Answer
See answer.
Work Step by Step
Apply the equation of continuity to the openings.
$$A_2v_2=A_1v_1$$
$$v_2=v_1\frac{A_1}{A_2}$$
Now apply Bernoulli’s equation. The pressure at each opening is atmospheric pressure.
$$P_1+\frac{1}{2}\rho v_1^2+\rho g y_1= P_2+\frac{1}{2}\rho v_2^2+\rho g y_2$$
$$ v_1^2- v_2^2=2 g( y_2-y_1)=2gh$$
Substitute.
$$ v_1^2- (v_1\frac{A_1}{A_2})^2=2gh$$
$$ v_1^2(1- (\frac{A_1}{A_2})^2)=2gh$$
$$ v_1=\sqrt{\frac{2gh }{(1- (A_1/A_2)^2)}}$$
This was to be shown.