The shock-wave cone created by a space shuttle at one instant during its reentry into the atmosphere makes an angle of 58.0° with its direction of motion. The speed of sound at this altitude is 331 m/s. What is the Mach number of the shuttle at this instant?

Verified step by step guidance

1. Understand that the Mach number is a dimensionless quantity representing the ratio of the speed of an object moving through a fluid and the local speed of sound. It is given by the formula: \( \text{Mach number} = \frac{v}{v_s} \), where \( v \) is the speed of the object and \( v_s \) is the speed of sound.
2. Recognize that the angle of the shock-wave cone, known as the Mach angle, is related to the Mach number by the formula: \( \sin(\theta) = \frac{1}{\text{Mach number}} \), where \( \theta \) is the Mach angle.
3. Given that the Mach angle \( \theta \) is 58.0°, use the formula \( \sin(\theta) = \frac{1}{\text{Mach number}} \) to find the Mach number. Rearrange the formula to solve for the Mach number: \( \text{Mach number} = \frac{1}{\sin(\theta)} \).
4. Substitute the given angle into the equation: \( \text{Mach number} = \frac{1}{\sin(58.0°)} \).
5. Calculate the sine of 58.0° and then compute the Mach number using the rearranged formula. This will give you the Mach number of the shuttle at the given instant.

Key Concepts

Mach Number

Mach number is a dimensionless quantity representing the ratio of the speed of an object to the speed of sound in the surrounding medium. It is used to categorize the speed regime of an object, such as subsonic, transonic, supersonic, or hypersonic. Calculating the Mach number involves dividing the object's speed by the speed of sound at the given altitude.

Shock Wave Cone

A shock wave cone is formed when an object travels through a fluid medium, like air, at a speed greater than the speed of sound, creating a conical region of compressed air. The angle of the cone is determined by the Mach number, with a smaller angle indicating a higher Mach number. This phenomenon is crucial for understanding supersonic flight and the effects of high-speed travel on aerodynamics.

Speed of Sound

The speed of sound is the rate at which sound waves travel through a medium, influenced by factors such as temperature, pressure, and the medium's composition. At sea level, the speed of sound in air is approximately 343 m/s, but it decreases with altitude due to lower temperatures and pressures. Understanding the speed of sound is essential for calculating the Mach number and analyzing the dynamics of objects moving at high speeds.