This is one of my favorite stories. In short, one of John Burk's (@occam98) students wanted to launch a space balloon. If you want all the details, this post at Quantum Progress pretty much says it all. The part that makes this story so cool is that it was the student who did all of the set up and fundraising and stuff. Love it. Oh, and the student is apparently named "M." I wonder if the student is either one of the Men in Black or a James Bond scientist.
Ok, you know what I do, right? I need to add something. Here is a very nice video of the space balloon launch.
Think about the things you do as a faculty member or a scientist or a writer or a home maker. You know what all of these people do? Organize stuff. They plan, they make things happen. They arrange a field trip for a group of kids to the local zoo. They coach soccer and plan games. They host conferences. When do you learn how to do this stuff? For me, it was as an undergraduate student when I took the course Make-Stuff-Happen 101. No, there was no such course. I learned on the job. These students will have an advantage. They already have experience making a project happen.
Enough about the project. I want to add something. When I watch the video from the balloon, I think "hey, I wonder if you could get altitude data just from the video?" I think you can. I am sure these space cats collected altitude data with some device, but what if it failed? How would I measure the height of the balloon? Angular size, that's how. If I know how big something is in real life AND I know the angular size, I can estimate the distance to that object. Here is a simple diagram.
If the angle is small enough, then the length of the object (L) is pretty close to the arc-length of the segment of the circle described by the angle θ. Hopefully, my diagram isn't too confusing. Here I have the object a distance r away from the observer. This would give the following relationship:
This seems pretty simple. If I know the angular size of an object and the actual length of the object, I can get the distance from this object. Two small problems: what object and what is the angular size of the images from the camera? First, the object. That is pretty obvious. Here it is:
According to Google maps, the selected points on this building are 67.5 meters apart. As the balloon gets higher, I can choose a different set of points (like two separate buildings) to calculate the height.
Great. But what about the angular size? This is a bit of a problem. First, the video could be edited and scaled down (or up). Second, I have no idea what kind of camera they used (or I could just look up the angular field of view). Just as an example, the iPhone 4 camera has a horizontal angular field of view of around 56°. If this was the camera used, I could go from there. However, I will need some other "trick".
I am going to have to guess at some sizes and distances to find the angular size. Yes, I know this isn't idea - but it is what I am going to do. Here is my best guess for distances show in the video from the camera right before launch.
This other frame gives an estimate for the starting height of the camera.
From this, I am going to guess that the camera starts about 1 meter above the ground. This would put the angular size of the camera's field of view at:
An angular size of 44.7° seems pretty reasonable. Oh, I know what you are saying. I can hear it all the way from here. "Why don't you just email this student and ask what kind of camera they used? Really, it's simple." My answer is "no". This is like saying "oh, you are having difficulty with a level on Angry Birds? Just use this cheat code or the Mighty Eagle." What fun is a game if you have to cheat?
Ok, one more thing on the angular size. How about angular size with uncertainties? Suppose the length in the video has an uncertainty of about +/- 5 cm and the distance to the ground has an uncertainty of about +/- 15 cm (those are just guesses). In that case, I could do a Monte Carlo calculation for uncertainty. This would give an uncertainty in the angular camera size of 0.14 radians (8 °).
Video Analysis
Now for the fun part. I can just mark the locations of the building in the frame and find the angular size of the building as a function of time. Knowing the size of the building, I can get the height as a function of time (with uncertainty of course). I hope that is obvious by now that I will use Tracker Video to get the angular data. Here is my first plot. This shows the angular size of two objects (the building and then later the distance from the building to the baseball field) using units of percent of the angular camera width.
Let me just be clear how I got this plot. After marking two locations on the building, I get (x,y,t) data for each point. The actual values for x and y don't really matter. To find the distance between these two points, I use:
Since I put the scale of the video with a width of 100 units, the distance between the points will essentially be the angular size in units of percent of the camera angle. See.
Ok, but we (by "we" I mean "I") really want the distance to the object. I just need to slightly modify my equation from before. Remember, I am calling s the angular size of the object in units of percent of the camera angle.
Here is a plot of the distance from the camera as a function of time. Remember in this case, L is the length of the building at 67.5 meters and the width of the camera angle is 0.78 radians.
That turned out a little bit better than I expected (I have low expectations sometimes). This plot says that after about 10 minutes, the balloon was just under 3000 meters high. The other thing I like is that for the time that I used two objects on the ground, the calculated heights agree fairly well. One other thing, this looks like the balloon ascended at a fairly constant speed. Interesting.
But what about the uncertainty? What is the lowest and highest values for the height that I could reasonably get? For the low end, I could say that the camera angle is at the higher value of 0.78 + 0.14 radians. Suppose I further assume that the uncertainty due to the length of the points in real life is pretty small compared to the camera angle. Then for the high end of altitude estimation, I could use the smaller camera angle, 0.78 - 0.14 radians. Here is a plot showing these upper and lower estimations.
This doesn't look too bad. But notice that as the balloon gets higher, the uncertainty in the height also gets larger. Ok, one more thing. What if I assume the the balloon ascends with a constant speed? I can find the slope of the height vs. time plot to get this value. Here is what that would look like. Oh, here is a quick refresher for linear regression in python.
I fit two different linear functions for the two sets of data. These give vertical speeds of 3.2 m/s and 4.5 m/s.
Homework
Here are your homework questions. They are due before I get around to blogging about them (you know if you are slow, I'll do - I will).
- What is the uncertainty in the vertical speed? Could you use a Monte Carlo uncertainty calculation?
- Is a linear fit the best for this data? Theoretically, should a balloon ascend at a nearly constant speed? This is while the air density is getting smaller and balloon radius is getting larger. Do these two effects cancel to produce a constant "upward" terminal speed?
- How well does this altitude data match altitude data from a pressure sensor? (I suspect you need the other data to answer this question).
- Did you see it? At around the 12:33 time in the video, there is a jet that flies into the field of view. Based on the angular size of the plane, how high is the plane flying? You will probably need to guess the actual type of plane and look up the size. This example might be useful.
- Similar to the question above, how fast was this plane flying?
- Similar to both of the previous questions, who was flying this plane? Where were they going? What did the pilot have for breakfast?
- If you assume a constant ascending speed, how long would it take the balloon to get to the height of the Red Bull Stratos space jump at 120,000 feet?
That should keep you busy for a while.
