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Note: I have noted that one of the formulas I have provided does not show up on my webpage in preview mode and so have also made it as an attachment.

Quick summary: I have found two different websites that offer the equation that I am looking for in different forms, but I am starting to think that the equation is actually incorrect. So I'm really asking if any rocket engineers could assure me that the equation I am using is indeed incorrect and (if I'm lucky) point me to a reference where I can get find the actual equation. I have actually spent a long time (two days) looking for a better formula but the web has not proven very helpful and the books I have read have not gone into enough detail for actual calculations. I do not actually have a textbook (one does not exist) because this is not strictly coursework.

Find the ideal mass distribution and total mass of a three stage rocket assuming:

1. A specific impulse of 300 seconds for all three stages

2. A ##\Delta## V of 9000 m/s

3. A dry mass fraction for each stage of 0.35

4. There are three stages

5. Payload is allowed to be left as an unknown

1.

https://www.physicsforums.com/cid:F649DCE6-2FDC-453C-830C-5D2F46A91771@guardedsystems.com

Where

M = final mass of the rocket (all three stages added together)

A = payload mass

S = dry mass fraction

Vf = final velocity: (considering the rocket starts from 0 m/s) is ##\Delta## V

C = exhaust velocity

https://math.la.asu.edu/~nbrewer/Fall2007/MAT267/RobertWagner/RobWagner%20Footnote%20181.html

- This assumes a constant inert-mass fraction throughout the rocket, which for my analysis is a given. It is clean because it uses a final formula rather than me having to partially derive one by adding up all of the stages of the rocket.

2.

## V_{exh} = I_{sp} * g_0## (For a rocket blasting off earth) (I'm using 9.8 for ##g_0## for now)

I can calculate the ##V_{exh}## to be 2940 using formula 2.

The rest is just plugging into formula 1 from which I can get a ratio between the final mass of the rocket and the mass of the payload. Here's the calculation:

## M = A \bigg( \Big((1 - .35) e^{9000/(3 * 2940)}/\big(1 - .35*e^{9000/(3 * 2940)}\big)\Big)^3 - 1 \bigg) ##

The end result is that M = 240805 A or that the total mass is 240 thousand times the payload. The Saturn V has a mass that is only roughly 17 times its payload (borrowed from mass data on Wikipedia here: https://en.wikipedia.org/wiki/Saturn_V). When I make ##g_0 ## only 9.3 I suddenly get a negative number for my mass payload ratio. This is telling me that I'm working with a very finicky formula and that it is probably incorrect.

I have done this equation in a more tedious fashion using a stage by stage analysis provided here:

http://www.projectrho.com/public_html/rocket/multistage.php#inertmass

Unfortunately this has led me to the same unrealistic answer, but prevents me from assuming my formula is incorrect (since two different sites seem to agree on the process).

Thanks for taking the time to read this.

Quick summary: I have found two different websites that offer the equation that I am looking for in different forms, but I am starting to think that the equation is actually incorrect. So I'm really asking if any rocket engineers could assure me that the equation I am using is indeed incorrect and (if I'm lucky) point me to a reference where I can get find the actual equation. I have actually spent a long time (two days) looking for a better formula but the web has not proven very helpful and the books I have read have not gone into enough detail for actual calculations. I do not actually have a textbook (one does not exist) because this is not strictly coursework.

1. Homework Statement1. Homework Statement

Find the ideal mass distribution and total mass of a three stage rocket assuming:

1. A specific impulse of 300 seconds for all three stages

2. A ##\Delta## V of 9000 m/s

3. A dry mass fraction for each stage of 0.35

4. There are three stages

5. Payload is allowed to be left as an unknown

## Homework Equations

1.

https://www.physicsforums.com/cid:F649DCE6-2FDC-453C-830C-5D2F46A91771@guardedsystems.com

Where

M = final mass of the rocket (all three stages added together)

A = payload mass

S = dry mass fraction

Vf = final velocity: (considering the rocket starts from 0 m/s) is ##\Delta## V

C = exhaust velocity

https://math.la.asu.edu/~nbrewer/Fall2007/MAT267/RobertWagner/RobWagner%20Footnote%20181.html

- This assumes a constant inert-mass fraction throughout the rocket, which for my analysis is a given. It is clean because it uses a final formula rather than me having to partially derive one by adding up all of the stages of the rocket.

2.

## V_{exh} = I_{sp} * g_0## (For a rocket blasting off earth) (I'm using 9.8 for ##g_0## for now)

## The Attempt at a Solution

I can calculate the ##V_{exh}## to be 2940 using formula 2.

The rest is just plugging into formula 1 from which I can get a ratio between the final mass of the rocket and the mass of the payload. Here's the calculation:

## M = A \bigg( \Big((1 - .35) e^{9000/(3 * 2940)}/\big(1 - .35*e^{9000/(3 * 2940)}\big)\Big)^3 - 1 \bigg) ##

The end result is that M = 240805 A or that the total mass is 240 thousand times the payload. The Saturn V has a mass that is only roughly 17 times its payload (borrowed from mass data on Wikipedia here: https://en.wikipedia.org/wiki/Saturn_V). When I make ##g_0 ## only 9.3 I suddenly get a negative number for my mass payload ratio. This is telling me that I'm working with a very finicky formula and that it is probably incorrect.

I have done this equation in a more tedious fashion using a stage by stage analysis provided here:

http://www.projectrho.com/public_html/rocket/multistage.php#inertmass

Unfortunately this has led me to the same unrealistic answer, but prevents me from assuming my formula is incorrect (since two different sites seem to agree on the process).

Thanks for taking the time to read this.

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