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Rocket in deep space

  1. May 18, 2012 #1
    Hey,

    I have a question on a rocket in deep space (all external forces negligible), basically I'm doing something wrong the latter part of the question - maximizing the momentum via differentiation, here's the question:

    Rocket.png

    So the momentum at a given mass 'm' is :

    [tex]p=mv_{i}+muln(\frac{m_{i}}{m})[/tex]

    I attained a derivative of respects to 'm' as:

    [tex]\frac{\partial p}{\partial m}=v_{i}+u(ln\frac{m_{i}}{m}-1)=0[/tex]

    Giving 'm' as :

    [tex]\LARGE m=m_{i}e^{\frac{v_{i}}{u}-1}[/tex]

    Which is wrong according to the solutions unless I assume v(i)=0 which I don't think I should.

    Where am I going wrong?

    Thanks guys,
    SK
     
  2. jcsd
  3. May 18, 2012 #2
    Remember that momentum is a vector not a scalar.
     
  4. May 18, 2012 #3

    Filip Larsen

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    Gold Member

    Are you sure that is a maximum? (hint: ∂p/∂m = 0 is a necessary but not sufficient condition for a maximum).
     
  5. May 18, 2012 #4

    D H

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    Staff Emeritus
    Science Advisor

    That's correct.

    That is a reasonable assumption for this problem.
     
  6. May 19, 2012 #5

    Filip Larsen

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    Gold Member

    I can see that my earlier comment, made on the assumption that you did have an error somewhere, could be read to imply that your solution is not a maximum, when in fact it is. I apologize for any confusion my comment may have caused.

    After doing the actual calculation I too concur that your solution gives maximum momentum in the interval 0 ≤ m ≤ mi or, equivalently, when vi ≤ u.
     
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