The Milk Problem

Finally, yesterday, I got to teach. On the first day, as discussed previously, I get into real physics: equilibrium situations in AP, position-time graphs in general.

I found a new problem appropriate for the first night’s assignment. I want to give students a sense of how physics will be different from math class, but I’m not yet ready to assign conceptual questions about the current topics – we haven’t gotten far enough. I use problems that require serious reasoning, especially those which lend themselves to order-of-magnitude estimates. The following is based on a problem from the Young and Freedman text:

Milk is often sold by the gallon in plastic containers. You are to solve by calculation and reasoning, not through research.

(a) Estimate the number of gallons of milk that are purchased in the United States each year. (Obviously, your answer should include both verbal and mathematical reasoning.)
(b) What approximate weight of plastic does this represent? Compare this weight to something with which you are familiar.

What an excellent question! Even if students WANTED to try to answer through library or google research, that’s a daunting task… as I found out.

My own reasoning: with 300 million people in the USA, figure about a gallon per week for a family of four. That’s around 70 million gallons per week, times 52 weeks, or somewhere near 4 billion gallons of milk per year sold in the US.

A bunch of googling produced

this article from the US General Accounting Office, suggesting that about 7 billion gallons of milk per year were sold in 2001. Hey! I’m well within a factor of 10, which is the goal of such a problem, anyway.

As for the weight of plastic… I originally guessed about 20 g from an empty milk jug, based on my experience with my hanging weights. This gives 108 kg of plastic for the billion gallons sold each year, or about 100,000 tons.

My chemistry colleague, the Atlanta Cracker, Mr. Paul Vickers weighed an empty half-gallon milk carton, getting 47 g. Woodberry Forest Librarian Phoebe Warmack turned up a claim that nowadays gallon milk jugs are less than 60 g. So my estimate was definitely good enough, because it gives the same ~100,000 tons of plastic as does a weight of 50 g or so.

And this is the whole point of the exercise. Not only do I want my students to gain their first exposure to “Fermi problems” and order-of-magnitude estimation, I also want to make a preemptive strike against the arguments I inevitably hear in class: “You said the answer was 5.9 N, but I got 5.8 N. What did I do wrong?” Or, as always, “isn’t g 9.8, not 10?” Hopefully they will see that a 2% difference is meaningless when making everyday measurements.