(II) An airplane travels at Mach 2.0 where the speed of sound is 310 m / s,(a) What is the angle

(II) An airplane travels at Mach 2.0 where the speed of...

(II) An airplane travels at Mach 2.0 where the speed of sound is $310 \mathrm{m} / \mathrm{s},(a)$ What is the angle the shock wave makes with the direction of the airplane's motion? $(b)$ If the plane is flying at a height of $6500 \mathrm{m},$ how long after it is directly over- head will a person on the ground hear the shock wave?

(II) An airplane travels at Mach 2.0 where the speed of sound
is $310 \mathrm{m} / \mathrm{s},(a)$ What is the angle the shock wave makes with
the direction of the airplane's motion? $(b)$ If the plane is flying at a height of $6500 \mathrm{m},$ how long after it is directly over-
head will a person on the ground hear the shock wave?

Key Concepts

Propagation Delay of Sound

The propagation delay of sound refers to the time it takes for sound waves to travel from their source to an observer. This concept is particularly important in scenarios where the source is moving at high speeds, as there is a noticeable time lag between the source's position and when the sound is perceived by an observer, due to the finite speed of sound.

Mach Number

The Mach number is a dimensionless quantity representing the ratio between the speed of an object and the speed of sound in the medium through which it travels. It is essential in determining the behavior of objects in airflow, particularly when transitioning between subsonic and supersonic speeds.

Mach Angle

The Mach angle is the half-angle of the conical shock wave generated by an object moving at supersonic speeds. It is determined using the relation sin(?) = 1/Mach number, which geometrically describes the angle between the direction of motion and the shock wave, illustrating how disturbances propagate in a supersonic regime.

Transcript

00:01 A, what is the angle the shockwave makes with the direction of the airplane motion? and b, if the plane is flying at a height of 6 ,500 meters, how long after it is directly overhead will a shock wave? i wrote down what we were given here.

00:15 We're told that the speed of sound, which i write as v .s, is 310 meters per second, that the velocity of the airplane, which i write as v .p.

00:24 P is 2 .0 mach speed.

00:29 Okay, so that's two times the speed of sound.

00:31 So it's 2 .0 times v .s.

00:33 And the plane is flying at a height of 6 ,500 meters.

00:36 So for part a, we're asked to find the angle of the shock wave.

00:41 Well, we can use equation 16 -12 to find that, which says that the sign of the angle is equal to the ratio of the speed of sound to the speed of the object in question.

00:58 Which is in this case the plane so v .p right but v .p is 2 .0 times v.

01:08 Sub s so the v .s .s.

01:09 Is canceled so this is going to be 1 over 2 .0 okay so solve for theta we find that theta is equal to the inverse of the sign which can be written as sine to the minus 1 of 1 over 2 .0 okay so if you take the inverse of one half, you find that this is equal to 30 degrees.

01:48 Okay, so that's the solution for part a.

01:52 Now, for part b, we are asked to find the time it will take for a person to hear the shock wave after the plane flies directly overhead.

02:06 Okay, and we're told that the height of the plane is 6 ,500 meters.

02:10 So we can just draw out a little diagram here.

02:12 We have the, this is the direction to the ground.

02:20 It's the height of the plane.

02:23 This is velocity of the plane times the time...