Please bear with me - I'm a lapsed mathematician and I'm self-studying these concepts.
A question asks the following:
Water flows in a pipe of diameter 5 m at a velocity of 10 m/s. It then flows down into a smaller pipe of diameter 2 m. The height between the centre of pipe sections is 5 m. The density is assumed to be uniform over the cross sections. The gauge pressure at Boundary 1 is 120 kPa. Calculate the velocity at the smaller pipe section.
There is no reason I can see to assume mass flow continuity doesn't apply, and using $v_1A_1 = v_2A_2$ one obtains $v_2 = 62.5$ m/s. However, using the Bernoulli equation while assuming atmospheric pressure at the smaller section, one gets $$ \frac{p_1}{\rho} + \frac{1}{2}v_1^2 + + gz_1 = \frac{p_2}{\rho}+\frac{1}{2}v_2^2 + gz_2 $$ $$ 120+ 50 + 5g = 0 + \frac{1}{2}v_2^2 + 0 $$ $$ v_2 = 20.93, $$ and in fact this is what the textbook answer gives. I am quite confused as to why mass continuity applies in other situations, even with changes in pressure, but doesn't seem to apply here.
My question is: is this textbook question then ill-posed? I feel as though by providing too much information about the pipe section without checking the calculations, the question is bound to create a contradiction. The 5m/2m diameters don't actually make it into the final answer.
EDIT: I have added below the only answer I'm satisfied with that makes sense of the data. Please correct me as you see fit.