Figure P $3\mathrm{.}4$

Problem $3\mathrm{.}4$

Rain falls on a houseroof that is inclined at angle $=60$ to the vertical, as shown in

figure $P3\mathrm{.}4.$ The mass flow rate of rain per unit horizontal area is $m=1E(-3)k\frac{g}{m^2}s.$

A very thin film of water forms on the roof, flowing parallel to the roof surface in

the direction of the $x$ axis as shown in figure $P3\mathrm{.}4.$ The flow speed $V$ in the water is

uniform across the thickness $h$ of the film and depends only upon the vertical distance

$z$ from the apex of the roof, varying as:

$V=kz$

where $k$ is a constant having the value $1E(-3)s^{-1}.$

$($a$)$ Show that in a steady flow the thickness $h$ of the film does not depend upon

the distance $x,$ and calculate its value. $($b$)$ If the rain starts to fall at time $t=0$ at a

steady rate of $m$ when the roof is dry, derive and solve a differential equation for the