Finding the mach number of an aircraft

[jmm5872]

A supersonic aircraft is flying parallel to the ground. When the aircraft is directly overhead, an observer sees a rocket fired from the aircraft. 10.4 s later the observer hears the sonic boom, followed 3.37 s later by the sound of the rocket engine. What is the mach number of the aircraft?

I think I was too simple in my assumptions, but I used trig to find the half-angle of the mach cone. I used the adjacent side as 10.4, and the opposite side as 13.77, and took the inverse tan. I got an angle of 52.9375.

Then I used this to find the mach number by using this equation:

Mach # = 1/sin($$\theta$$)

=1/sin(52.9375) = 1.25317

 

[willem2]

Those 2 sides are not the 2 sides next to the 90 degree angle, so taking an inverse tan was wrong.

I calculated the distance from the point from where the sonic boom is first heard to the point where the rocket is released with pythagorars. (the other 2 sides are 13.77c and 10.4c where c is the speed of sound).
This distance is covered by the plane in 3.37s, so it's equal to 3.37v, where v is the speed of the plane.

 

[jmm5872]

I'm still a little confused by this. I am trying to set the triangle up and can't get past this diagram...I know you said it was wrong, but I can't picture the triangle you are describing.

I tried to attach a drawing I made, I hope it works

 

[willem2]

Your triangle is actually Ok, (and mine wasn't) but to get $\tan{\phi}$ you need to find the length of the sides in meters, and not in seconds.
The vertical distance is $13.77 v_{sound}$ and the horizontal distance is $10.4 v_{plane}$

When you have an expression for $\tan{\phi}$, you can combine that with

$$\frac { v_{plane} } { v_{sound} } = \frac {1} {\sin{\phi}}$$