Question

The density of the atmosphere varies with height h above the ground as per relation rho= rho knot e^-kh where k is a constant then pressure at a height h is (assume g is constant)

Answer 1

Answer:

Pressure at height h is $P = P_{o}+\rho _{o}g (e^{-kh} -1)$

Explanation:

Change in pressure above direction is given as

$dP = - \rho \times g\times dh$      

$dP = - \rho_{o}e^{-kh} \times g\times dh$

$dP = - \rho_{o}g\times e^{-kh}dh$      ...1)

integrating above equation 1)

$\oint_{P_{o}}^{P} = - \rho _{o}g\oint_{0}^{h}(e^{-kh}dh)$

where $P_{o}$ is pressure at ground surface.

$P - P_{o} = -\rho _{o}g(\frac{e^{-kh}}{-k})_{0}^{h}$

$P - P_{o} = \rho _{o}g (e^{-kh} -1)$

$P = P_{o}+\rho _{o}g (e^{-kh} -1)$