How to calculate the Mach Number?

Mach number is calculated by finding the ratio of speed of an object to the speed of sound in the surrounding medium. Mach number is unitless (dimensionless quantity). The formula for calculating Mach number is M = v/c , where M is Mach number, v is the velocity of object (in meters per second, feet per second, etc.) and c is speed of light (in meters per second, feet per second, etc.).

Mach number is a fundamental concept in aerodynamics that compares the speed of an object to the speed of sound in the surrounding medium. It is a dimensionless quantity denoted by ‘M’ and calculated as the ratio of the object’s velocity to the speed of sound. The Mach number classification includes subsonic (M < 1), sonic (M = 1), supersonic (M > 1), and hypersonic (M >> 1) speeds.

The Mach number (M) is calculated using the formula:

M = v / c

where:

• M is the Mach number,
• v is the velocity of the object,
• c is the speed of sound in the medium.

Key Points

• Subsonic (M < 1): Object moves slower than the speed of sound.
• Sonic (M = 1): Object’s speed equals the speed of sound.
• Supersonic (M > 1): Object exceeds the speed of sound.
• Hypersonic (M >> 1): Object travels at extremely high speeds.

Mach numbers categorize speeds of objects relative to the speed of sound in a particular medium.

Subsonic (Mach number < 1):

• Speeds slower than the speed of sound.
• Common for most everyday objects and vehicles.
• Example: Passenger airplanes during cruising.

Transonic (Mach number ≈ 1):

• Speeds near the speed of sound.
• Objects experience both subsonic and supersonic airflow.
• Example: Some fighter jets during maneuvers.

Supersonic (Mach number > 1):

• Speeds faster than the speed of sound.
• Shockwaves form due to compressed air in front of the object.
• Example: Concorde passenger aircraft.

Hypersonic (Mach number > 5):

• Extremely high speeds, several times faster than the speed of sound.
• Significant aerodynamic heating due to air compression.
• Example: Spacecraft during re-entry into the Earth’s atmosphere.

Transhypersonic (Mach number > 10):

• Speeds beyond hypersonic, often associated with specialized aerospace vehicles.
• Extreme aerodynamic and thermal challenges.
• Example: Experimental scramjet engines in research vehicles.

Related Article:

• Reynolds Number
• Laminar and Turbulent Flow
• How to calculate the 5 Number Summary?
• Critical Velocity Formula

Question 1: If the speed of an object is given as 480 m/s at a temperature of 30 degrees, then calculate the Mach number for the object.

Answer: 

Given:

Object speed (v) = 480 m/s

Temperature (T) = 30°C = 303 K (conversion for temperature unit)

Step 1: Calculate the speed of sound (c) at 30°C

The speed of sound in air varies with temperature. We can use the following formula:

c = sqrt(k * R * T)

Where :

k = specific heat ratio (1.4 for air)

R = gas constant (287 J/kg*K)

T = temperature in Kelvin

Calculation:

c = sqrt(1.4 * 287 * 303) ≈ 347 m/s

Step 2: Calculate the Mach number (M)

M = v / c

Calculation:

M = 480 m/s / 347 m/s ≈ 1.38

Therefore, the Mach number of the object at 30°C is approximately 1.38.

Question 2: An Airplane travels in air at 20°C at a speed of 2400 km/hr. Find the Mach number. Take k=1.4 and R=287 J/Kg K.

Answer:

Given:

Airplane speed (v) = 2400 km/hr

Temperature (T) = 20°C (needs conversion to Kelvin)

Specific heat ratio (k) = 1.4

Gas constant (R) = 287 J/kg*K

Step 1: Convert speed from km/hr to m/s

v = 2400 km/hr * (1000 m/km) * (1 hr/3600 s) = 667 m/s

Step 2: Convert temperature from Celsius to Kelvin

T = 20°C + 273.15 = 293.15 K

Step 3: Calculate the speed of sound (c) using the formula:

c = sqrt(k * R * T)

c = sqrt(1.4 * 287 J/kg*K * 293.15 K) ≈ 343.11 m/s

Step 4: Calculate the Mach number (M):

M = v / c

M = 667 m/s / 343.11 m/s ≈ 1.94

Question 3: If the speed of an object is 500 m/s at a temperature of 10 degrees and the Mach number for the object is 2. Then calculate the velocity of sound in air?

Answer:

Given:

Object speed (v) = 500 m/s

Mach number (M) = 2

Formula:

M = v / c

Rearranging for c:

c = v / M

Calculation:

c = 500 m/s / 2 = 250 m/s

Therefore, the speed of sound in air in this scenario is approximately 250 m/s.

Question 4: If the aircraft is moving with a speed of 5600 m/s and its velocity of sound in air is 800 m/s. Calculate the Mach number for that aircraft?

Answer:

Given:

Aircraft speed (v) = 5600 m/s

Speed of sound (c) = 800 m/s

Formula:

Mach number (M) = v / c

Calculation:

M = 5600 m/s / 800 m/s = 7

Therefore, the Mach number of the aircraft is 7. This indicates that the aircraft is traveling seven times faster than the speed of sound, placing it firmly in the hypersonic regime.

Question 5: A pipeline transports methane gas (CH₄) at 40°C with a velocity of 560 m/s. What is the Mach number of methane gas and what type of flow does it have?

Answer:

Given that, 

Velocity v = 560 m/s 

Temperature T = 40°C

T = (40 + 273)K = 313 K.

Local speed sound in air in an ideal gas c = √kRT

where k = specific heat of methane gas = 1.264

R= gas constant of methane 

= 518.25 J/ Kg K 

Therefore c= √(1.264)(518.25)(313)

c = 452.81 m/s.

Mach number of methane gas 

M = v/c 

= 560/452.81

= 1.23

Therefore Mach number of methane gas is 1.23

Mach number of methane > 1

Therefore the flow is supersonic.

Mach numbers provide a key framework for understanding the velocity of objects relative to the speed of sound. This classification system, spanning from subsonic to trans hypersonic, guides the design and exploration of vehicles across various industries, from aviation to space travel. As technology evolves, Mach numbers will remain a cornerstone for pushing the boundaries of speed and advancing aerospace engineering.

What is Mach number?

The Mach number (M) is a unitless quantity representing how fast something is moving relative to the speed of sound in the surrounding medium (usually air). It’s essentially a speed ratio.

How is Mach number calculated?

The formula for Mach number is:

M = v / c

M = Mach number

v = object’s velocity

c = speed of sound in the medium

What are the different Mach regimes?

Mach number is used to classify different speed regimes:

• Subsonic (M < 1): Object travels slower than the speed of sound.
• Transonic (M ≈ 1): Object approaches the speed of sound. This is a challenging regime for flight due to compressibility effects.
• Supersonic (M > 1): Object travels faster than the speed of sound. This is where sonic booms occur.
• Hypersonic (M >> 5): Object travels at extremely high speeds, much faster than the speed of sound.

Why is Mach number important?

Mach number is important because the behavior of objects in fluids (gases and liquids) changes significantly as they approach the speed of sound.

Does the speed of sound change?

Yes, the speed of sound depends on temperature. For example, the speed of sound in air at sea level is about 343 meters per second (767 mph), but it will be different at higher or lower altitudes. Mach number doesn’t provide an absolute speed value because of this variation.