Problem 56 - University Physics with Modern Physics

Question

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 in \(\mathrm{mi} / \mathrm{h} )\) 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 \(\mathrm{m} / \mathrm{s} ?\)

Step-by-Step Solution

Answer

(a) Mach 1.18; (b) 390.58 m/s or 873.47 mi/h; (c) Mach 1.14, angle 61.1°.

Step 1: Understanding the Relationship

The angle of the shock-wave cone, known as the Mach angle \( \theta \), is related to the Mach number \( M \) by the formula \( \sin(\theta) = \frac{1}{M} \). We are given \( \theta = 58.0^{\circ} \). First, convert the angle into radians, which is unnecessary here directly since we are using trigonometric functions assuming they handle degree inputs or convert within calculator settings.

Step 2: Calculate Mach Number

Using the formula \( \sin(58.0^\circ) = \frac{1}{M} \), we find \( \sin(58.0^\circ) = 0.848 \). Thus, the Mach number is given by \( M = \frac{1}{0.848} \approx 1.18 \).

Step 3: Calculate Speed of Shuttle

The speed of the shuttle \( v \) can be found by multiplying the Mach number \( M \) by the speed of sound at that altitude, i.e., \( v = M \times 331 \ \mathrm{m/s} \approx 1.18 \times 331 \ \mathrm{m/s} \approx 390.58 \ \mathrm{m/s} \).

Step 4: Conversion to Miles per Hour

Convert the speed from \( \mathrm{m/s} \) to \( \mathrm{mi/h} \) using the conversion factor \( 1 \ \mathrm{m/s} = 2.237 \ \mathrm{mi/h} \). Thus, \( v \approx 390.58 \times 2.237 \ \mathrm{mi/h} \approx 873.47 \ \mathrm{mi/h} \).

Step 5: Mach Number at Low Altitude

At low altitude, the speed of sound is \( 344 \ \mathrm{m/s} \). Assuming the shuttle travels at the same speed \( 390.58 \ \mathrm{m/s} \), the Mach number is recalculated as \( M = \frac{390.58}{344} \approx 1.14 \).

Step 6: Mach Angle at Low Altitude

Using the formula \( \sin(\theta) = \frac{1}{M} \) again, \( \sin(\theta) = \frac{1}{1.14} \approx 0.877 \). Solving for \( \theta \), \( \theta = \sin^{-1}(0.877) \approx 61.1^\circ \).

Key Concepts

Shock-Wave Cone

When an object travels through the air at speeds faster than the speed of sound, it creates a series of pressure waves known as shock-waves. These waves merge into a single, powerful shock-wave, with a distinct cone shape trailing behind the moving object. This is known as the shock-wave cone. The angle of this cone, called the Mach angle, can tell us a lot about the object's speed relative to the speed of sound.

For instance, when the space shuttle makes a 58-degree angle with the shock-wave, it tells us that the shuttle is traveling faster than sound. The equation for the Mach angle is given by \( \sin(\theta) = \frac{1}{M} \), where \( M \) is the Mach number. This relation shows how the angle decreases as the object's speed increases. It's a fascinating insight into high-speed aerodynamics!

Speed of Sound

The speed of sound is a crucial parameter in determining how fast an object is moving in relation to the medium it is traveling through. In the air, this speed varies with altitude and air temperature. At sea level, the speed of sound is approximately 343 meters per second (m/s), but it can decrease as the altitude increases.

At the altitude where the original exercise takes place, the speed of sound is given as 331 m/s. Knowing the speed of sound allows us to compute the Mach number, which tells us how many times faster than the speed of sound an object is moving. To find this, simply divide the object's speed by the speed of sound at that specific altitude. The result lets scientists categorize the speed into different regimes: subsonic, transonic, supersonic, and hypersonic, each explaining different flight characteristics.

Atmosphere Reentry

Reentry into the Earth's atmosphere is a critical phase for any spacecraft. During reentry, a vehicle usually travels at high speeds, significantly exceeding the speed of sound. This high-speed travel through atmospheric layers can generate enormous heat and pressure. The spacecraft design must accommodate these extreme conditions to protect it and its occupants or cargo.

During reentry, phenomena such as shock-wave formation play a crucial role. The angle and behavior of the shock-wave can impact the forces experienced by the vehicle. Understanding the aero-thermodynamics of reentry ensures that engineers can design spacecraft that withstands the immense forces and heat experienced during this phase, ensuring a safe return to Earth. This is where precise calculations about Mach number and shock-wave interactions become vital.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles and sides of triangles. They are instrumental in various fields, including physics and engineering, especially when dealing with wave and oscillation concepts.

In the context of the exercise, understanding the sin function is crucial. It's used to relate the Mach angle \( \theta \) to the Mach number \( M \) via the equation \( \sin(\theta) = \frac{1}{M} \). This function takes an angle in the shock-wave cone equation and allows the calculation of the Mach number, demonstrating how trigonometric concepts are key in solving problems involving angles and ratios.

• Sine: Opposite side over the hypotenuse in a right-angle triangle.
• Cosine: Adjacent side over the hypotenuse.
• Tangent: Opposite side over the adjacent side.

Mastering these functions helps in various aerospace calculations, depicting how mathematics describes real-world scenarios.