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Homework Help: Hot Air Baloon

  1. Feb 4, 2015 #1
    1. The problem statement, all variables and given/known data

    Kindly see the attachment.

    2. Relevant equations

    3. The attempt at a solution

    As with all such questions, its in setting everything up that I'm having some trouble.

    I know that F = mdv/dt + vdm/dt. And also that F = R - m(t)g, but R = M0g. From here though I don't know how to proceed to obtain the differential equation to solve for v. Any hints?

    Thank you very much.

    Attached Files:

  2. jcsd
  3. Feb 4, 2015 #2


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    Gold Member

    Careful there. Is the m in the first equation the same as the m in the second equation?

    Also, you have been given m(t), right?
  4. Feb 4, 2015 #3


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    Science Advisor

    How sure are you that F = m dv/dt + v dm/dt? Using principles of conservation of momentum in systems that are explicitly not closed is a tricky thing to get right.

    In the frame of reference in which the balloon is momentarily at rest, v = 0 and the v dm/dt term goes away. Acceleration does not change depending on the choice of reference frame. It follows that F = m dv/dt or, more familiarly, F = ma.
  5. Feb 8, 2015 #4
    Thanks for all the replies. I apologise for not getting back until now, my internet has been on the fritz. Ok, so I'm trying to be more careful with my m's. M is for the mass of the balloon, and m for the mass of the sand. So I have F = Mdv/dt = R - M(t)g, where R = M0g, and m(t) is the mass of the sand in the balloon. I calculated m(t) and got m(t) = m0(1 - t/T). Is that right so far? I'm just about to plug that into my force equation and I'll get back to you all on how that goes. Thanks

    Update: Ok so in my force equation I have M(t)g, but I have only worked out m(t). Would it be valid to make the assumption that since the mass of the balloon << mass of sand, that their combined mass is approximately the mass of the sand. I.e. M ~ m?
    Last edited: Feb 8, 2015
  6. Feb 8, 2015 #5
  7. Feb 9, 2015 #6


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    Yes. This all looks correct.
    No. The stated problem is careful to distinguish between m and M. You must be equally careful.
  8. Feb 12, 2015 #7
    Thanks for all the help!
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