B Separation of variables - rocket equation

1. Nov 27, 2016

Januz Johansen

hello there
Im trying to do a derivation of tsiolkovsky's rocket equation, but i got stuck at the step when i have to use separation of variables (marked with red in the pic), i used maple to solve it, so i could get on with it, but i want to understand what is happening to solve this, so can anyone explain how to solve this step with separation of variables?
Thanks :)

2. Nov 27, 2016

Orodruin

Staff Emeritus
It is exactly what they have done. What step in particular do you have problems with?

3. Nov 27, 2016

Januz Johansen

ok thanks so i have done some right ;)

im having trouble explaining what is happening, or i think i do.
I can explain the first steps, just isolate the variables on each side of the equation.
But what rules are used/how is this integrated (the bordered step)
Thanks :D

4. Nov 27, 2016

Orodruin

Staff Emeritus
You integrate both sides between the same points. It is essentially making an integration and then making a change of variables.

5. Nov 27, 2016

Januz Johansen

Thanks i see now :D

6. Nov 27, 2016

Orodruin

Staff Emeritus
Oh, and the integration boundaries on the LHS should be $v_f$ to $v_i$. In this particular example it does not matter for the result because the integrand is constant. You then make a change of variables to $m(v)$ and use $m_f = m(v_f)$ and $m_i = m(v_i)$.

7. Nov 27, 2016

Januz Johansen

Hello
Do you mean like this?

im not 100% sure what you mean with the change of variables
Again thank you for helping

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8. Nov 27, 2016

Orodruin

Staff Emeritus
Yes. Consider the integral
$$\int_{v_f}^{v_i} dv.$$
Now, you know that $m$ is a function of $v$ so change variables to $m$. The integral changes to
$$\int_{m(v_f)}^{m(v_i)} \frac{dv}{dm} dm.$$
Insert the known differential equation and perform the new integral.

9. Nov 27, 2016

Januz Johansen

so i have it like so:

or do i get -1/u*m(vi)-m(vf)?
thank you for your patience and help

10. Nov 27, 2016

Orodruin

Staff Emeritus
No, you have the wrong integration boundaries in the first integral. They are what I said in my post.