Question

At what altitude is the density of air one half that of at the surface of the earth assuming temperature is constant and 1 m^3 of air weighs 1.25kg at the surface of the earth. (this question comes from a chapter in an ODE text book by Arnold, involving the autonomous differential equation x'=v(x))

Answer 1

This equation is given preceding this example: \[\int\limits_{x_{0}}^{x} dx/v(x) = t - t_{0}\]

Answer 2

one derivation given is as follows

dp/dy =- rho g

dp/p = - g (rho0/p0) dy

ln (p/p0) = - g (rho0/p0)y

works out to p = (p0) exp(-ay)
a = 0.116/km.

If you have another answer, you can compare it against this.

Answer 3

The answer i have is 5.6km

Answer 4

ln(0.5) = -.693= -0.116 y
y = 5.98 km

almost identical.

Answer 5

Thanks for your help

Answer 6

Glad to do so.