This homework is intended to give you practice with dimensional analysis and scaling, with some applications to biological systems. It is due in my mailbox (in mailroom, 2nd floor of NPB) before 5:00 pm, February 8, 2002. You may work together on the homework, but you should hand in your own solution, of course.

1. During World War II, the physicist G.I. Taylor (1886-1975) was asked to consider the mechanical effect of nuclear explosions. Such an energetic explosion generates a very large blast wave that expands rapidly with time through the atmosphere. Taylor realized that he could use dimensional analysis to estimate how the size of the resulting fireball (radius r) varies according to the energy E released in the explosion and the time t elapsed since the explosion. In fact, he realized that this size could really depend only on the energy, the time elapsed, and (perhaps) the density (kg/m3) of the atmosphere.

   Following Taylor, estimate the functional dependence of r on these parameters.

   You may be amused to learn that Taylor then compared his simple result with some declassified photos of the New Mexico atomic test, and from that he estimated the energy yield of that test -- previously a top secret.

2. Birds can fly because their wings generate an upward force, called lift. Ignoring little details like the design and angle of the wing, and the effects of flapping the wings, we can imagine that the amount of lift force generated depends only on S (the surface area of the wing), v (the airspeed of the bird), and ro (the density of air at sea level).
   • Use a dimensional argument to obtain an approximate formula for the lift force generated by the wing, in terms of these parameters.
   • The ratio of this lift force to the weight W of the bird is obviously an important dimensionless number for a bird. Based on that fact, obtain the relationship between the weight of a bird, the velocity with which it flies, and its wing surface area.

3. Now consider the size of the bird. Small birds are not precise scale models of large birds, but they aren't really shaped that differently. Let's assume that all birds are roughly scale replicas of each other. In that case, write down a (simple) equation that relates the weight of a bird to the surface area of its wings. You will need to define or introduce a numerical constant whose value you probably don't know -- that's OK. Use your knowledge of birds to estimate a reasonable value for this constant. (This constant is not a dimensionless number.) You are looking for the scaling law that relates the surface area S of a bird's wings to the weight W of the bird.
4. Since you can write S in terms of W, use your results from questions #2 and #3 to solve for the power-law relationship that connects the weight W of a bird with the density of air ro, and the airspeed v of the bird.
5. Based solely on this result,
   • Estimate the cruising speed of a 3 lb chicken. (This would be a happy wild chicken, not the overfed, non-flying farm variety.) Do you consider this to be a reasonable estimate? (Note: The density of air at sea level is about 1.25 kg/m3.)
   • The "correct" airspeed of that chicken is evidently about 15-20 m/s. What do you think accounts for your error (if any)?

6. A house wren (a small bird) has a mass of about 10 grams and flies at about 7 m/s.
• Is this consistent with the above value for the airspeed of a chicken? Why or why not?
• Use this fact and your result from question #4 to estimate the cruising speed of a Boeing 747 aircraft, which has a weight (at takeoff) of about 3.5x10^6 Newtons.
• In fact, you should have obtained an accurate value for the speed of the 747 at takeoff. But when it cruises over the ocean at 39,000 feet this aircraft actually travels about twice as fast. What do you think accounts for this discrepancy?

Note: Data on flight obtained from work of H. Tennekes