Having trouble with Rocket concepts

[Demon117]

Hello, I am trying to investigate single-stage rockets and I've come across a particular situation I don't know how to handle. The situation I have is that the rocket in question is burning it's fuel not at a constant rate but at a rate $R(\dot{m})$. So to find the equations of motion shouldn't be much different than for that of a constant rate. Assuming no external forces I should have:

$\frac{dp}{dt}=m_{o}\frac{dv}{dt} + V\frac{dm}{dt} = 0$

Here, V is given by $V=v-v_{ex}$, where $v_{ex}$ is the velocity of the particulates with respect to the motion of the rocket. But where does $R(\dot{m})$ enter into the picture? Or am I missing something here? Any good references for this type of question? Thanks in advance.

 

[SteamKing]

Remember m-dot = dm/dt, so R(m-dot) must be worked into the second term.

 

[Demon117]

Remember m-dot = dm/dt, so R(m-dot) must be worked into the second term.

It would seem that I have misrepresented the equation of motion in light of that. Modifying it would give me something more like

$m_{0}\frac{dv}{dt}+V R(\dot{m})=0$
$m_{0}\frac{dv}{dt} = -V R(\dot{m})$

From here how would you find v(t), this problem doesn't seem solvable analytically.

 

[BobG]

Are you sure you're burning your fuel at a non-constant rate?

I think the true situation is that the mass of your rocket is changing as you burn fuel at a constant rate.

The momentum of the fuel coming out the back is equal to the change in momentum of your rocket. Except, since you're burning fuel, not only is the velocity of the rocket changing; but its mass is changing too. By looking at the energy it took to move from the launch pad to some altitude, you can calculate how much fuel you burned and how that affected the rocket's mass.