Sunday, November 13th, 2011...8:37 pm

polynomials & Angry Birds

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They’re a phenomenon! They’re addictive. They’re angry…really, angry! But, they’ll be happier when they see how math can help them. Yes, we’re going to use ideas often introduced in algebra, specifically polynomials, to help those Angry Birds take out their aggression on the sneaky pigs.

Much of angry birds comes down to choosing the angle to sling the bird through the air! Note that a bird’s initial angle out of the slingshot equals the angle between the horizon and the stretched sling at the moment of release. The angle determines the path the bird will follow in the air.

Can math help us determine an angle to shoot the bird? Keep in mind, though, it can be difficult to shoot exactly at, say 35 degrees versus 40 degrees.

So, we will restrict ourselves to the path of a red angry bird with 3 different initial angles – 30, 45 and 60 degrees. Which of them would land the bird closest to a desired point? This can give us a sense of how to shoot the angry bird.

Trajectory depends on the angle and the initial velocity of a projectile. How fast does an angry bird travel? To answer this, we need a sense of scale. Rather than using a meter or inch, we’ll use a scale of 1 slingshot, which we’ll denote as 1 AB.

A key component to determining the path of a slung angry bird
is knowing the underlying model of motion used within the programming of the game. In the article, The Physics of Angry Birds, Rhett Allain, an Associate Professor of Physics at Southeastern Louisiana University, determined that an angry bird enjoys a frictionless flight, not feeling any effects of the air through which it passes. So, the bird follows parabolic motion. In particular, the path an angry bird follows for different launching angles are:

  begin{array}{rcl}  y & = & 1 + 0.5773503x - 0.0704629x^2 mbox{~for~} 30^circ\  y & = & 1 + x - 0.105694x^2 mbox{~for~} 45^circ \  y & = & 1 +  1.7320508x - 0.211389x^2 mbox{~for~} 60^circ \  end{array}

Given we are trying to get an estimate of an initial angle, let’s round our formulas and use:

  begin{array}{rcll}  y & = & 1 + 0.577x - 0.070x^2 mbox{~for~} 30^circ\  y & = & 1 + x - 0.106x^2 mbox{~for~} 45^circ \  y & = & 1 +  1.732x - 0.211x^2 mbox{~for~} 60^circ \  end{array}

These are pleasing mathematically but can be a bit difficult to visualize. So, let’s graph them and see what the trajectories look like?

The graph of the trajectories helps us see that of these three initial angles only 30 degrees results in a hit that will teeter the tower! Suppose instead, we want to know where a bird will be at some point along its flight. Let’s pose this type of question in this way: What height will a bird be when it crosses the vertical line seen below if shot at 45 degrees?

First, we need to know how far the dotted line is from the slingshot or, in terms of graphing, the coordinate along the x-axis. Remember 1 unit = length of slingshot. So, we lay down slingshot after slingshot and find that the dotted line is (conveniently) 5 slingshots away. So, we need only to set x = 5 in the parabola the angry bird will follow, specifically, y = 1 + x - 0.106x^2 to know the height of the bird at that point, which equals 3.35. Again, laying down slingshots shows this height below.

Compare this to the red dotted trajectory earlier. How did we do? How else could you use this information to help you play Angry Birds? What other mathematical questions come to mind? Ponder it a bit, you might find posing the questions and the search for answers to be a bit addictive, just like the game you’ll be analyzing.

The topic of polynomials fall within a variety of areas of mathematical study. This blog posting is cast within algebra as a way of thanking Kristianna Luce and Maryellen Magee, fellows in the 2011 Charlotte Teachers Institute Math through Popular Culture seminar. Both teachers will be using these ideas in their curriculum units. Soon, I’ll be posting an entry on deriving such polynomials with the use of Calculus.

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