• Support PF! Buy your school textbooks, materials and every day products Here!

Deriving an equation for water flowing from a tap

  • Thread starter Woolyabyss
  • Start date
  • #1
143
1

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.
 

Answers and Replies

  • #2
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
32,767
5,036
.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?
 
  • #3
143
1
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
 
  • #4
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
32,767
5,036
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?
 
  • #5
143
1
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.
 
  • #6
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
32,767
5,036
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.
 

Related Threads on Deriving an equation for water flowing from a tap

Replies
2
Views
1K
Replies
4
Views
25K
Replies
1
Views
1K
Replies
12
Views
5K
Replies
43
Views
2K
  • Last Post
Replies
11
Views
35K
  • Last Post
Replies
1
Views
2K
Top