A Bouncing Angry Bird

Whenever new levels are released for Angry Birds, I feel compelled to play right away. There was a recent update to the Angry Birds Seasons with new levels. Something weird happened on one of the levels. A blue bird ended up bouncing vertically on one of the rubber mats. The weird part was that the bird kept bouncing higher and higher. Maybe this isn’t weird in the Angry Birds world, but it’s weird here on Earth. If you want to see what I am talking about (or analyze it yourself), here is the video.

How about an analysis. Here is the initial motion of one of the bouncing blue birds. I set the length of sling shot at 4.9 meters since this gives the acceleration of 9.8 m/s².

Here you can see that the quadratic fit for one of the bounces has a t² coefficient of 4.92 m/s². Since this corresponds to the (1/2)a term in the kinematic equation, the acceleration would be 9.84 m/s². So, at least for these low level bounces the vertical acceleration is constant and the scale seems to be set correctly. Now, let’s look at all the bounces. Here is a plot of the bird just at the lowest and highest point.

The data looks quadratic-like, so I fit a quadratic function. It’s just what I do. But I’m not too happy. I would like to see if this trend keeps going. How can I get more data when the blue bird goes off the screen for bounces higher than about 35 meters. Well, let me go ahead and mark the time the bird leaves and then comes back on the screen. Here is the data from Tracker Video Analysis:

For the motions that go off screen, can I treat them as plain old projectile motion? If so, I can just look at the time in the air to get the maximum height. Here is a more detailed plot of one of the later bounces (that goes off screen).

This looks nice. A constant acceleration of around 10 m/s² means that I can just use the time to determine the height. How would you do this? Instead of just using a kinematic equation, let me start with the definition of acceleration for the vertical direction (which has a constant value of g). Since I want to know how high it goes, let me take the time interval that starts with the bird on the ground moving up and ends at the highest point (where the bird has zero velocity). Also, I am just talking about 1-dimension here so I will drop the y-subscript on the velocity.

Don’t forget that Δt here is the time just to go UP, not up and down. But what about the height? That’s what I want. Let me use the definition of average velocity:

Now I can put in my expression for v₁ from before:

So, if I just measure the time between the bounces for the ones that go off the screen I can get the maximum height. Here is my adjusted bounce-height plot. Now with more data (but with the same function from before):

Oh snap. That’s not what I expected. It looks like the bouncing gets to a maximum of around 45.5 meters (above the ground). Actually, I suspect something else is happening. It’s not that there is some magical ceiling in the game that the birds hit (well, there might be). But if the bird was hitting something at the top, the time of flight would be different for the faster balls. Let me calculate the starting speed for each bounce (calculated from the height).

I like this better. Why? First, a limit on the maximum velocity would mean that each bouncing bird would still have a constant acceleration. Second, in a previous analysis I found that the yellow bird can reach a maximum speed of 30 m/s. Notice the maximum speed in this plot. Boom. 30 m/s.