Calculating Apollo 13 Command Module Drag Coefficient

[petitericeball]

So, after looking at all the stuff those geniuses up at NASA came up with, I'm trying to figure out how to get the drag coefficient for the Apollo 13 command module reentry.

Heres the stuff I thought were important:

Entry-interface
400,000 ft
36,129fps

Drogue chutes deploy at 23,000 ft, slowing module down from 300mph to 175 mph.

So, 300mph is 1584000fph or 26400 fps. So, 36,129fps - 26400fps = 9729fps.

If the Ei is at 0:00 then the Drogue chutes open at 8:16, or 496 seconds, divide 9729 by 496 and you get the speed lost per second, which would be 19.61fps.

The equation is Cd=drag/(.5*pAV^2)

I have no idea where to go next. I don't know how to convert what I have into drag, and the density of the fluid is always changing. I have a chart that shows the relation between altitude and air density, but I can't find a way to put all of it in without doing a new equation for each step in altitude. So both velocity and air density would be constantly changing, the only constants are the .5 and the A.

This project is due on Thursday, so any help would be awesome.

 

[rcgldr]

You have more unknowns here, the re-entry angle, and the mass of shuttle, plus the coefficient of drag which you're trying to determine, all of which affect the path. The density of the atmosphere versus altitude is available at a few web sites, typically there are 3 equations used, depending on the altitude. The path is a complex curve, making it more difficult to numerically solve. You'll need to use numerical integration, such as Runge Kutta:

http://en.wikipedia.org/wiki/Runge-Kutta

If your trying to solve for coefficient of drag, I can only think of an iterative process that makes an initial guess, then "binary" searches (trying higher / lower steps in drag) until the results match the speed versus altitude versus time at the two known points.

Note that NASA knew in advance what the coefficient of drag was, since they don't get to do repeated re-entries to determine re-entry angles for the capsule to end up within a desired target zone.

On a side note, the Lunar Module had a plutonium button / thermal condcutor power source, and this is the only one ever to return to Earth (it's now at the bottom of some ocean, probably still generating electricity).