Saturn V specific impulse issue in velocity modelling with Tsiolkovsky

[unicornication]

Hi!

I've been trying to model the Saturn V's velocity using Tsiolkovsky's ideal rocket equation, and in the process, I think I may have made a mistake with regards to the specific impulse?

I've come up with the following equation, taking the change in gravity into account. (a(t) is the altitude function derived from actual values, with a very small error range, c(t) is the fuel consumption of the rocket per second)

(note: what appear to be powers after the end of most lines are footnote references, apologies for the confusion!)

And this a plot of the model I created versus the actual values plotted against time-

The percentage error here between the two sets of values vary from ~80% to ~14%, and the graph shape is vastly different.

I'd like to ask if there is a change in specific impulse? Or have I done anything else wrong in modelling the equation above?

I think specific impulse does change, but is there a mathematical equation by which I can rewrite this equation to take that into account?

Also, I apologize if I've made a stupid mistake, I'm a HS student doing some fun research, quite new to this!

Thank you very much!

edit: title was shortened, "Saturn V specific impulse issue in velocity modelling with Tsiolkovsky's equation against actual values?" was the original title

 

[mfb]

Some effects I don't see in your formula:
- the exhaust velocity is the specific impulse multiplied with g at the surface, not at the current position of the rocket. Anyway, that is a small effect
- gravity is pulling the rocket down, this gives -g acceleration in the vertical direction
- the rocket launch is not purely vertical, so the interaction between acceleration and gravity is not trivial
- air resistance is not negligible
- the specific impulse depends a bit on the air pressure

I assume your calculation is for the first stage only.

 

[D H]

Some effects I don't see in your formula:
- the exhaust velocity is the specific impulse multiplied with g at the surface, not at the current position of the rocket. Anyway, that is a small effect

Of the effects you listed, this is the only one included in the original post. As you mentioned, this is a small effect compared to the others that you listed.unicornication, gravitation, a non-vertical launch, air resistance, and altitude-dependent specific impulse are much bigger effects than the one effect that listed by mfb that you did incorporate in your model.

 

[mfb]

Of the effects you listed, this is the only one included in the original post.

What does the denominator do then? It looks like a modification of g to the gravitational acceleration at the current height.

 

[D H]

I read your response incorrectly, and also the original post. You are correct, mfb. The relation between specific impulse in seconds and exhaust velocity is g₀, not the value of g at altitude. That factor shouldn't be there.unicornication, where did you get that equation?

 

[unicornication]

Some effects I don't see in your formula:
- gravity is pulling the rocket down, this gives -g acceleration in the vertical direction
- the rocket launch is not purely vertical, so the interaction between acceleration and gravity is not trivial
- air resistance is not negligible
- the specific impulse depends a bit on the air pressure

Hi mfb,
Thank you for your feedback, greatly appreciated!

Though, I'm kind of stuck as to what the relationships are between the variables you stated and, say, time- in other words, I'm not exactly sure how to factor these changes into the equation.

I assume your calculation is for the first stage only.

That is correct!

unicornication, where did you get that equation?

Hi D H,
Thank you for your response, again, greatly appreciated! I thought about why the numbers are as large as they are and thought that the constant 9.81m/s gravity could be what's making the numbers as large as they are, and I knew that gravity changes with altitude, and so I thought that was a factor, apparently not! I got the equation http://www.mansfieldct.org/Schools/MMS/staff/hand/lawsgravaltitude.htm.

 

[SteamKing]

When a rocket lifts off, the direction of the initial trajectory is vertical, but as the rocket climbs, its trajectory assumes a shallower angle with respect to the horizontal so that the velocity of the rocket will send it into orbit.

The aerodynamic drag force on a rocket will be proportional to the density of the atmosphere (which also changes with altitude) and the square of the velocity of the rocket, among other factors.

 

[mfb]

Though, I'm kind of stuck as to what the relationships are between the variables you stated and, say, time- in other words, I'm not exactly sure how to factor these changes into the equation.

For a purely vertical launch, gravity would just give another summand in the formula (downwards, of course). For a real launch, you need data about the angle between the rocket and gravity, and you have to keep track of horizontal and vertical velocity at the same time.

For aerodynamic drag, see SteamKing.

- the specific impulse depends a bit on the air pressure

I guess if you don't want to neglect this, you need more data about the rocket.