# Approximately certain

## Problem

In this problem we look at several sets of physical quantities. Some of the quantities are very precisely stated and will be possible to work out exactly. Others might be clearly stated, yet will defy an exact calculation. Others might not be clearly stated: you will need to state some more assumptions or do some research before a meaningful approximation might be made (be scientific about
this process).Whilst it might not be possible to calculate exactly each value, each does have a value: **you are required to arrange each of these in order of magnitude** . Be sure to justify your ordering with scientific and mathematical rigour.

The energy:

- Used to walk up the steps of the Burk Dubai skyscraper ($818$m)
- Contained in a full-sugar can of coke
- Contained in a single atom of lead (according to Einstein's equation $E=mc^2$)
- Needed to boil a kettle of tap water

- For a radio wave to travel halfway around the world
- For a top sprinter to run $1$mm at top speed
- For the end of the second hand on a watch to move a distance of $1$ micron
- For a test tube of hydrogen gas to fully combust when exposed to a flame

- You could jump vertically up on the surface of the moon
- You could throw a tennis ball-sized lump of lead
- Between peaks of two sound waves caused by two successive hand claps in your fastest possble round of applause
- You can run in $1$ second

- Of the atmosphere
- Of all of the people in the world
- Of the north polar ice cap
- Of all of the living bacteria presently living on Earth

Can you make up other similar quantities? Be inventive!

## Getting Started

For some questions, you may need to be more accurate in your estimations than others to arrange the list in order of magnitude.

## Student Solutions

## Teachers' Resources

### Why do this problem?

This problem gives an excellent workout in estimation and calculation using a wide range of physical equations and situations. It is rather open, and will particularly benefit those students used to following recipes in their work. It also highlights the fact that in science it is rather hard to calculate anything without making some sort of assumptions. Good science will clearly state and be aware of these assumptions; bad science will ignore them.

### Possible approach

### Key questions

- What is precisely stated and what is not precisely stated?
- What factors would complicate the most accurate calculation? How can we deal with these? Which factors can we neglect and which are important?
- Can you give quick, sensible lower and upper bounds on the quantities before attempting a computation?
- Is a detailed computation necessary for all of the parts of the problem?