Algebra 2 Problem

The following problem was recently assigned in Mr. Glasbrenner's Algebra 2 Class and one student was challenged to consider how accurate the math might be. Carter Freymiller gladly accepted this challenge and you will see his explanation and slow motion video of his Hawk Brim below.

Brim's wingspan is 48 inches, which is pretty large considering she only weighs 43 oz or roughly about the weight of 4 large bananas. I have noticed that as height increases, the less lift she needs to get to me. Yet she still beats her wings. The first clip that I have is a perfect example as it's 3 seconds long and she only beats her wings 3 times. She only does this in the last 0.25 seconds. This form of wing beating is to slow down and lose momentum, versus the next clip that shows her in flight for 6 seconds. She only does this in the last 0.25 seconds. This form of wing beating is to slow down and lose momentum, versus the next clip that shows her in flight for 6 seconds.

This time she beats her wings roughly 26 times. She uses her wing beats to gain height in the first 0.3 seconds so it's easier at the end when you see her glide to the fist. If we take the chart in question 45 and compare it to the largest bird on the chart, being a lesser black-backed gull with a rate of 2.8 wing-beats per second with a wingspan of 42 inches. The K value in that equation was 118.16. 48 inches isn't much different, so if we take Brim's 2.46 beats per second and put her on level ground for her to fly to me she would have to beat her wings only 15 times. So why would she beat her wings 26 times in the 6 seconds it took for her to get to me?

What about this third video where she beats her wings 10 times in 2 seconds? The first video she beats her wings 3 times in 3 seconds of flight, the second time 26 beats in 6 seconds, and finally 10 times in 2 seconds.

Why is there so much variation? I have 3 reason for all of these.

  1. The first video is because she's so far up. If she came down to an exact point (height) she could glide down exactly to my fist without any effort. But if she were to go higher she would have to slow down using her wings more. If she came down lower than her key point she would have to use her wings more to get to me or else she won't make it. So the specific equation must contain height and how far they are traveling.
  2. The second video is different. Here Brim uses lift. She powers herself as fast as she can in the first 50 feet to give herself milliseconds of breaks and then she propels herself 20 more feet to give her a millisecond break. She is so close to the ground because when she gets half a wingspan from the ground the air cushion she creates lets the air do all the work and create minimal work. This took us centuries to figure out and Brim's doing it at 4 months old (in the video). In this equation we need to take into account the 4ft and air cushion she uses.
  3. In the third clip you can notice that we are much closer. You can see as she takes off she doesn't tuck her feet in to conserve energy because it's such a close distance. This is awkward for her to keep in flight so she has to work much harder by beating her wings faster to get to me. In this equation we have to account for body position and how high up/close we are. Therefore, I can tell you her beats per second in each individual clip.