The velocity profile in a fluid stream is approximated using a three$-$part model, where the velocity is $1\mathrm{.}6\frac{m}{s}$ in $15$ percent of the cross$-$sectional area, $3\mathrm{.}2\frac{m}{s}$ in $35$ percent of the area, and $4\mathrm{.}5\frac{m}{s}$ in the rest of the area.

$($a$)$ How much does the total momentum of the fluid differ from the product of the mass flow rate and the average velocity?

$($b$)$ Can you think of a situation where there is a velocity profile in the flow stream and the momentum flow is exactly equal to the mass flow times the average velocity?