Re: Which experiences more air resistance: book, pencil, marble, feather?

The question: "Which experiences more air resistance: book, pencil, marble, feather? This was a question on my students' end-of-quarter criterion referenced test (CRT), which accounts for 20% of their total science grade. My students said the book would experience greater air resistance because of its larger surface area, but the correct answer was supposed to be a feather. My students knew that the feather would hit the ground last, but they reasoned that it wasn't what the question asked. I believe the feather falls slower because of a lesser terminal velocity, but I still believe the book experiences more air resistance. Can you help?"

There is insufficient information given to unambiguously answer the test question. If the book falls with its largest face parallel to the ground it probably experiences the most "air resistance", while, as you say, the feather obtains the slowest "terminal velocity" and is the probable reason that the test's answer is the feather. However, if the book falls edge first and the feather is a large (for instance, wing) feather then the feather might experience a larger air resistance, depending on the objects' speeds! This being said, my opinion is that a book usually would experience a larger air resistance than a feather, due to two factors discussed below, in the usual understanding of this "experiment".

"Air resistance" is usually synonymous with "drag force", and you can find information about that at many sites on the Internet, such as this one at nasa.gov. There you will find that the drag force is proportional to the area of the object and the drag coefficient:
F_d = -0.5 rho v² A C_d
where "A" is the cross-sectional area and
C_d is the drag coefficient. The other factors are
rho, the density of the fluid (air, in this case), and
v, the velocity.

The drag coefficient of a book in the flat orientation is on the order of 1.2 or thereabouts. I have not been able to find the drag coefficient for a feather, though I guess that it is roughly in the ballpark of 1.0. (See, for instance, this page at nasa.gov or this listing for some typical values.) Clearly the cross-sectional area of a book (falling "flat") is much larger than that of a feather, unless one is discussing a huge wing feather from a large bird and a small book! When most people think of this experiment they assume a large book falling "flat" and a small feather.

At terminal velocity the drag force equals the weight of the object. So, assuming that the book weighs more than the feather, the drag force for the book is greater than the drag force for the feather, at terminal velocity. However, if the book is small and the feather large, and both objects are not yet falling very fast, then it might be the feather experiencing the larger air resistance.

Seeing all these caveats you can see why there is insufficient information given in the test question. However, with only the information given the best answer, in my opinion, would be the book.

John Link, MadSci Physicist