# Rocket in deep space

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

$$p=mv_{i}+muln(\frac{m_{i}}{m})$$

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

$$\frac{\partial p}{\partial m}=v_{i}+u(ln\frac{m_{i}}{m}-1)=0$$

Giving 'm' as :

$$\LARGE m=m_{i}e^{\frac{v_{i}}{u}-1}$$

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

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Remember that momentum is a vector not a scalar.

Filip Larsen
Gold Member
Are you sure that is a maximum? (hint: ∂p/∂m = 0 is a necessary but not sufficient condition for a maximum).

D H
Staff Emeritus
Giving 'm' as :

$$m=m_{i}\,e^{\frac{v_{i}}{u}-1}$$
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.

Filip Larsen
Gold Member
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.