"For Hummingbirds, It's Easy to Shift Into Reverse"

I was reading the Science section of the New York Times this weekend, and I came across this article that relates directly to the Physics that we have been studying in class recently: http://www.nytimes.com/2012/10/02/science/for-hummingbirds-its-easy-to-shift-into-reverse.html?ref=science&_r=0

In a very interesting study done by scientists at the Max Planck Institute for Ornithology, it was found that hummingbirds fly in reverse almost as easily as they fly forward. The head scientist on this study, Nir Sapir, hypothesized that these birds would invest much more energy when flying backwards. After conducting his experiment, however, Dr. Sapir found that his hypothesis was incorrect. He found that the rate at which the birds beat their wings and the rate of their oxygen consumption was roughly the same in both directions. Dr. Sapir was able to test this by capturing five hummingbirds, putting them into a wind tunnel, and having them feed on sucrose disguised as flower. As the birds were feeding, Dr. Sapir turned on the air in the tunnel and changed its direction multiple times, so that the birds would have to fly backwards in order to remain at the flower.

Movies of the experiment taking place: http://jeb.biologists.org/content/215/20/3603/suppl/DC1

While this is not quite an example of conservation of energy, because the energy exerted when flying forward was still slightly less than the energy exerted when flying backwards, the efficiency of these hummingbirds can be looked at.

This is a picture, taken from the NY Times website, of a hummingbird being tested.

Knowns (found from doing a little bit of research on this experiment):

e = 80% (when flying backwards)

v = 3 m/s (the speed when flight required the least amount of energy, in both directions) 

Equations:

e = P_out/P_in

P_out = flying in reverse

P_in = flying forward

P = work/time

P = F*v 

The Math:

0.80 = (F_r*v)/(F_f*v)

0.80 = (F_r*3 m/s)/( F_f*3 m/s)

F_f (force when flying forward, in the x-direction) = F_{air resistance}

*While we have not yet learned about air resistance, for the purposes of the Physics of this problem, I have estimated this force to equal 34 N.

F_{air resistance} = 34 N = 0.5*(1.29 g/L)(3 m/s)²(0.47)(π(2 m)²) 

Therefore, 0.80 = (F_r*3 m/s)/(34 N*3 m/s)

And F_r = 27.2 N. 

So, when the hummingbird is flying at 3 m/s, the hummingbird’s efficiency is 80%, and thus the force forward is only 6.8 N greater than the force reverse.