# How to account for burned fuel mass when calculating spacecraft acceleration?

I thought I could simply remove half of the burned fuel mass to account for the mass lost during the engine burn.

But I could not find a Newton's Second Law formula Calculator that would allow that, or any other source online that would support my logical conclusion.

Instead, I can see "Propellant mass fraction" and it's logarithmic application in the Tsiolkovsky rocket equation.

I need a rough fuel mass estimation when accelerating spacecraft form NEO asteroid towards Earth.

I would greatly appreciate if someone could clarify that for me without complicated math.

• If the math behind the Tsiolkovsky rocket equation is too much, this question could not be answered. Did you read en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation very carefully? You may need several hours and some days.
– Uwe
Commented Jan 14, 2023 at 22:49
• There is a story about Wernher von Braun. He was interested in Tsiolkovsky's book but could not understand it. He asked his teacher for help and was told : you got to learn the math. The career of von Braun would not been possible without understanding the math behind the rocket equation.
– Uwe
Commented Jan 14, 2023 at 22:54
• Build it in KSP and just use MechJeb. Commented Jan 15, 2023 at 5:49
• I was wrong, the book about rockets von Braun wanted to understand was written by Oberth, not Tsiolkovsky. (Die Rakete zu den Planetenräumen)
– Uwe
Commented Jan 15, 2023 at 21:07

The quantity you ultimately need when planning your manoeuvre is change in velocity, which in spaceflight terminology is called delta-v, $$\Delta v$$ (searching this term would give you a lot of explanations)

You are absolute right that accounting for burned propellant is required to account for the rocket accelerating quicker when nearing empty.

The reason your method of "using the average mass" doesn't work is that rockets accelerate much, much quicker when nearing empty.

Rocket acceleration curves look like this:

Finding the net change in velocity amounts to finding the area under this function.

In everyday life, we encounter many linear functions. These are easy to find the area under, we just use the halfway point, relying on the fact that the area under the curve is the same as the rectangle formed by the halfway value. Your method is this method.

This is calculus, which is math that's absolutely essential to understanding rockets.

However: Your method is fine as a rough estimate, because the blue area and the red area aren't too different if the spacecraft isn't mostly fuel and little else. It will just always be too pessimistic, underestimating how much you would really accelerate. You'll have a little extra left in your tanks when the burn is done, which is fine unless you're using taxpayer money.

Using the formulas, on the other hand does not require calculus.

$$\Delta v = v_e \cdot ln\left(\frac{m_{start}}{m_{end}}\right)$$

"Change in velocity is rocket exhaust speed times funny function of start mass divided by end mass." The funny function takes care of calculating the area under rocket curves.

You do perhaps need the reverse formula:

$$\frac{m_{start}}{m_{end}} = e^{\Delta v / v_e}$$

The funny number $$e$$ takes care of the curvy rocket curve. "How much fuel do I need" is the difference between $$m_{start}$$ and $$m_{end}$$

You can in fact do a straight reaction mass calculation if you have the exhaust velocity - rocket+fuel at steady state then fuel at speed X in one direction and rocket in the other.

The issue is that it will only be sorta accurate where fuel burned is a small fraction of the total craft mass, if 90% of the starting mass is fuel then burning the last 10% of fuel will generate a lot more DV than the first 10%. See beautiful graphs in SE - stop firing the good guys answer

It may be useful to break this down and do a spread sheet where you start with your final vehicle mass and speed, work out how much fuel would have been needed to get from 99% to your target velocity, add that fuel to the vehicle mass and find the fuel needed to push craft+final fuel from 98% to 99% velocity and work back 100 steps, then total up your fuel masses. Alternatively if your source data is a thrust and fuel consumption rate you can do 1 second burns adding fuel mass each time until you get your target velocity.

This will be incorrect, being higher than a proper Tsiolkovsky implementation but then you can do your log math in the same spread sheet and have a sanity check you got it right.

• Intuitively, it feels like a tradeoff between ease of understanding and mathematical precision. If you write an article for a wider audience about the feasibility of a certain space mission and use Newton’s: a = F/m almost everyone will understand it. But if you put in “Tsiolkovsky rocket equation”, a lot of people will lose attention. Commented Jan 15, 2023 at 15:01
• @TheMatrixEquation-balance if your goal is to be more engaging than correct, please don't write articles about space. Commented Jan 17, 2023 at 18:46