Delta V Calculations for Science Fair

[Tank Buddy]

I am going to do some rocket science for my science fair project! I want to find what volume of fuel I have to burn for the Lv-909 and 48-7S (lighter but lower Isp) I have gotten the basis down for the equation. Exhaust velocity*In(mo/m1) = exhaust velocity*In(Mo/M1). I'm just having trouble figuring out how to make the equations equal when there are two variables (Mo and M1). Any help without spoiling the result?

I will add a photo of my math so far soon.

[Pbhead]

2 equations, two variables (mass before burn, mass after burn). easy sauce. that is the correct quantity of variables and equations.

[Supernovy]

I'd say solve for the mass ratio, then determine the dry/wet mass either from making up one of them, or by using the stock fuel tank masses.

[Tank Buddy]

I was planning on looking into the ratio of those masses, I'll make that a priority. Thanks.

[SkyRender]

Also, it's not In, it's ln (math shorthand for "natural logarithm").

[blizzy78]

Why not use a common payload fraction? In KSP, you can get away with 15% (and even more.) So for example, if your payload is 10 t, the complete rocket would be about 10 / 15 * 100 = 66.7 t. Which means that 57.7 t of that would be fuel and engines (ie. anything not being payload.)

[Superluminaut]

I am going to do some rocket science for my science fair project! I want to find what volume of fuel I have to burn for the Lv-909 and 48-7S (lighter but lower Isp) I have gotten the basis down for the equation. Exhaust velocity*In(mo/m1) = exhaust velocity*In(Mo/M1). I'm just having trouble figuring out how to make the equations equal when there are two variables (Mo and M1). Any help without spoiling the result?

I will add a photo of my math so far soon.

Hey cool project, but what exactly are the specifics here? Are you trying to determine how much fuel each engine needs to accelerate a given payload to x velocity?

[Yasmy]

Hi Tank. I'll go over the math for you.

A) The exponential function, exp, and the natural logarithm, ln, are inverses: exp(ln(x)) = x, and ln(exp(x)) = x

Try it on a calculator. exp(x) is also written e^x.

Logarithms remove exponents: ln(a^ = b * ln(a)

Note that if a = e, we get back the identity from above, since ln(e) = 1: ln(exp() = ln(e^ = b * ln(e) = b

Assume upper case letters for ship 1, lower case letters for ship 2:

1) V * ln(M0/M1) = v * ln(m0/m1)

Bring the powers inside using rule B above:

2) ln( (M0/M1)^V ) = ln( (m0/m1)^v )

Exponentiate both sides: If a = b, e^a = e^b:

3) exp(ln( (M0/M1)^V )) = exp(ln( (m0/m1)^v ))

Use rule A: exp(ln(a)) = a

4) (M0/M1)^V = (m0/m1)^v

Take the vth root of both sides: (the power of 1/v)

((M0/M1)^V)^(1/v) = ((m0/m1)^v)^(1/v)

This simplifies to:

5) (M0/M1)^(V/v) = m0/m1

Now you know how to find the ratio of initial to final mass,

m0/m1 = (M0/M1)^(V/v),

that ship 2 needs in order to have the same delta-v as ship 1.

[Yasmy]

Hi again, Tank.

Now you know the math. Let's talk design.

m0/m1 = (M0/M1)^(V/v) for equal delta-v.

Quiz: Figure out which of these scenarios is preferable: Which produces enough delta-v for ship 2?

A) m0/m1 < (M0/M1)^(V/v)

m0/m1 > (M0/M1)^(V/v)

Let's assume now that you have designed ship 1, and are happy its delta-v, but want a version with at least as much delta-v, changing only the engines and the amount of fuel.

In KSP, most liquid fuel tanks have a full to dry mass ratio of 9. In the real world, the ratios depend on a lot of things, such as fuel type and cryogenic needs.

I'm going to use 9 below like KSP, but you could use a variable or a different constant for a different tank mass ratio.

m0 = rocket mass with full fuel

m1 = rocket mass with no fuel

Let's define a few more variables:

mP = mass of the payload (minus the tanks): everything on the rocket that is not a fuel tank, fuel, or an engine

mT = mass of the empty fuel tanks

mE = mass of the engines

empty rocket: m1 = mP + mT + mE

full rocket: m0 = mP + 9*mT + mE

Presumably, mP = MP: the payload is the same on ship 1 and ship 2.

mE and ME are not the same: mE = 48-7S mass. ME = LV-909 mass, for example.

mT and MT are also not the same: ship 2 will need a different number of fuel tanks

I'm leaving the final bit of algebra to you: Solve for mT, the mass of the empty

tanks, (or 9*mT, the full fuel tank mass), (or 8*mT, the fuel mass).

Good luck!

[Tank Buddy]

I have decided my question is "How much fuel must two rockets burn in order to have the same delta v, where on rocket has a lighter but less efficient engine, and another has a heavier but more efficient engine." This will answer several questions:

1. What Delta-V the rockets overlap at.

2. How much fuel is burned.

3. When one engine is better suited.

[Superluminaut]

I have decided my question is "How much fuel must two rockets burn in order to have the same delta v, where on rocket has a lighter but less efficient engine, and another has a heavier but more efficient engine." This will answer several questions:

1. What Delta-V the rockets overlap at.

2. How much fuel is burned.

3. When one engine is better suited.

Very concise, I like it.

So here are some tools you’ll need

Things you can look up in math books or talk to math teachers about

Solving systems of linear equations

Properties of logarithms

Natural logarithms

This one you already know

The Tsiolkovsky rocket equation

Also making your project look clean and organized will make people believe you are a genius.

So type out everything in a professional font. In your math and labels use “Îâ€Vâ€Â, don't use “Delta-V†or "change in velocity", and use computer generated graphs with short labels so that someone who looked at it for 5 seconds knows exactly what everything means. You can use excel to graph by entering all your domain and range values. Should be some stuff on youtube that shows how to do it.

Also you probably know more about spaceflight from KSP than the people who will evaluate your project. While it may seem obvious to you why ÃŽâ€V is important, they probably don't know. So don't change your question, but include a paragraph explaining why it is important.