At some point on its trip a Concorde aircraft flies 520 m/s at an altitude where the temperature is -56 degree C. Determine the Mach angle (in degrees) of its shockwaves at that instance. Determine the Mach angle (in degrees) of its shockwaves at that instance.
To determine the Mach angle of the shockwaves, we need to use the formula:
Mach angle (θ) = sin^(-1)(1/Mach number)
The Mach number can be calculated using the formula:
Mach number (M) = velocity/speed of sound
First, we need to calculate the speed of sound at an altitude where the temperature is -56 degrees Celsius. The relationship between temperature and the speed of sound is given by:
Speed of sound (v) = √(γ * R * T)
where:
γ = ratio of specific heats = 1.4 (for air)
R = gas constant = 287 J/(kg*K)
T = temperature in Kelvin = -56 + 273 = 217 K
Plugging in the values, we get:
v = √(1.4 * 287 * 217)
v ≈ 294.8 m/s
Next, we can calculate the Mach number:
M = 520 / 294.8
M ≈ 1.764
Finally, plugging the Mach number into the formula for the Mach angle:
Mach angle (θ) = sin^(-1)(1/1.764)
θ ≈ 38.9 degrees
Therefore, the Mach angle of the Concorde's shockwaves at that instance is approximately 38.9 degrees.
To determine the Mach angle of the shockwaves, we first need to find the Mach number of the Concorde aircraft.
The Mach number (M) is the ratio of the aircraft's speed (v) to the speed of sound (c) in the medium it is traveling through.
Given:
Speed of the Concorde aircraft (v) = 520 m/s
Temperature (T) = -56 degree C
To find the Mach number, we need to find the speed of sound (c) at that temperature. The speed of sound in air can be calculated using the following equation:
c = sqrt(gamma * R * T)
where:
gamma is the ratio of specific heats of air (approximately 1.4)
R is the specific gas constant for air (approximately 287 J/kg.K)
Let's calculate the speed of sound (c) first:
c = sqrt(1.4 * 287 * (T + 273.15)) [Converting temperature to Kelvin]
c = sqrt(1.4 * 287 * (217.15)) [Substituting T = -56 degree C]
c ≈ 314.977 m/s
Now, we can calculate the Mach number (M):
M = v / c
M = 520 / 314.977
M ≈ 1.651
Now it's time to calculate the Mach angle (θ). The Mach angle can be determined using the following equation:
θ = arcsin(1 / M)
θ = arcsin(1 / 1.651)
θ ≈ 34.86 degrees
Therefore, the Mach angle of the shockwaves at that instance is approximately 34.86 degrees.
To determine the Mach angle of the shockwaves created by a Concorde aircraft flying at a given speed and altitude, we need to use the formula for the Mach angle:
Mach angle (θ) = arcsin(1/Mach number)
First, we need to calculate the Mach number (M) using the given speed. The Mach number represents the ratio of the aircraft's speed to the speed of sound:
Mach number (M) = velocity of the aircraft / speed of sound
To determine the speed of sound, we can use the formula:
speed of sound = √(γ * R * T)
Where:
- γ is the specific heat ratio (approximately 1.4 for air)
- R is the gas constant for air (approximately 287 J/(kg·K))
- T is the temperature in Kelvin (convert from Celsius to Kelvin by adding 273)
Now, let's calculate the Mach number:
Given:
- Velocity of the Concorde aircraft = 520 m/s
- Altitude temperature = -56 degrees Celsius
First, convert the temperature from Celsius to Kelvin:
Temperature (T) = -56 + 273 = 217 K
Next, calculate the speed of sound:
Speed of sound = √(1.4 * 287 * 217) = 320.33 m/s (approximately)
Now, calculate the Mach number:
Mach number = 520 / 320.33 = 1.625 (approximately)
Finally, calculate the Mach angle:
Mach angle (θ) = arcsin(1 / 1.625) ≈ arcsin(0.615) ≈ 37.09 degrees (approximately)
Therefore, at the given instance, the Mach angle of the shockwaves created by the Concorde aircraft is approximately 37.09 degrees.