Mass to orbit calculations

[Axiom]

I'm trying to rearrange the rocket equation (for 2 stages, I want to add more though) to give me payload to a given delta-v, but I get stuck on this step.

Is there any way to rearrange for X? If not, is there any way to get a close approximation of X?

A is the wet mass of stage 1 and stage 2
B is the dry mass of stage 1 and wet mass of stage 2
C is the wet mass of stage 2
D is the dry mass of stage 2

 

[Altaïr]

The equation looks too complicated, I don't see how you could rearrange and solve for x analytically... It would be doable with a single stage, but not two. You'll have to plot it for the solution.

A trick I use to make the rocket equation easier to deal with is that I express it using fuel mass instead of wet mass. As wet mass = fuel mass + dry mass, it can be expressed that way:
ΔV = Ve × Ln(1 + fuel_mass/dry_mass)

For your second stage for example, fuel_mass is (C-D), while dry_mass is (D+x). C-D is known, so you only have an unknown variable at the denominator:
ΔV2 = Ve2 × Ln(1 + (C-D)/(D+x))

That still won't save you unfortunately, but I find this form easier to deal with in practice

 

[Axiom]

Ah ok, I guess I'm stuck with plugging in values and seeing what works

I could use a graphs but the problem is that excel can't plot exact graphs for these types of equations

 

[Altaïr]

You can use desmos or any other graphing calculator for this. Otherwise you can test values by yourself manually indeed.

 

[Axiom]

Yeah, I kinda do that sometimes, but the problem is that Excel is better for planning the rocket and Desmos is better for graphing Delta-V, I guess I could get Excel to generate the values for Desmos but that's a problem for not today