Deriving an equation for water flowing from a tap

[Woolyabyss]

Homework Statement

A stream of water flows downward from a tap. Assume that the water is in free fall once it leaves the tap, at which point its speed is v1, and the initial diameter of the water stream is A1.
(i) Find an expression for the speed v2 of the liquid as a function of the distance, h, it has fallen.
(ii) Combine this with the equation of continuity, to find an expression for the speed v2 in terms of the distance h and the cross-sectional areas at A1 and A2.
(iii) The cross-sectional area of the stream changes from 1.2 cm2 to 0.35 cm2 between the positions A1 and A2. Calculate the volume rate of flow of water from the tap, in cm3 s

Homework Equations

Continuity equation
Kinematic equations

The Attempt at a Solution

(i)

.5(v₁)² + gh = .5(v₂)²
therefore v₂ = √(2*(.5(v₁)²)+gh)

(ii)
.5(v₁)² + gh = .5(v₂)²

therefore using A₁v₁ = A₂v₂

.5(v₁)² + gh = .5(v₁)²*A₁/A₂

and so
.5(v₁)²((A₁/A₂)² - 1) = gh

therefore v₁ = √(2gh/(A₁/A₂)² - 1)

(iii)
I'm not sure how you can get a numerical value for this question but the question seems to imply you can.

any help would be appreciated.

 

[haruspex]

.5(v1)2 + gh = .5(v2)2
therefore v2 = √(2*(.5(v1)2)

Try that step again.

therefore v1 = √(2gh/(A1/A2)2 - 1)

It asks for v2 in terms of h, A1, A2, not for v1.

Your continuity equation involved terms like A1v1. What physical aspect does that represent?

 

[Woolyabyss]

Try that step again.

It asks for v2 in terms of h, A1, A2, not for v1.

Your continuity equation involved terms like A1v1. What physical aspect does that represent?

Ok, so

v2 = (2gh/(1-(A2/A1)^2))^.5

A1v1 represents the volume flow rate

 

[haruspex]

Ok, so

v2 = (2gh/(1-(A2/A1)^2))^.5

A1v1 represents the volume flow rate

Yes, you seem to be missing a piece of data. E.g. the height difference between A1 and A2. Is there any more info, maybe in a diagram?

 

[Woolyabyss]

Yes, you seem to be missing a piece of data. E.g. the height difference

Yes, you seem to be missing a piece of data. E.g. the height difference between A1 and A2. Is there any more info, maybe in a diagram?

between A1 and A2. Is there any more info, maybe in a diagram?

There is a diagram but it doesn't contain anymore information. I guess you're meant to leave your answer in terms of h.

 

[haruspex]

There is a diagram but it doesn't contain anymore information. I guess you're meant to leave your answer in terms of h.

Maybe, but it seems odd to provide numbers for the areas but not for h.