Speed of sound with temperature gradient

[lukas123]

Hi,
could you please help me with my homework? I want to determine the height of mountain (from foot to peak) using the speed of sound.

Homework Statement

Known data: time delay, height1, temp1 plus known dependence between the height and temperature.
What I want to determine: height2, temp2, speed of sound cmin and cmax

Homework Equations

I used article on wikipedia: http://en.wikipedia.org/wiki/Speed_of_sound

The Attempt at a Solution

I tried to make some kind of integral from height1 to height2, but I failed, because I don't know the height2 (top height).
Could you please give me a clue, how to solute it?

 

[Delphi51]

Welcome to Physics Forums, Lukas.
I played with your interesting problem for a while and came up with these thoughts. You have known dependence between height and temperature and velocity, so there is some known function V(h) that gives the velocity as a function of height. Considering a bit of height dh, the time to travel that distance would be dt = dh/V(h). Assuming this could be integrated,
t = integral (h1 to h2) of dh/V(h) = f(h2) - f(h1)
where f is the integral of dh/V(h) and could be found.
So f(h2) = t + f(h1).
If the known function f can be inverted to F, you have
h2 = F[t + f(h1)]
I don't know if this is useful; it depends on being able to integrate 1/V and being able to find the inverse function of the result.

I also wonder about the dependence of the speed of sound on air pressure as it goes up the mountain.