How to Calculate Shock Wave Cone Angle and Plane Speed in a Sonic Boom Situation

[negatifzeo]

Homework Statement

You look directly overhead and see a plane exactly 1.4·km above the ground, flying faster than the speed of sound. By the time you hear the sonic boom, the plane has traveled a horizontal distance of 2.4·km.

(a) Find the angle of the shock wave cone.

(b) Find the speed of the plane (Mach number).

Homework Equations

Sin (shock wave cone angle)=(c/v)

PLane Speed= v/c

v=velocity of source

c=speed of sound

The Attempt at a Solution

The plane is 1400 meters high. When the shockwave is heard, the plane has traveled 2400 meters horizontally. Sound travels at 343 m/s. 1400/343 gives me 4.08 seconds. 2400/4.08 gives the plane a speed of 588 m/s.

Sin-1(343/588)=35.7 degrees. Of course, I'm here because these are not the correct answers and I don't know why. Any clues as to what I did wrong would be greatly appreciated.

 

[tiny-tim]

Hi negatifzeo!

You're not thinking clearly.

The sound leaves the plane before it's overhead.

Draw a diagram. There's a right-angled triangle in it. You know one of the angles has sin c/v (which one?). Then use Pythagoras' theorem!

 

[negatifzeo]

I don't understand. It makes sense that the planr makes a sound before its directly overhead, of course. But how can you solve for sin c/v without knowing the aircraft's speed?

 

[tiny-tim]

But how can you solve for sin c/v without knowing the aircraft's speed?

Because v is an unknown, just like x in ordinary algebra equations.

Draw the diagram … use v, or a multiple of v, where you would normally use a number … and you'll get an equation in v (maybe linear, maybe quadratic) … which you can then solve.

Try it (and start with the diagram)!

 

[Anony-mouse]

I'm having problems with this question too.

I understand that the angle is Tan-1 (1.4/2.4) = 30.25 degrees

But how do you go about finding the mach number? I also calculated the speed as 588 m/s, making the Mach speed 1.78, but that wasn't right.