# 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?