Physics makes use of numbers both small and large. Try these questions involving big and small numbers. You might need to use pieces of physical data not given in the question. Sometimes these questions involve estimation, so there will be no definitive 'correct' answer; on other occasions an exact answer will be appropriate. Use your judgement as seems appropriate in each context. Feel free to attempt them in any order; some will seem easier than other dependent on your knowledge of physics.

Your goal is to provide the best, sensible approximation to the questions taking into account the precision to which each question is stated. Along with finding a numerical answer, clearly express any scientific or modelling assumptions made and which formulae you used along the way.

1. It is known that the value of $g$ on the moon is about one-sixth that on earth. How high do you think that you would be able to jump straight up on the surface of the moon?
2. The mass of an atom of lead is $3.44\times 10^{-22}$g. Lead has a density of $11.35$ g cm$^{-1}$. How many atoms of lead are found in a single cubic centimetre of lead?
3. The earth orbits the sun on an almost circular path of average radius about $149\,598\,000\,000$m. How fast is the earth moving relative to the sun?
4. The tallest buildings in the world are over $800$m high. If I dropped a cricket ball off the top of one of these, estimate how fast it would be moving when it hit the ground.
5. What weight of fuel would fit into a petrol tanker?
6. The charge on a proton is $1.6\times 10^{-19}$C. What is the total sum of the positive charges in a litre of Hydrochloric acid of pH 1.0?
7. What is the mass of a molecule of water?
8. How many molecules of water are there in an ice cube?
9. Around 13.4 billion years ago the universe became sufficiently cool that atoms formed and photons present at that time could propagate freely (this time was called the surface of last scattering). How far would one of these old photons have travelled by now?
10. How much energy is contained in the matter forming the earth?

NOTES AND BACKGROUND
An obvious part of the skill with applying mathematics to physics is to know the fundamental formulae and constants relevant to a problem. By not providing these pieces of information directly, you need to engage at a deeper level with the problems. You might not necessarily know all of the required formulae, but working out which parts you can and cannot do is all part of the problem solving process!

Approximation problems can involve sophisticated application of mathematics, especially when clearly stated in the form: given these assumptions, the following numerical consequences follow.