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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:

velocity versus 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!

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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. –  Michael Brown Jul 30 '13 at 7:36
    
Yes, specific impulse changes during ascent due to reduction of pressure on the outside. –  Deer Hunter Jul 30 '13 at 7:56
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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 :) –  Chris White Jul 30 '13 at 7:56
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What about drag?? –  Deer Hunter Jul 30 '13 at 7:58
    
Thank you for the links, @MichaelBrown! :) Will definitely read up! –  Karan K Jul 30 '13 at 8:21
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