Propellant mass calculation

Question

I have been given the following problem:

To solve it I used the formulas:

$v_e = I_{sp}.g_0$

$M_{propellant} = (e^{\Delta_v/v_e} - 1).M_{final}$

And I assumed that the apogee kick manoeuvre was executed first. So I started by calculating the propellant required for the second manoeuvre:

$v_e = 200\times9.81 = 1962m/s$

$M_{p2} = (e^{1050/1962} - 1).3000 = 2123.21Kg$

And then for the first manoeuvre I got:

$v_e = 320\times9.81 = 3139.21m/s$

$M_{p2} = (e^{2300/3139.2} - 1).(3000+2123.21) = 5536.29Kg$

Giving a total propellant mass of:

$M_p = 7659.5Kg$

However, this answer differs from the correct answer. Furthermore, if I consider the manoeuvres being performed in the reverse order, I get different masses for each propellant, but somehow, the total mass is the same, which seems strange, since the hint suggests that the order matters in the total amount of fuel required.

Could you please point out where did I make a mistake? Thank you.

Answer 1

As discussed in the comments, your calculated value is correct and the test solution is wrong.

I will however make a note about this part:

Furthermore, if I consider the manoeuvres being performed in the reverse order, I get different masses for each propellant, but somehow, the total mass is the same, which seems strange, since the hint suggests that the order matters in the total amount of fuel required.

This is expected!

let's rearrange your formulas slightly:

$$\frac{M_{start}}{M_{final}} = e^{\Delta v / v_e}$$

That is, each burn just describes a mass ratio of before and after the burn. Now consider the total mass of the rocket:

$$ M_{total} = M_{dry} \cdot e^{\Delta v_1 / v_{e1}} \cdot e^{\Delta v_2 / v_{e2}}$$

Which is completely equal to doing the burns in the opposite order:

$$ M_{total} = M_{dry} \cdot e^{\Delta v_2 / v_{e2}} \cdot e^{\Delta v_1 / v_{e1}}$$