Speed of jet plane

1. Feb 27, 2013

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?

2. Feb 27, 2013

haruspex

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

3. Feb 28, 2013

Von Neumann

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.

4. Feb 28, 2013

haruspex

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