"Hey, did you know there is a new *Angry Birds *game coming out? Angry Birds Space?"

Well, of course I am going to look at the physics here. But how? The game won't be released until March 22. Oh, how about I find a video online. Here is some sample gameplay.

So, what can I figure out from this video? Let the physics begin.

Gravity ——-

Before looking at real Angry Birds data, let me talk about gravity. If the moons exert gravitational forces on the birds, what would that be like? The usual model for the gravitational interaction between two masses looks like this:

This says that if you have two masses (m 1 and m 2 ), there will be a gravitational force pulling them together. If the vector r is from the center of the moon to the other mass, the force will be in the opposite direction (so towards the moon). Also, the magnitude of this force will increase the closer the centers of the object get to each other. Oh, I forgot to say that G is the gravitational constant.

For Earth-based Angry Birds, I could look at the x-position vs. time and the y-position vs. time to get an idea about the forces on the birds. That won't work so well here. Why? For the Earth-based motion, there was a constant force on the birds – a downward gravitational force that didn't change in direction or magnitude. With this moon, neither of these will be true.

One alternative will be to look at the energy. If I assume there are no external forces on the objects, I can say that the total energy is constant. In this system, I could say there are two types of energy, kinetic and gravitational potential energy. This would be written as:

So, if I look at the kinetic energy of one of the objects as a function of distance from the center of the moon, I can get an estimate for the gravitational potential energy of the rock-moon system (or bird-moon). Also, it is important to note that I am assuming there is no recoil motion from the moon. Just looking at the video, this seems to be reasonable. This would be fairly close to true if the mass of the moon is significantly larger than the mass of the objects.

Actual Data ———–

First, the launched bird. Here is the trajectory of that bird. Of course, I used Tracker Video Analysis to get this data.

Clearly, I should just look at the first part of the motion. Who knows what is going on during that "special" motion. But, like I said, I really need a plot of kinetic energy vs. radial distance. Actually, this will be kinetic energy per mass of the yellow bird (even though it doesn't look like it is the color yellow, the shape looks like that bird).

Is this graph what I was expecting? Really, it is difficult to say. There is lots of noise - which is sort of excepted (even if unwanted). When you start with position-time data and take numerical derivatives, you get noise. However, this graph does show that when the bird is farther away from the center of the moon it has less kinetic energy. That is what I would expect. It is unfortunate that I can't really get a shape of the gravitational potential energy from this plot. Let me just get some rough values.

The lowest value of r is 12.6 meters (scaling based on my previous Angry Birds scales). At this lowest value, the bird has a K/m of about 450 J/kg. When the bird was first launched, it has a K/m of about 200 J/kg at a distance of 37 meters. If I assume at this starting instant all the energy was from the launch (it really hadn't had a chance to speed up), that would mean that the change in potential would be the opposite the change in kinetic energy. So, from 37 meters to 12.6 meters, the gravitational energy per kg decreased by about 250 J/kg.

Let me just assume that this just like real gravity. In that case, I could find the mass of the moon. Let me write it like this:

Ok, that is a pretty massive moon for its size (radius about 6.3 meters). Before I do some more stuff, let me repeat this EXACT same calculation, but for another object. Actually, two object. First, when the bird flies off and hits something, it looks like a rock falls straight down towards the moon. Here is the plot of K/m vs. r for that object. Forget that. Instead, this is a plot of distance from the center of the moon vs. time.

This is odd. It starts off moving at 12.3 m/s towards the moon and then it slows down to about 9.58 m/s. At the end, it is moving about 16.1 m/s. It really looks like it has three discrete speeds and doesn't continuously change. Odd. Well, if I use the same idea as above, this starts 47 meters from the center of the moon and ends at 8 meters from the center (it doesn't make it all the way to the surface). This would give a moon mass of 7.8 x 1012 kg. Weird. It is off by a factor of 10.

Here is the last object. It is a rock that is shot off the surface of the moon and returns back to the moon. Here is a plot of K/m vs. r for that rock.

The problem here is that the rock gets back to about r = 7 meters, but appears to have less kinetic energy that the last time it was at that level. If this is a closed system (with no air drag) the value of K/m should be the same for the same distance from the center. Perhaps this is just a noise in the data problem. But perhaps not. If I say the rock has about 100 J/kg at a distance of 7 meters and just 10 Joules/kg at 20.2 meters, then the mass of the moon would be 1.45 x 1013 kg. Hmmmmmm.

I think I will have to wait for the game to come out so I can set up my own experiments and collect more data. Really, the best test for the gravitational force would be to get the bird to orbit the moon. That would be cool.

What Is the Moon Made Of? ————————-

Let me go with my lowest calculation for the mass of the moon. Remember, this mass is based on the assumption that this is a real moon with real gravity. Double remember that I really haven't confirmed that it is real-ish gravity. So, I will start with a mass of 7.8 x 1012 kg. With this, I can find the density of the moon. Assuming a radius of 6.3 meters, this would be a density of 7.4 x 109 kg/m3.

Compare this to the density of THE moon at about 3,300 kg/m3. Not even close. The Earth has a density of 5,500 kg/m3. Well, what about something super-dense on Earth? Lead only is at around 11,000 kg/m3. Ok, so this thing is just crazy dense.

Numerical Model —————

Since my data isn't the best, let me see if I can reproduce some of these motions by assuming normal gravity. This really isn't that difficult to do. Here is my numerical recipe.

Create the bird and the moon as objects. State all the constants. Take a small time step and calculate the following: Based on the position of the moon and bird, calculate the gravitational force on the bird. (ignore the gravitational force on the moon since the mass is probably too large) During this time step calculate the change in momentum of the bird due to this force. From the momentum, calculate the change in position of the bird. Update the time and go back to step 2.

Really, it's that simple. If I use my highest value for the mass of the moon (7.17 x 1013 kg), and a bird launched at the same location with the same speed I get this trajectory:

Not too bad, but also not the same as the Angry Birds shot. What about a plot of K/m vs. r, like I did in the video analysis?

Of course, there is no noise in this plot - also it doesn't go to as high of a value for kinetic energy since it doesn't get as close to the moon. Here are the two sets of data on plotted together (data from the video plus data from the numerical calculation):

Ok, I can't stop. What if I use a launch velocity of 23 m/s. Why that value? Well, that is the launch velocity of the birds in the Earth-based game. (as I found from a previous analysis) And what about the launch angle? From the trajectory plot in Tracker, I get a launch angle of about 39.5°. This would give an x- and y-components of the initial velocity with values of 17.75 m/s and 14.63 m/s.

No. That doesn't work.

Conclusions ———–

Clearly, I need more data. If I could set up my own experiments, that would help. But does Angry Birds in Space (I keep thinking of PIGS IN SPACE) use the 1/r2 form of the gravitational force? Really, I am not sure. If it does, the mass of the planet would be HUGE! From my simple analysis and models, it seems like the motion is pretty close to being consistent with typical gravity. The data just isn't that great.

What other questions are there? Well, I could look at the other moon. Does it have a gravitational interaction with the birds and rocks and stuff? What about those circles around the moons. Is that supposed to be an atmosphere? Does something special happen when an object crosses that boundary? Of course, the most important question to answer: why are there clouds in space?