s what happens during a sonic boom:

• Air pressure waves build up around the object as it travels at supersonic speeds.
• The pressure from the shock waves creates the explosive sound.
• It's often compared to the sound of thunder during a storm.

The angle of elevation to the plane determines how and when we experience the sonic boom in relation to the object's Mach number. Understanding this is key in areas like military aviation and engineering where managing sound levels is crucial.

Arcsine Function

The arcsine function is one of the inverse trigonometric functions, often denoted as \(\arcsin\). It is the inverse of the sine function, providing the angle that corresponds to a given sine value. Key points include:

• Its range is from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
• It is typically used to calculate angles from given ratios.

When solving exercises involving angles of elevation for sonic booms, \(\arcsin\) helps find the angle \(\alpha\) by rearranging the sine equation. This function is essential for interpreting trigonometric relationships, especially when exact angle measures dictate physical events, such as the precise hearing of a sonic boom as affected by the Mach number.