Math and the Avenger video

Model rockets are more than just fun to fly - you can actually do math and science stuff with them. For example, I was wondering how high my Avenger flew yesterday. There was no altimeter in the payload section, so I had no direct measurement. But I did have a video of the flight, consisting of a couple of thousand frames taken at 30 frames per second. It occurred to me that a simple application of photogrammetry could help determine the altitude achieved by my two stage beauty. Time to do some math!

Frame 493 from the Avenger video with objects identified (click to enlarge)

Looking through the frames, I found the above frame showing the launch area about three quarters of a second before apogee. It was easy enough to locate Duane's SUV, and a Google search revealed that all models of this vehicle are 188.4 inches (15.7 feet) in length. I know the horizontal field of view of this camera is 37.8 degrees, having measured it about a year ago with a simple set up involving a couple of rulers. This should be enough to determine the altitude of the rocket at the time this frame was taken.

The relevant formulas are:

where theta is the angle subtended by the object (in this case, Duane's SUV) in radians, d is the length of the object in pixels, FoV is the field of view of the camera in degrees, and L is the horizontal width of the frame in pixels (720 for this camera). D is the actual length of the object, and H is the altitude of the rocket in the same units as D. It is easy enough to measure d, the length of the SUV in pixels with software - 21.9 pixels, according to the GraphClick program on my Mac. FoV is 37.8, and L is 720. D is the length we found from Google, 15.7 feet. Plugging in the numbers, we find

Is this reasonable?

Lacking measurements from an altimeter, the best we can do to check the above number is to look at the output of computer simulations. I constructed a CAD model of the Avenger in the Rocksim and Open Rocket software (the first is the best and costs about 100 bucks, the second is pretty good and free) and added in the camera, which I weighed at 0.5 ounces.  I like the Open Rocket interior view, which shows all the components:

Avenger CAD model in Open Rocket (click to enlarge)

Open Rocket gave a peak altitude of 789 feet, whereas Rocksim calculated an apogee of 833 feet on the C6-0/B6-4 motor combination. Rocksim usually produces altitudes that are a bit on the high side, but given the crudeness of our image analysis, the 782 foot altitude agrees very well with the computer simulations. Not bad for a little ciphering, eh?

Jethro Bodine would be proud.