Fluid dynamics calculation -- Draining a barrel through a pipe

[swiper122]

Homework Statement

A barrel of cylinder shape, with radius 6 cm, is filled with water 10 cm high. You open a pipe at the bottom with area of 1 cm^2. How much time is needed for water to drop 5 cm in height?

Relevant Equations

Bernoulli equation: p + (⍴ *v^2)/2 + ⍴gh = constant.
Volumetric flow rate = Q= dV/dt = A*v

v1- velocity of water at the top of the barrel
v2- velocity of water at the the pipe (bottom of cylinder)

p + (⍴ *(v1)^2)/2 + ⍴gh=p + (⍴ *(v2)^2)/2 + ⍴gh
atmosferic pressure cancels out

(⍴ *(v1)^2)/2 + ⍴gh = (⍴ *(v2)^2)/2 + ⍴gh
density doesn't affect the result(cause its in every term)

(v1)^2/2 + gh = (v2)^2/2 + gh
(finaly i can set the height of the pipe to zero)

(v1)^2/2 + gh = (v2)^2/2
(i am left with this equation and 2 unknowns)

Q= dV/dt = A*v
(Now, I am not sure how to use this equation, since i think Volumetric flow is not constant, because velocity is not constant (potential energy is decreasing as height lowers with time)
I feel like i need to integrate but i don't know how to set it up. The result is supposedly 63 s. Thank you

 

[haruspex]

i am left with this equation and 2 unknowns

You can take the fluid as incompressible, so there's a relationship between the two velocities.
There's also a differential relationship between the velocities and the remaining height.