I am honestly not quite sure how many blog posts I have about Thanksgiving. It’s probably about 1 per year for 8 years. I’m going to guess it’s 8. Here goes my internet search.

This is really just for me so that I won’t forget. I mean, I will forget—but then I can look back at this post and remember stuff. Here’s to you Future Rhett.

But this isn’t the best system. The problem is that there are two coordinates—the angle for the top bar and the angle for the bottom bar. Sure, it’s cool—but what if you want to plot angle vs. time or something. You have to plot both angles vs. time and that’s a bummer.

OK, how about a model of bounded population growth? That’s just one dimensional, right? Actually, it doesn’t even have to be population, it’s just an equation—something like this.

In this expression, r is some parameter—it really doesn’t matter what. Let’s just model this expression for different values of r. I’ll use a starting x value of 0.1 and r values of 0.7 and 0.9. Here is the code.

Notice that when r = 0.7, the population reaches some stable value—but this is not true for r = 0.9.

Bifurcation Diagram

Now for another way to look at a chaotic systems—the bifurcation diagram. Honestly, I didn’t really understand these things until I made one. Here’s what we are going to do.

Start with some initial value of x (just pick something—I’m going to use 0.5). Pick a value for r also. Let’s just start at 0.1.

Run the model for 200 iterations and throw out that data. This should allow us to look at the long term behavior for that particular value of r (throws out the transient behavior).

Now run the model for 100 additional iterations and save these.

Create a plot of these final x values vs. r.

Next increase the r value a little bit (I will increase it by 0.001)

Repeat until you get bored.

So if the model is stable after the initial stuff, then it will just keep plotting the same value of x after the first 200 iterations and you will just get a dot. If it’s not stable after the first stuff, then you will get a bunch of dots with different x values.

OK, let’s do it. Here is the code. Oh, I made a function to iterate the model. I probably should put more comments in there.

This is what it looks like.

Up to an r value of about 0.75, you only get one final x value. After that, you get two different values . With r over 0.9, it gets crazy.

OK, that’s enough for now. I just want to make sure future Rhett knows how to make a bifurcation diagram.

There should be a grave yard for blog posts that start, but never get published. Fortunately, I have this site. Here I can share with you my failed posts. Get ready.

It starts with this epic video from the Soyuz MS-10 failed launch.

That’s pretty awesome. It’s doubly awesome that the astronauts survived.

Ok, so what is the blog post? The idea is to use video analysis to track the angular size of stuff on the ground and from that get the vertical position of the rocket as a function of time. It’s not completely trivial, but it’s fun. Also, it’s a big news event, so I could get a little traffic boost from that.

How do you get the position data? Here are the steps (along with some problems).

The key idea is the relationship between angular size, actual size and distance. If the angular size is measured in radians (as it should be), the following is true where L is the length (actual length), theta is the angular size, and r is the distance.

Problem number 1 – find the actual distance of stuff on the ground. This is sort of fun. You can get snoop around with Google maps until you find stuff. I started by googling the launch site. The first place I found wasn’t it. Then after some more searching, I found Gagarin’s Start. That’s the place. Oh, Google maps lets you measure the size of stuff. Super useful.

Finding the angular size is a little bit more difficult. I can use video analysis to mark the location of stuff (I use Tracker Video Analysis because it’s both free and awesome). However, to get the angular distance between two points I need to know the angular field of view—the angular size of the whole camera view. This usually depends on the camera, which I don’t know.

How do you find the angular field of view for the camera? One option is to start with a known distance and a known object. Suppose I start off with the base of the Soyuz rocket. If I know the size of the bottom thruster and the distance to the thruster, I can calculate the correct angular size and use that value to scale the video. But I don’t the exact location of the camera. I could only guess.

As Yoda says, “there is another”. OK, he was talking about another person that could become a Jedi (Leia)—but it’s the same idea here. The other way to get position time data from some other source and then match that up to the position-time data from the angular size. Oh, I’m in luck. Here is another video.

This video shows the same launch from the side. I can use normal video analysis in this case to get the position as a function of time. I just need to scale the video in terms of size. Assuming this site is legit, I have the dimensions of a Soyuz rocket. Boom, that’s it (oh, I need to correct for the motion of the camera—but that’s not too difficult). Here is the plot of vertical position as a function of time.

Yes, that does indeed look like a parabola—which indicates that it has a constant acceleration (at least for this first part of the flight). The term in front of t^{2} is 1.73 m/s^{2} which is half of the acceleration. This puts the launch acceleration at around 2.46 m/s^{2}. Oh, that’s not good. Not nearly good enough. I’m pretty sure a rocket has an acceleration of at least around 3 g’s—this isn’t even 1 g. I’m not sure what went wrong.

OK, one problem won’t stop me. Let’s just go to the other video and see what we can get. Here is what the data looks like for a position of one object on the ground.

You might not see the problem (but it sticks out when you are doing an analysis). Notice the position stays at the same value for multiple time steps? This is because the video was edited and exported to some non-native frame rate. What happens is that you get repeating frames. You can see this if you step through the video frame by frame.

It was at this point that I said “oh, forget it”. Maybe it would turn out ok, but it was going to be a lot of work. Not only would I still have to figure out the angular field of view for the camera, but I need to export the data for two points on the ground to a spreadsheet so that I can find the absolute distance between them (essentially using the magnitude of the vector from point A to point B). Oh, but that’s not all. When the rocket gets high enough, the object I was using is too small to see. I need to switch to a larger object.

Finally, as the rocket turns to enter low Earth orbit, it no longer points straight up. The stuff in the camera is much farther away than the altitude of the rocket.

OK, that’s no excuse. I should have kept calm and carried on. But I bailed. The Soyuz booster failure was quite some time ago and this video analysis wouldn’t really add much to the story. It’s still a cool analysis—I’ve started it here so you can finish it for homework.

Also, you can see what happens when I kill a post (honestly, this doesn’t happen very often).

Actually, there is one other reason to not continue with this analysis. I have another blog post that I’m working that deals with angular size (ok, I haven’t started it—but I promise I will). That post will be much better and I didn’t want two angular size posts close together.

One sentence labs. Leave the procedure up to the students. I think I will need some type of turn in sheet for these labs though. What about informal lab reports?

After the summer session of physics (algebra-based), I have the following comments.

It seems like every other video has a problem with vector notation. Students often set a vector equal to a scalar. Frustrating.

Students seem to confuse two standards: The Momentum Principle and Collisions. I have students submit videos for the momentum principle that are just a collision. The key point is that the momentum principle deals with force, time and change in momentum. I guess this is my fault. I offered suggested homework problems from a textbook and it covered momentum and collisions in the same chapter. I guess they thought they were the same thing.

Students are not very skilled at picking problems to solve. They like the lowest level of something like “mass is 2 and velocity is 3, what is the momentum?”. I tried to help them, but it didn’t seem to work. I showed a bunch of questions in class and had them “rate” them then discuss what makes a good problem. (I think I wrote about that here on my blog).

I’m still not happy with the “student review”. I want students to watch other student videos – but I don’t know how to implement that.

Students like to procrastinate. I’m getting a bunch of redos on the last day of submissions. That sucks to grade.

I hate vertical videos – but I hate videos that are recorded sideways even more. I stopped accepting the sideways videos since they can fix it and send it back to me.

I try to give meaningful feedback in my responses – but sometimes I just give a grade (score out of 5 points).

I’m trying to give higher scores. If they do well on the in-class assignment and submit multiple videos that aren’t wrong, I typically will at least give a 4/5.

Part of the reassessment process has students pick problems to solve that they think are good demonstrations of their understanding of the material (or the standard).

For me (as the evaluator), I can learn quite a bit about what a student thinks just based on the problem they pick to solve. However, it seems that students really don’t want to pick problems. They would prefer to have me just tell them what problems to solve.

OK, let’s do this. Let’s look at some problems and see which ones are good and which ones are not so good. In this case, it will be for the Position-Velocity-Acceleration standard. For this standard, students should show that they understand and can use the definitions of position, velocity, and acceleration in 1 dimension. So here are some questions. You get to pick which one is the best. Actually, why don’t you score them from 0-10 (11 being the best).

Problem A.

A plane has a mass of 1120 kg and is landing on a runway. The landing speed of the plane is 50 m/s and the runway is 2140 meters long. What is the acceleration of the plane?

Problem B.

Your car is the fastest all around. No one can beat you. It has an acceleration of 8.2 m/s^{2}. Suppose you start from a rest (because, don’t all drag racers do this). How long would it take your awesome car to get to a speed of 55 m/s? What is this speed in mph? What is the average speed during this time? How far did you go?

Problem C.

A police car starts from rest and can accelerate at 5.5 m/s^{2}. The police car starts accelerating as soon as a speeding car passes by with a speed of 25 m/s. Assuming the police car has a constant acceleration and the other car has a constant speed, where does the police car catch up to the other car?

Problem D.

Can you have a hang time of over 2 seconds when jumping?

Problem E.

A rocket is in space traveling with a speed of 328 m/s. It fires its rockets to create an acceleration of -10.7 m/s^{2} (slowing down). What is the speed after 5.8 seconds?