Question

The shock-wave cone created by a 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 m/s. (a) What is the Mach number of the shuttle at this instant, and (b) how fast (in m/s and in mi/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 m/s ?

Short answer

(a) Mach 1.179, (b) 390.3 m/s (873.4 mi/h), (c) Mach 1.135, angle 61.9°.

Step-by-step solution

Understand the Problem

The problem gives us the angle of the shock-wave cone, the speed of sound, and asks us to find the Mach number at a given altitude, the shuttle's speed, and the Mach number and angle at a different speed of sound.

Find the Mach Number (a)

The Mach angle (\( \theta \)) is related to the Mach number (\( M \)) by the formula: \( \sin \theta = \frac{1}{M} \). Given \( \theta = 58.0^\circ \), we calculate the Mach number: \[M = \frac{1}{\sin(58^\circ)} \approx 1.179 \].

Calculate the Speed of the Shuttle (b)

Using the Mach number, we calculate the speed of the shuttle: \[v = M \times v_{sound} = 1.179 \times 331 \text{ m/s} \approx 390.3 \text{ m/s} \].To convert this speed into miles per hour (mi/h), use the conversion factor: \[1 \text{ m/s} = 2.237 \text{ mi/h} \]. Thus, \[390.3 \text{ m/s} \times 2.237 \approx 873.4 \text{ mi/h} \].

Calculate Mach Number at Lower Altitude (c)

At lower altitude, where the speed of sound is 344 m/s, the Mach number is given by: \[M_{low} = \frac{v}{v_{sound\_low}} = \frac{390.3 \text{ m/s}}{344 \text{ m/s}} \approx 1.135 \].

Determine the New Shock-Wave Cone Angle (c)

With the new Mach number at low altitude, use \( \sin \theta_{low} = \frac{1}{M_{low}} \) to find the new angle: \[\sin \theta_{low} = \frac{1}{1.135} \approx 0.881 \].Thus, \[\theta_{low} = \sin^{-1}(0.881) \approx 61.9^\circ \].

Key concepts

Shock-Wave Cone

When a space shuttle or any object travels faster than the speed of sound, it creates a shock-wave cone. This is a three-dimensional wave pattern that forms due to the abruptness of pressure changes in the surrounding air. Essentially, as the object moves through the atmosphere, it compresses the air molecules in front of it, forming a cone-shaped wave that trails behind.
This shock-wave cone is closely related to the Mach number, which is the ratio of the object's speed to the speed of sound.

• If the Mach number is greater than 1, the object is traveling supersonically, and a shock-wave cone forms.
• The angle of the cone (\( \theta \)) is directly linked to the Mach number using the formula: \( \sin \theta = \frac{1}{M} \), where \( M \) is the Mach number.

As the Mach number increases, the angle of the shock-wave cone narrows, indicating faster speeds. For instance, a shuttle with a Mach number of 1.179 at a specific altitude would have a shock-wave cone angle of 58 degrees. This visual effect is prominent during high-speed flight, like reentry into Earth's atmosphere, when shock waves become visible as part of a bright plasma trail.

Speed of Sound

The speed of sound is a critical factor in understanding how shock waves and Mach numbers work. It is the speed at which sound waves travel through a medium, such as air, and is affected by various factors including temperature and pressure.
During reentry or high-speed flight, the local speed of sound determines how fast an object can travel before it becomes supersonic. In the provided example, the speed of sound at a particular altitude is 331 m/s.

• The speed of sound increases with temperature; hence, at lower altitudes where it's warmer, the speed would typically be higher, at around 344 m/s.
• This difference affects the Mach number calculation. An object traveling at the same speed but experiencing different local speeds of sound would have different Mach numbers.

The interplay between the vehicle's speed and the speed of sound determines whether it creates a sonic shock wave, which is why variations in altitude and temperature can alter the character and intensity of these waves.

Reentry into the Atmosphere

Reentry is a critical phase during a space shuttle’s journey. When the shuttle reenters Earth's atmosphere from space, it transitions from very high vacuum-like conditions to the dense layers of the atmosphere. This drastic change necessitates slowing down from orbital speeds exceeding the local speed of sound, leading to the creation of intense shock waves.
During reentry, the shuttle must handle severe aerodynamic heating caused by compression of atmospheric gases at incredibly high speeds.

• The Mach number, representing the shuttle's speed divided by the local speed of sound, is crucial in analyzing these conditions.
• At higher altitudes, the air is thinner, and the speed of sound is lower; as a result, the Mach number can be quite high before the shuttle reaches the denser lower atmosphere where it decreases.

Understanding the mechanics of shock-wave cones and the speed of sound helps engineers design better thermal protection systems and improve the shuttle's aerodynamic stability during this challenging phase. Thus, the study of vibration, heat, and energy dissipation during reentry plays a significant role in ensuring the success and safety of space missions.