Rocket in deep space

  • Thread starter Sekonda
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  • #1
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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
 

Answers and Replies

  • #2
549
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Remember that momentum is a vector not a scalar.
 
  • #3
Filip Larsen
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Are you sure that is a maximum? (hint: ∂p/∂m = 0 is a necessary but not sufficient condition for a maximum).
 
  • #4
D H
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Science Advisor
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Giving 'm' as :

[tex]m=m_{i}\,e^{\frac{v_{i}}{u}-1}[/tex]
That's correct.

Which is wrong according to the solutions unless I assume v(i)=0 which I don't think I should.
That is a reasonable assumption for this problem.
 
  • #5
Filip Larsen
Gold Member
1,274
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Are you sure that is a maximum? (hint: ∂p/∂m = 0 is a necessary but not sufficient condition for a maximum).
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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