Problem $04\mathrm{.}004-$ Calculate velocity using a venturi tube in a convergent$-$divergent duct

References

An instrument used to measure the airspeed on many early low$-$speed airplanes, principally during $1919$ to $1930,$ was the venturi tube.

This simple device is a convergent$-$divergent duct. $($The cross$-$sectional area of the front section decreases in the flow direction, and

the cross$-$sectional area of the back section increases in the flow direction. Somewhere between the inlet and exit of the duct, there is

a minimum area called the throat.$)$ See figure below. Let $A_1$ and $A_2$ denote the inlet and throat areas, respectively. Let $p_1$ and $p_2$ be

the pressures at the inlet and throat, respectively. The venturi tube is mounted at a specific location on the airplane $($generally on the

wing or near the front of the fuselage$)$ where the inlet velocity $V_1$ is essentially the same as the free$-$stream velocity$-$that is$,$ the

velocity of the airplane through the air. With a knowledge of the area ratio $\frac{A_2}{A_1}($a fixed design feature$)$ and a measurement of the

pressure difference $p_1-p_2,$ we can determine the velocity of the airplane. For example, assume $\frac{A_2}{A_1}=\frac{1}{4}$ and $p_1-p_2=106l\frac{b}{f}{t}^2.$ If

the airplane is flying at standard sea level, what is its velocity?