Problem 9 - College Physics

Question

On March \(27,2004,\) the United States successfully tested the hypersonic X-43A scramjet, which flew at Mach 7 (seven times the speed of sound) for 11 s. (a) At this rate, how many minutes would it take such a scramjet to carry passengers the approximately \(5000 \mathrm{~km}\) from San Francisco to New York? (Use the speed of sound at \(\left.0^{\circ} \mathrm{C}, 331 \mathrm{~m} / \mathrm{s} .\right)\) (b) How many kilometers did the scramjet travel during its 11 s test?

Step-by-Step Solution

Answer

(a) Approximately 36 minutes. (b) Approximately 25.49 km.

Step 1: Calculate Mach 7 speed

Mach 7 means travelling at 7 times the speed of sound. Given that the speed of sound is 331 m/s, the speed of the scramjet is: \[ 7 \times 331 \, \text{m/s} = 2317 \, \text{m/s} \]

Step 2: Convert speed to km/s

To convert the speed to kilometers per second, divide the speed in meters per second by 1000:\[ \frac{2317 \, \text{m/s}}{1000} = 2.317 \, \text{km/s} \]

Step 3: Determine travel time for 5000 km

To find the time in seconds to travel 5000 km at the speed of 2.317 km/s, use the formula:\[ \text{Time (s)} = \frac{5000 \, \text{km}}{2.317 \, \text{km/s}} \approx 2158.67 \, \text{s} \]

Step 4: Convert travel time to minutes

Convert the travel time from seconds to minutes by dividing by 60:\[ \frac{2158.67 \, \text{s}}{60} \approx 35.98 \, \text{minutes} \]

Step 5: Calculate distance traveled in 11 seconds

For part (b), calculate the distance the scramjet traveled during its 11 s test. Use the speed of 2.317 km/s:\[ \text{Distance} = 2.317 \, \text{km/s} \times 11 \, \text{s} = 25.487 \, \text{km} \]

Key Concepts

Understanding Mach Number

The Mach number is a critical concept in aerodynamics and physics that helps us understand how fast an object is moving relative to the speed of sound. It's a dimensionless unit, meaning it doesn't have a specific measurement. Instead, it is a ratio. A Mach number of 1 means an object is traveling at the speed of sound, whether in air or any other medium.

When a vehicle exceeds the speed of sound, it achieves supersonic speeds, and when it reaches Mach 5 or more, it is considered hypersonic. The Mach number is calculated by taking the speed of the object and dividing it by the speed of sound in that medium, expressed in the formula: \[ \text{Mach Number} = \frac{\text{velocity of the object}}{\text{speed of sound}} \]

• Subsonic: Less than Mach 1
• Transonic: Approximately Mach 1
• Supersonic: More than Mach 1 but less than Mach 5
• Hypersonic: Mach 5 and above

Decoding Speed of Sound

The speed of sound is the velocity at which sound waves travel through a medium, such as air. At standard atmospheric conditions at sea level and a temperature of 0 degrees Celsius, the speed of sound in air is precisely 331 meters per second. This speed can fluctuate slightly based on atmospheric conditions such as temperature and pressure.

In practical terms, the speed of sound is crucial in various fields like meteorology, aviation, and audio technology. Increasing the temperature or pressure of the medium generally increases the speed of sound because molecules move faster under such conditions, facilitating quicker wave transmission.

Exploring Scramjet Engines

Scramjets, short for Supersonic Combustion Ramjets, are a type of jet engine designed for use at hypersonic speeds, typically over Mach 5. Unlike traditional jet engines, scramjets do not have rotating parts like turbines, which allows them to operate effectively at extreme speeds.

Scramjet engines compress the incoming air in-flight (thanks to their speed), mix it with onboard fuel, and ignite the mixture to produce thrust. Some of their notable features include:

• No moving parts: Enhancing reliability and reducing mechanical complexity.
• Operates at extreme altitudes: Utilizing atmospheric oxygen for combustion instead of carrying heavy oxidizers, making them efficient for deep space exploration.
• Cost-effective: Potentially more economical compared to rockets for short bursts of hypersonic travel.

Defining Hypersonic Flight

Hypersonic flight refers to speeds faster than Mach 5, which is five times the speed of sound. It opens up exciting possibilities for military, space travel, and commercial aviation due to its speed and efficiency.

Characteristic phenomena of hypersonic flight include intense aerodynamic heating, making thermal management a critical challenge. This heating, and the resulting ionization of gases around the vehicle, creates a distinct challenge in design and material usage. Additionally, navigating at hypersonic speeds requires advanced guidance and control systems due to the rapid pace and precision needed.

As hypersonic technology advances, the potential for point-to-point travel on Earth or quick, low-cost access to space becomes more realistic, with innovations like the X-43A scramjet paving the way for these incredible journeys.