# Saturn V specific impulse issue in velocity modelling with Tsiolkovsky's equation against actual values?

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: $$\Delta v(t) = I_\mathrm{sp} g_0 \left(\frac{R_\oplus}{a(t)+R_\oplus}\right)^2 \ln\left(\frac{m_0}{m_0-c(t)}\right).$$ $a(t) = 0.0033 t^2 - 0.1462 t + 1.9487$ is the altitude function derived from actual values, with a very small error range, and $c(t) = 12579.9 t$ is the fuel consumption of the rocket per second. Also I have \begin{align} \Delta v & && \text{change in velocity} \\ t & && \text{time elapsed} \\ I_\mathrm{sp} & = 263\ \mathrm{s} && \text{specific impulse time} \\ g_0 & = 9.81\ \mathrm{m/s^2} && \text{acceleration of gravity on Earth} \\ R_\oplus & = 6371\ \mathrm{km} && \text{radius of Earth} \\ m_0 & = 2{,}800{,}000\ \mathrm{kg} && \text{gross mass of Saturn V rocket.} \end{align}

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?

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

• Note that this site uses MathJax for rendering LaTeX equations. I don't have time at the moment to edit your post but you can have a look at the MathJax site or here or click "edit" on any of the questions with equations on this site to get an idea how it works. Jul 30 '13 at 7:36
• Yes, specific impulse changes during ascent due to reduction of pressure on the outside. Jul 30 '13 at 7:56
• At any time there is guaranteed to be at least one grad student around who is looking for ways to avoid research. Karan K: if you aren't familiar with Latex, it's a great thing to learn. You can click edit on this question now to see how I formatted things - and at least 95% of what I did is "proper" and not a hack :)
– user10851
Jul 30 '13 at 7:56
• What about drag?? Jul 30 '13 at 7:58
• Thank you for the links, @MichaelBrown! :) Will definitely read up! Jul 30 '13 at 8:21