Speed of Jet Plane - Solving the Problem

[Von Neumann]

Problem:

You see a jet plane flying and you think that it is flying at a constant altitude h. Say you hear the sonic boom at a time t after the plane passes directly overhead. Show that if the speed of sound v is the same at all altitudes, the speed of the plane is

v_s=hv/sqrt(h^2-v^2*T^2)

Partial Solution:

The sound travels a distance vT and the plane travels a distance v_s*T before the boom is found. Correct? However, I need to relate this to the height. If you assume h^2=(vT)^2+(v_s*T)^2. Since the speed of sound is the same at all altitude then the boom forms a perfect cone with angle θ with the ground. Then substituting in sinθ=v/v_s you can easily get the expression. However I don't fully understand the assumption h^2=(vT)^2+(v_s*T)^2 because geometrically it doesn't make sense to me. Any insight?

 

[haruspex]

If you assume h^2=(vT)^2+(v_s*T)^2

Why would you assume that? Pythagoras wouldn't have.

 

[Von Neumann]

Why would you assume that? Pythagoras wouldn't have.

As I stated, with that assumption I get the identity as given. If you also go back and read the question, I ask for insight on the geometric significance of that assumption.

 

[haruspex]

As I stated, with that assumption I get the identity as given. If you also go back and read the question, I ask for insight on the geometric significance of that assumption.

Looks like my hint was too subtle. Draw the diagram on which you based that equation. Which is the hypotenuse?

 

[Nickie]

Thank you for providing your partial solution. Your approach is correct in using the relationship between the distance traveled by the plane and the distance traveled by the sound to determine the speed of the plane. As for the assumption h^2=(vT)^2+(v_s*T)^2, this is based on the Pythagorean theorem which states that in a right triangle, the square of the length of the hypotenuse (h) is equal to the sum of the squares of the lengths of the other two sides (vT and v_s*T). In this case, we are considering the triangle formed by the plane's flight path, the distance traveled by the sound, and the altitude of the plane. Therefore, this assumption is valid in determining the speed of the plane. I hope this helps clarify any confusion you may have had. Good luck with your problem-solving!