PROBLEM $3[40$ points$]$

Wind is blowing towards a mountain range as shown in the figure below.

Upstream of the mountain: the wind on the ground is calm but its speed increases linearly from $0$ to $20$ mph in the first $1000$ ft above the ground. The wind speed above $1000$ ft becomes uniform at $20$ mph $.$

At the mountains: the wind rises up to get over the mountains. The mountain range is $3000$ ft high and several miles long $($into the page$).$

a$.(35$ points$)$ Find the wind speed in the $3000$ ft to $5000$ ft region directly above the mountains.

Assume:

The wind speed is uniform in this region.

steady$-$state conditions and uniform air density of $1\mathrm{.}0k\frac{g}{m^3}.$

Above $5000$ ft the wind speed is $20$ mph everywhere.

b$.(5$ points$)$ Repeat part a$,$ if the density of the air above the mountains is $0\mathrm{.}9k\frac{g}{m^3}$ but remains $1\mathrm{.}0k\frac{g}{m^3}$ upstream of the mountains.

$6$