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. (a) What is the Mach number of the shuttle at this instant

Verified step by step guidance

1. Identify the given angle (θ) between the shock-wave cone and the direction of motion of the shuttle, which is 58.0°.
2. Recall the relationship between the Mach number (M), the speed of the object (v), and the speed of sound (a) in the medium, which is given by the formula: \( M = \frac{v}{a} \).
3. Use the formula for the angle of the shock wave cone in relation to the Mach number: \( \sin(\theta) = \frac{1}{M} \).
4. Rearrange the formula to solve for the Mach number: \( M = \frac{1}{\sin(\theta)} \).
5. Substitute the given angle (θ = 58.0°) into the formula to calculate the Mach number of the shuttle.

Key Concepts

Mach Number

The Mach number is a dimensionless quantity representing the ratio of an object's speed to the speed of sound in the surrounding medium. It is calculated by dividing the object's speed by the speed of sound. A Mach number greater than 1 indicates supersonic speed, while a Mach number less than 1 indicates subsonic speed. Understanding Mach number is crucial for analyzing high-speed aerodynamics, especially in aerospace applications.

Shock Waves

Shock waves are disturbances that travel through a medium, such as air, at supersonic speeds. They occur when an object moves faster than the speed of sound, compressing the air in front of it and creating a cone-shaped wave pattern. The angle of the shock wave cone is related to the Mach number, with larger angles corresponding to higher Mach numbers. This concept is essential for understanding the behavior of objects like space shuttles during reentry.

Speed of Sound

The speed of sound is the speed at which pressure waves propagate through a medium, such as air, and is influenced by factors like temperature and altitude. At sea level and at 20°C, the speed of sound is approximately 343 m/s, but it decreases with altitude due to lower temperatures. Knowing the speed of sound is vital for calculating the Mach number and understanding the dynamics of objects moving through the atmosphere.