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Bernoulli equation (watering can)

  1. Mar 29, 2016 #1
    1. The problem statement, all variables and given/known data

    Hi everybody! Here is a classical Bernoulli problem, which I'd like you to review to check if I (finally!) make a proper use of Bernoulli's equation!

    A gardener waters a bush with the help of a watering can (see attached pic). The water level in the can lays at height H = 1m, and the water flows horizontally out of the can at height h.
    a) at which distance x from the bush must the water flow when the mouth of the can is located at h = 0.8m, in order for the water to reach the bush?
    b) for which height h is x maximum? (we consider H is constant)

    2. Relevant equations

    Bernoulli equation, free fall and maybe a little bit of derivative?

    3. The attempt at a solution

    I think I get the point of the problem, but I often get confused :biggrin: Hopefully this time I get it right:

    a) I first set up a Bernoulli equation between points (0) (behavior at height H) and (1) (mouth of the can at height h):

    ρg(H - h) = ½⋅ρ⋅v12
    (Here I considered P0 = P1 = 0 at gauge pressure, v0 = 0 and h the reference height)
    ⇔ v1 = √(2⋅g⋅(H - h)) = √(0.4⋅g)

    Now from basic equations of motion I know:

    horizontal displacement: x = v1⋅t
    vertical displacement: -h = -½⋅g⋅t2

    I solve for t using the second equation and get:

    t = √(2h/g)

    ...and substitute it in the first one:

    x = v1⋅t = √(0.8⋅h) = 0.8m

    Is that correct?

    b) Here I used the equation I found for velocity in a):

    x = v⋅t = √(2g⋅(H - h))⋅√(2h/g) = 2√(hH - h2)

    Here I wasn't sure what I should do so I simply took the derivative of (hH - h2) with respect to h and solved for which h it is equal to 0 (that is, when the function reaches a local critical point):

    f'(h) = (hH - h2)' = H - 2h
    f'(h) = 0 ⇔ h = ½⋅H = 0.5 m

    Does that make sense? I get x = 1m, which is at least indeed bigger than in a). I'm sure there is also an easier method to get that result, but I couldn't figure it out yet. Any idea?


    Thank you very much in advance for your answers, I appreciate it.


    Julien.
     

    Attached Files:

  2. jcsd
  3. Mar 29, 2016 #2

    haruspex

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    All looks good, well done.
     
  4. Mar 29, 2016 #3
    @haruspex thank you, that is definitely thanks to your huge help! :) I hope this post can help other people too!

    Julien.
     
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