A few weeks ago we had a small, informal competition in the department – guess the maximum gradient on one of the roads on campus. I think the motivation for this this small stretch of hill (or what passes for a hill here in Hamilton) was going to be used as part of a dynamics experiment, and so one of our technicians was about to go out and measure the gradient.
I’m happy to say that I won the competition, without even going out to the road and looking at it carefully. The prize was simply to feel smug. I predicted a maximum gradient of 9.5 degrees; I think from memory the measured gradient was 10.1. Being a physicist, I estimated rather than guessed. I simply thought "What is the average gradient?". This wasn’t too difficult. Thinking about how the buildings are laid out on campus, I thought about how many floors that the road drops by. That gave me an estimate of the drop distance. Then I compared it in my head to the length of the swimming pool to estimate the length of that stretch of road. Divide the former by the latter, take the inverse tangent, and I get the average gradient in terms of an angle.
Then came the bit that was rather more vague. I needed the maximum gradient, but had the average. Clearly the maximum is higher than the average. So to go from one to the other, I need to multiply by a number that’s bigger than 1. So I picked 2. Though even that wasn’t a wild guess. The road starts off level, and ends level, so if we assume it gains gradient uniformly then loses it uniformly, the maximum gradient will be about double the average.
Perhaps I shouldn’t have been all that surprised that I was very close. The point is that I estimated rather than guessed. It’s a skill that is very important in physics, but it’s one that often gets overlooked during teaching. The difference is that an estimate is based on what we do know about the situation – even if it’s only approximate knowledge – rather than a guess which is simply a number plucked out of the air.
Some fun things to get students to estimate include the number of carbon atoms that are worn off the soles of their shoes during a day’s wear and the mass of the building they are sitting in.