if a plane is flying at a speed so that the apex half-angle of its shock wave is 30 degrees, what is the speed if the speed of sound is 320 m/s?
sin 30 = 1/M = 1/2
M = 2 = V/a (the Mach number)
v = 2 a
a is the sound speed
To find the speed of the plane, we need to use the trigonometric relationship between the shock wave angle and the Mach number.
The Mach number (M) is the ratio of the speed of an object to the speed of sound in a particular medium. It is given by the equation:
M = v / c
where:
M is the Mach number,
v is the velocity of the object, and
c is the speed of sound.
In this case, we want to find the speed of the plane, given that the apex half-angle of its shock wave is 30 degrees and the speed of sound is 320 m/s.
Step 1: Convert the apex half-angle to the full angle.
Since the apex half-angle is given as 30 degrees, the total angle of the shock wave will be double that, which is 60 degrees.
Step 2: Convert the angle to radians.
To use trigonometric functions, we need to convert the angle from degrees to radians.
1 radian = (180 degrees / π)
So, 60 degrees = (60 / 180) * π radians ≈ π/3 radians
Step 3: Use the inverse tangent function to find the Mach angle.
The Mach angle (θ) can be calculated using the equation:
θ = arctan(1 / M)
Since we are given the total angle of the shock wave (60 degrees), we can use its complementary angle (the Mach angle) to find the Mach number.
θ = π / 3 radians
Step 4: Solve for the Mach number (M).
Using the equation from step 3, we can rearrange it to find the Mach number:
M = 1 / tan(θ)
M = 1 / tan(π / 3)
Using a calculator, we can evaluate tan(π / 3) ≈ √3
So, M = 1 / √3 ≈ 0.577
Step 5: Calculate the speed of the plane.
Now, we can use the Mach number to find the speed of the plane using the equation:
v = M * c
v = 0.577 * 320 m/s ≈ 184.58 m/s
Hence, the speed of the plane is approximately 184.58 m/s.
To determine the speed of the plane, we can make use of the information provided about the shock wave's apex half-angle and the speed of sound.
The apex half-angle of a shock wave, denoted by θ, can be related to the Mach number (M) of the aircraft by the equation:
M = 1 / sin(θ)
In this case, the apex half-angle is 30 degrees or π/6 radians.
Let's calculate the Mach number using the formula:
M = 1 / sin(30°)
M = 1 / sin(π/6)
Using a calculator, we can determine that sin(π/6) is approximately 0.5.
M = 1 / 0.5
M = 2
The Mach number (M) represents the ratio of the speed of the plane to the speed of sound. Therefore, we can calculate the speed of the plane (v) by multiplying the Mach number by the speed of sound (c):
v = M * c
v = 2 * 320 m/s
v = 640 m/s
Hence, the speed of the plane is 640 m/s.