How to Calculate Mach Number: Stepwise Guide with Examples
The Mach number is a fundamental concept in Physics, especially in the study of aerodynamics and motion in fluids. It is defined as the ratio of an object's speed to the speed of sound in that medium. This dimensionless quantity helps us understand whether the flow of air around a moving object can be considered compressible or not.
For example, the Mach number is crucial in the analysis of aircraft speeds and behaviors at different altitudes and temperatures.
When an aircraft moves through air, it disturbs the air particles. If the velocity is high, a compressing effect occurs in the surrounding air. Using the Mach number, we can determine how the air or any fluid responds—whether as a compressible or incompressible fluid. This makes Mach number an important parameter in both flight dynamics and fluid mechanics.
Mach Number: Concept and Definition
- Mach number (M) compares the speed of an object (such as an airplane) to the speed of sound in the same medium.
- At standard room temperature, the speed of sound in air is approximately 343 m/s. The Mach number tells us how "fast" something is moving relative to the speed needed to trigger compressibility effects in the surrounding air.
- If a plane is flying exactly at the speed of sound, its Mach number is 1 ("Mach 1"). If it is flying faster or slower, the Mach number is greater than or less than one, respectively.
- The value depends on air temperature, pressure, and the nature of the gas itself, which may change with altitude or local environment.
Mach Number Formula
The formula for Mach number is:
- Where M = local Mach number
- v = speed of the object (in m/s)
- c = speed of sound in the given medium (in m/s)
The speed of sound depends on properties of the medium, mainly temperature. For a gas, the speed of sound is given by:
- K: Ratio of specific heat at constant pressure (Cp/Cv) of the gas
- R: Specific gas constant
- T: Temperature of the gas (in Kelvin)
Types of Flow Based on Mach Number
The Mach number helps classify different regimes of fluid flow and corresponding vehicle or object design characteristics. These regimes are:
| Mach Number Range | Type of Flow (Regime) | Description / Example |
|---|---|---|
| Below 0.8 | Subsonic | Aircraft designs feature rounded nose and leading edges |
| 0.8 – 1.3 | Transonic | Aircraft typically built with swept wings |
| 1.3 – 5.0 | Supersonic | Sharp-edged designs; example: modern fighter jets |
| 5.0 – 10.0 | Hypersonic | Small, cooled wings; example: US X-15 aircraft |
| Greater than 10.0 | Hypervelocity | Thermal control becomes crucial in design |
Importance of Mach Number
- Helps in designing aircraft shapes and structures for optimal performance at given speeds.
- Determines whether a fluid (typically air) around an object should be considered compressible.
- Is a unitless parameter, making it easy to compare different objects' speeds relative to the speed of sound.
Mach Number: Solved Examples
Let's solve typical problems to illustrate application of Mach number formula.
| Example | Solution Steps | Result |
|---|---|---|
| 1. An object travels at 240 m/s. If speed of sound is 340 m/s, what is its Mach number? |
v = 240 m/s; c = 340 m/s M = v / c = 240 / 340 = 0.70 |
Mach number = 0.70 (Subsonic) |
| 2. An aircraft has Mach number 1.35. If speed of sound is 340 m/s, what is its speed? |
M = 1.35; c = 340 m/s v = M × c = 1.35 × 340 = 459 m/s |
Aircraft speed = 459 m/s |
Stepwise Approach to Solving Mach Number Problems
| Step | Action | Note |
|---|---|---|
| 1 | Write known values (speed of object and speed of sound) | Use correct units (m/s) |
| 2 | Apply the Mach number formula | M = v / c |
| 3 | Calculate the result | Perform division |
| 4 | Classify the flow regime | Refer to the range table above |
Key Formulas at a Glance
| Formula | Description | Variables |
|---|---|---|
| M = v / c | Mach number as ratio of object speed to speed of sound | M = Mach number; v = speed of object; c = speed of sound |
| c = √(K R T) | Speed of sound in a gas | K = Cp/Cv; R = gas constant; T = temperature (in K) |
Related Concepts and Further Learning
- Expand your understanding of Speed of Moving Objects and Velocity.
- Learn about Projectile Motion for practical applications of speed and Mach number.
- Explore Acceleration and Relative Speed to deepen your concepts.
- Try stepwise practice questions using the tables and formulas provided above.
Continue to practice and apply these concepts in numerical and theory-based questions for complete mastery of motion and aerodynamics in Physics.
FAQs on Mach Number in Physics: Definition, Formula, and Applications
1. What is Mach number?
Mach number is a dimensionless quantity that expresses the speed of an object relative to the speed of sound in the same medium.
Key points:
- Mach number (M) = Object’s speed (v) / Speed of sound (a)
- M < 1: Subsonic | M = 1: Sonic | M > 1: Supersonic
- Used to classify object speeds in aerodynamics, aviation, and fluid mechanics.
2. What is the formula for Mach number?
The Mach number (M) formula is:
M = v / a
Where:
- v = Speed of the object (in m/s or km/h)
- a = Speed of sound in that medium (in m/s or km/h)
This formula is as per the official Physics (JEE/NEET/CBSE 2025) syllabus.
3. What does Mach 1 mean?
Mach 1 means the object is traveling at the speed of sound in the given medium. At sea level, in air at room temperature, Mach 1 is approximately 340 m/s (1,224 km/h).
4. How does Mach number change with altitude?
Mach number depends on the speed of sound, which decreases as altitude increases (due to lower air temperature and density).
- As altitude increases, speed of sound decreases.
- The same airspeed yields a higher Mach number at higher altitudes.
- For example, Mach 1 is about 340 m/s at sea level but drops to ~295 m/s at 11 km altitude.
5. What are subsonic, sonic, supersonic, and hypersonic speeds?
Speed ranges based on Mach number:
- Subsonic: M < 1 (e.g., typical passenger jets)
- Sonic: M = 1 (speed of sound)
- Supersonic: 1 < M < 5 (fighter jets, bullets)
- Hypersonic: M > 5 (spacecraft re-entry)
These terms are used in aerodynamics and fluid mechanics to describe different flow regimes.
6. Who is Mach number named after?
Mach number is named after Ernst Mach, an Austrian physicist and philosopher who researched supersonic motion and shock waves.
7. How do you calculate the speed of an object if its Mach number is given?
To find the object's speed:
Speed (v) = Mach number (M) × Speed of sound (a)
Steps:
1. Note the Mach number (M) and local speed of sound (a)
2. Multiply: v = M × a
3. Example: If M = 2, a = 340 m/s, then v = 2 × 340 = 680 m/s.
8. Why is Mach number important in Physics and Engineering?
Mach number indicates the compressibility of flow and helps in designing aircraft, engines, and predicting aerodynamic effects:
- Determines whether a flow is subsonic or supersonic
- Influences shape/design of aircraft and projectiles
- Important for safety, efficiency, and performance calculations in aerodynamics.
9. Does the Mach number formula change for different mediums?
The basic Mach number formula (M = v / a) remains the same for all mediums, but the speed of sound (a) will be different in each medium (air, water, etc.). Always use the correct value of 'a' for the medium the object is moving through.
10. How is the speed of sound in air calculated?
The speed of sound in air (a) can be calculated using:
a = √(γRT)
Where:
- γ (gamma): Ratio of specific heats (about 1.4 for dry air)
- R: Specific gas constant (287 J/kg·K for air)
- T: Absolute temperature in Kelvin
This formula shows speed of sound depends mainly on air temperature.
11. Can Mach number be used for objects in water?
Yes, Mach number can be applied to any object moving in a medium where the speed of sound is known—including water, gases, and solids. The value of Mach 1 will be different in each medium depending on the speed of sound there.
12. What type of questions are asked about Mach number in competitive exams?
Competitive exams typically ask Mach number questions involving:
- Calculating Mach number from object and sound speeds
- Interpreting flow regimes (subsonic, supersonic, etc.)
- Applying Mach number to aircraft, bullets, or wave scenarios
- Understanding effect of temperature/altitude on Mach values
Practice previous year and sample Mach number numericals for best results.
