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

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

 

[rdl]

Thank you for sharing your approach to finding the mach number of the aircraft. Your use of trigonometry to calculate the half-angle of the mach cone is a valid method. However, it is important to note that there may be other factors at play that could affect the accuracy of this calculation, such as wind speed and direction. It would be beneficial to also consider these variables in your analysis. Additionally, it is important to keep in mind that the mach number is not a constant value for an aircraft and can vary depending on its speed and altitude. Overall, your approach is a good starting point for determining the mach number of the aircraft, but further analysis and consideration of potential variables may be necessary for a more accurate calculation.