Projectile motion with air resistance?

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The money issue was why I lacked confidence. I'm not gonna spend $50 to RENT a computer program for a semester, that's a load of bull. I'm trying to see if my teacher will let me use her copy (I'm doing just one calculation, so it's not like I'm really robbing wolfram research).

How did you do that in a spreadsheet? I think I'm going to try a couple different approaches to this, and hopefully at least one will work out reasonably well.

Since it's a long-range ballistics, my initial calculations (ignoring air resistance) had a max height of over 500 Km above the ground. I stopped a moment and realized that that's significant enough that you could no longer reasonably assume the force due to gravity to be constant. So now I'm going to try a lower angle (it will require a higher initial velocity of course, I previously fired at 45º) because I don't know how to account for a changing acceleration due to gravity.

What about Portage and Octave?

You don't need mathematica or matlab - I was just suggesting, if you had access to one of them, that would make it relatively easier - but it's not absolutely necessary by any means.

These folks here: http://octave.sourceforge.net/ claim to provide a mac os version of octave, which is very similar to matlab (no need for portage or fink or such, apparently)

With octave, you should be able to find plenty of differential equation tutorials using matlab, for the most part, it should be the same for octave.

Hmm. On looking back, I see that you want to...

Now, the first problem is that you haven't specified which country. The launch velocity wouldn't need to be very great, if the country were Vatican City.

If the country happens to be a larger one, I suspect that an answer of "approximately 11.2 kilometres per second" will suffice. That is escape velocity. In the absence of air resistance, if you can reach orbit, you can pretty much choose where to come down.

I'm in the United States, firing from Cincinnati, Ohio, to St Olaf College in Northfield, Minnesota. After correcting for the curvature of the Earth, it is a horizontal distance of slightly over 1000Km. This isn't exactly a realistic problem we're doing (as has been noted, the book would vaporize at such speeds, plus the max height reached is significant enough that we really shouldn't assume acceleration due to gravity to be constant) but my partner doesn't care about any of this and my teacher doesn't have enough of a mind for physics to notice.

Well, I'm feeling lazy...
http://en.wikipedia.org/wiki/Orbit_equation

Consider your book as just a satellite of the Earth. As such, it will have an elliptical orbit. The above gives your the formula for such orbits, which will form a family based on the eccentricity and the h^2/mu terms.

You require that set of the ellipses that intersect the surface of the Earth in two places 1,000km apart.

That will give you a set of possible firing angles/speeds. You could then choose the angle that requires the lowest speed. I'd guess it would be around the 45 degree mark, as we're not going to be wildly at odds with the case for normal ballistic parabolas.

I leave the detail to you...

don't know how helpful this would be, but search for references to the Paris Cannon the Germans used in WWI (fired about 100+ miles). There was also a guy who sold Saddam Hussein a cannon that would fire from Iraq to Tel Aviv (was actually built, maybe tested, but never used). This was prior to GWI (as opposed to the current GWII).

There may be some of his research posted on the Net, somewhere.

That's pretty amazing. I'll probably throw some of that in to my write-up on the history of theories of projectile motion.