Estimation in physics class

One of the things you will learn to do in this course is to "estimate." To a physicist, this means something very specific, so let's make sure we all understand. Estimation is the process of calculating an approximate answer to a problem when it is not possible to get an exact answer. For instance, you may not have all of the information you need to solve a problem. In this case, you use all the information you do have, and "educated guesses" for the information you do not have. The art is in how you make these educated guesses. It may also be that you have plenty of information, but the situation is extremely complex. In this case, you may make assumptions that simplify the situation. In any case, you must state what educated guesses or assumptions you have made. You may also round off the numbers a bit. Consider an example.

Question: You and a friend are going to drive from Indy to Los Angeles for a vacation. Estimate how long the trip will take.

Answer: Well, my road atlas says that it is 2088 miles. I assume an average speed of 70 miles per hour, so that is about 30 hours of driving. Assuming we drive straight through, taking shifts at the wheel, and that we make 4 stops of one hour each for gas and a bite to eat, the trip will take 34 hours.

An equally good answer: Well, I know the US is about 3000 miles across, and Indy to LA is about two thirds of that, so 2000 miles. If we average 60 miles per hour, that is 33 hours of driving. We hate to do more than 8 hours at a time, so it will take 4 full days, where we push it to nine hours on one of the days.

In this case, only information was missing. I filled it in with educated guesses. Please notice a few things. The final answers may vary widely, so long as they are consistent with the assumptions, which must be stated. Also, there is a calculation involved:

Question: Estimate the maximum kinetic energy you can get by running.

Answer: Kinetic energy = (1/2)mv², but I don't know either m or v in appropriate units. Here's what I do. I know I weigh about 160 lbs. I look up the conversion, and find that that is about 73 kg. I don't know how fast I can run, but I remember that the record for the 100 m dash is a bit under 10 seconds, or about 10 m/s. I am nowhere near that fast, so I divide by 2 to get 5 m/s. Now, I put these together to get about 900 Joules.

Sometimes, it is necessary to make an assumption that simplifies the situation. In the next example, I simplify the shape of a complicated object.

Question: Estimate the mass of an elephant.

Answer: Mass can be calculated from volume times density. I assume the density of an elephant is about the same as that of a person (a bit less dense than water): about 900 kg/m³. To get the volume, I assume the elephant is a cylinder 2 m in diameter and 4 m long. This gives a volume of 9.4 m³. This gives a mass of about 8500 kg, or 9.3 tons. (Note: I checked afterwards, and found that an average mass of a male elephant is about 5500 kg, or 6 tons, so I was a bit high, but within a factor of two.)

Estimation can get quite complicated, but it is highly useful. In the workplace, engineers must often do this to decide if a method is feasible, or if a completely different technique must be used. One of the great masters of this technique was Enrico Fermi, and difficult estimations are often called "Fermi Problems." In 1945, Fermi was present at the test of the first atomic bomb. He torn up a bit of a note he had in his pocket, and drop the bits of paper when he saw the flash. By observing how far the shock wave blew the paper, he was able to estimate the yield of the bomb to within about 10%. We won't go quite that far in this course, but we will gain skills in this area.

If you are interested, there are a number of web sites listing interesting Fermi problems. Doing a bunch of these would make a nice honors project!