Calculating Altitude from Sonic Boom and Angle of Sight

[rlc]

Homework Statement

You hear the sonic boom of a high-speed jet plane exactly 4.50 s after it passes directly overhead in level flight. At the time you hear the boom, you see the plane at an angle of 16.0 o above the horizon. Assume that the speed of sound at the altitude of the plane is 325 m/s.
How fast is the plane traveling?
What is the altitude of the plane?

Homework Equations

v(plane)=velocity of sound/sin(alpha)
But I don't know the equation to find altitude.

The Attempt at a Solution

I know how to get the first question:
325/sin(16)=1179.1 m/s -->which LONCAPA says is right.

What I don't know is how to get the second part. How do you find the altitude based off of the information given?

 

[BvU]

Well, if LONCAPA (who's that?) says it's right, it must be right...

Did you make a drawing ?

In fact, part b) is fairly simple: If you hear a bang 4.5 s after a flash, how far away was the lightning ?

Or am I dreaming ?

[edit] shooting at a rapid rate, eh? three-barrel ! Some catching up to do, or an exam coming up ?

 

[rlc]

LONCAPA is an online college homework site where you can plug your answers in and it tells you if that the answer it has as right.

325 m/s *4.5 s=1462.4 m
But the website says that this is incorrect. I'm assuming it is because of the angle. Do you know what equation to use or how to factor this information in?

 

[NTW]

Make a drawing of the sonic cone... And you'll see everything very clearly. It's a nice and easy problem.

 

[rlc]

I made a picture, but I think I did something wrong.
Does this make a triangle? I thought using 1179.1*tan(16 degrees) will give the altitude based off of calculus, but the homework website I am using says that this is the wrong answer. What am I missing?
(Also thank you for helping!)

 

[NTW]

LONCAPA is an online college homework site where you can plug your answers in and it tells you if that the answer it has as right.

325 m/s *4.5 s=1462.4 m
But the website says that this is incorrect. I'm assuming it is because of the angle. Do you know what equation to use or how to factor this information in?

It is incorrect because, by the time the sound reached your ears, the plane had moved away from the point where that sound originated.
Again: make a clear drawing and you'll see the solution... I remind you that alpha is the semi-angle between the axis of the cone (the trajectory of the plane) and any generatrix of the cone.

 

[rlc]

Ah! I got it!
I forgot to factor in time!
(1179.1m/s*4.50s)tan(16 degrees)=1521.46 m (which the website is right)

 

[BvU]

Well, sorry, can't think of anything better than what you came up with...

NTW: thanks for stepping in..

 

[NTW]

Ah! I got it!
I forgot to factor in time!
(1179.1m/s*4.50s)tan(16 degrees)=1521.46 m (which the website is right)

The answer may be right, but it appears to me that you reached it by manipulating the variables almost blindly. With a simple drawing of the sonic cone, the trajectory of the plane and the observer on the ground, everything is perfectly transparent...

 

[rlc]

It wasn't done blindly. I looked in my textbook, and it showed very clearly (like you said) how the cone created a triangle, where velocity times time was one side of the triangle, the angle was incorporated, and the altitude was another side of the triangle. Simple calculus allowed me to move on from there. Thank you for all of your help for this problem.