Big and small numbers in physics

6504 /www/nrich/html/content/id/6504/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <ul id="stemLinks"> <li><a href="http://nrich.maths.org/7278">Warm-up</a></li> <li><a href="http://nrich.maths.org/7428">Try this next</a></li> <li><a href="http://nrich.maths.org/6505">Think higher</a></li> <li><a href="http://nrich.maths.org/7605">Read: mathematics</a></li> <li><a href="http://www.peep.ac.uk/content/977.0.html">Read: science</a></li> <li><a href="http://scaleofuniverse.com/">Explore further</a></li> </ul> <div> </div> 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. <ol> <li>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?</li> <li>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?</li> <li>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?</li> <li>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.</li> <li>What weight of fuel would fit into a petrol tanker?</li> <li>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?</li> <li>What is the mass of a molecule of water?</li> <li>How many molecules of water are there in an ice cube?</li> <li>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?</li> <li>How much energy is contained in the matter forming the earth?</li> </ol> <div class="framework">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!</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Note that some of these questions involve making estimations and assumptions so these answers are not definitive! 1. Required extra data: Typical height a person can raise their centre of mass through when jumping on the Earth $h \approx 0.5 \textrm m$. $\therefore$ under a gravitational field of one-sixth of the strength height reached $h \approx 3.0 \textrm{ m}$ 2. Required extra data: None $ \rho_{Pb} = 11.35\ \textrm{g cm}^{-3} \Rightarrow 1 \textrm{ cm}^3$ of lead has mass $m = 11.35\ \textrm{g}$ Mass of one lead atom $m_{A:Pb} = 3.44 \times 10^{-22}\ \textrm{g}$ $\therefore$ number of atoms in $1 \textrm{ cm}^3$ \[n = \frac{m}{m_{A:Pb}} = \frac{11.35}{3.44 \times 10^{-22}} = 3.30 \times 10^{22}\] 3. Required extra data: 1 year = 365.25 days 1 year is one full rotation around Sun, therefore period of orbit in seconds is: $$T = 365.25 \times 24 \times 60 \times 60 = 31\ 557\ 600\ \textrm{s}$$ The angular velocity of the Earth's orbit is therefore: $$\omega = \frac{2\pi}{T} = \frac{2\pi}{31\ 557\ 600} = 1.99 \times 10^{-7}\ \textrm{rad s}^{-1}$$ The speed of the Earth relative to the Sun is therefore: $$ v = \omega R = 1.99 \times 10^{-7} \times 149\ 598\ 000\ 000 = 29\ 800\ \textrm{ms}^{-1}$$ 4. Required extra data: Acceleration due to gravity $g = 9.81\ \textrm{ms}^{-2}$ Assuming constant acceleration (i.e. ignoring drag effects), $u = 0 \textrm{ ms}^{-1};\ s = 800\textrm{ m};\ a =g$ $$v^2 = u^2 + 2as \Rightarrow v^2 = 0 + 2(g)(800) \Rightarrow v = 40 \sqrt{g} = 125 \textrm{ ms}^{-1}$$ Note air-resistance is likely to cause the ball to tend towards a terminal velocity below this, typically around $90 \textrm{ ms}^{-1}$ 5. Required extra data: Density of fuel (assume petrol) $\rho_{fuel} \approx 700\ \textrm{kg m}^{-3}$ Typical diameter of fuel tank $D = 4\ \textrm{m}$ Typical length of fuel tank $\ell = 15\ \textrm{m}$ Volume of fuel tank $$V = \pi \left(\frac{D}{2}\right)^2 \ell = \frac{\pi \times 4^2 \times 15}{4} = 60 \pi\ \textrm{m}^3$$ $\therefore$ weight of fuel in tank $$W = V \rho_{fuel} = 60 \pi \times 700 = 132\ 000\ \textrm{kg}$$ This seems to be an overestimate as typical values for capacity of a petrol tanker are between 20000 and 40000 litres. What values might be more sensible for the diameter and length? 6. Required extra data: Hydrochloric acid is a strong acid (complete dissociation) $\Rightarrow \textrm{pH} = -\log_{10} [\textrm{H}^+]$ Avogadro's constant $N_A = 6.022 \times 10^{23}\ \textrm{mol}^{-1}$ $$\textrm{pH} = 1.0 \Rightarrow [\textrm{H}^+] = 10^{-1.0} = 0.1 \textrm{ mol dm}^{-3}$$ $$\therefore 0.1 \textrm{ mol in } 1 \textrm{ dm}^3 (1\ \ell) \textrm{ which is } 0.1N_A = 6.022 \times 10^{22}\textrm{ hydrogen ions}$$ $$\therefore \textrm{ total sum of positive charges } = 6.022 \times 10^{22} \times 1.6 \times 10^{-19} = 9635 \textrm{ C}$$ 7. Required extra data: Relative molecular mass of water $m_{R} = 18.02\ \textrm{u}$ Avogadro's constant $N_A = 6.022 \times 10^{23}\ \textrm{mol}^{-1}$ 1 mole of water has a mass $m = 18.02\ \textrm{g}$ There are $N_A$ particles in 1 mole of a substance $\therefore$ one molecule of water has mass $$m = \frac{18.02}{6.022 \times 10^{23}} = 2.992 \times 10^{-23}\ \textrm{g}$$ 8. Required extra data: Answer to Q7. (mass of one molecule of water) $m = 2.992 \times 10^{-23}\ \textrm{g}$ Typical volume of an ice cube - assume dimensions $2\ \textrm{cm} \times 2\ \textrm{cm} \times 3\ \textrm{cm} \Rightarrow V = 12\ \textrm{cm}^3$ Density of ice $\rho_{ice} = 0.917\ \textrm{g cm}^{-3}$ Mass of ice cube $$M = \rho_{ice}V = 0.917 \times 12 = 11.004\ \textrm{g}$$ $\therefore$ number of molecules in one ice cube is $$n \approx \frac{11.004}{2.992 \times 10^{-23}} = 3.7 \times 10^{23}$$ 9. Required extra data: Speed of electromagnetic radiation (in free space) $c = 299\ 792\ 458\ \textrm{ms}^{-1}$ $$T = 13.4\ \textrm{billion years} = 13.4 \times 10^9 \times 365.25 \times 24 \times 60 \times 60 = 4.229 \times 10^{17}\ \textrm{s}$$ $$ \therefore d = cT = 299\ 792\ 458 \times 4.229 \times 10^{17} = 1.27 \times 10^{26}\ \textrm{m}$$ 10. Required extra data: Mass of Earth $M_E = 5.9742 \times 10^{24}\ \textrm{kg}$ Speed of electromagnetic radiation (in free space) $c = 299\ 792\ 458\ \textrm{ms}^{-1}$ Einstein's famous formula relating energy to mass is: $$E = mc^2 \therefore E_{Earth} = M_E c^2 = 5.9742 \times 10^{24}\times 299\ 792\ 458^2 = 5.369 \times 10^{41}\ \textrm{J}$$ </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><h3>Why do this problem?</h3> The ability to work with big and small numbers is a very important scientific skill. These <a href="http://nrich.maths.org/public/viewer.php?obj_id=6504&part=">questions</a> provide a variety of contexts where handling such numbers is necessary, while offering practice at identifying key information and making decisions about estimates and assumptions. <h3>Possible approach</h3> <div>These questions could be used individually as starters or fillers within lessons. Alternatively, a class could work on different questions in small groups and then present their answers with reasoning to the rest of the class to stimulate discussion. Some questions can be tackled with little specialist knowledge; others may require research and additional information, as well as some assumptions, in order to come up with an answer.</div> <div>It is important to stress that there is no "correct" answer for some of these questions - the value of this task comes from having a go at estimating and calculating with big and small numbers, and discussing assumptions that need to be made and extra data that need to be sought in order to find a solution.</div> <h3>Key questions</h3> <div>What assumptions have you made?</div> <div>What other information do you need?</div> <div> </div> Are there any questions which give an exact answer? Can you say anything about the accuracy of those answers which aren't exact? <h3>Possible extension</h3> This problem could be done in conjunction with <a href="http://nrich.maths.org/public/viewer.php?obj_id=6140&part=">Big and Small Numbers in Biology</a>. Similar questions could be created and shared within the class. <h3>Possible support</h3> Start with those questions which have familiar content, and encourage whole class discussion of the ideas in them, before tackling any of the questions which require extra information. It is well worth suggesting that students read the article <a href="http://nrich.maths.org/public/viewer.php?obj_id=6300&part=">Student Guide to Getting Started with Rich Tasks</a> before tackling a very open problem like this one. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Note that most of the ideas used here are typically covered at school before the age of 16, although possibly in mathematics, physics or chemistry. In estimation questions don't be afraid to have a go with a guess at some numbers in the problem and then to refine your estimate after checking it makes some sort of sense. Although there is no 'right' answer to an estimation, there are good or bad estimates and sensible or over detailed calculations. Think how you might make your estimation a good one, and think how it makes sense to ignore certain complexities in particular calculations. </mdoxml> 2 3 0 0 0 1 0 Big and small numbers in physics Work out the numerical values for these physical quantities. Numbers and the Number System Standard index form/Scientific notation Applications physics Applications chemistry Calculations and Numerical Methods Estimating and approximating Applications physics Applications Maths Supporting SET Applications physics Using, Applying and Reasoning about Mathematics Investigations Admin Discussion Admin Discussion Applications STEM - physical world Collections Fundamental Particles STEM mapping STEM - Scientific Calculation STEM mapping STEM Physical World STEM mapping STEM - Estimation STEM mapping STEM - Orders Of Magnitude STEM mapping STEM - Units STEM mapping STEM - Calculation STEM mapping STEM - Equations And Formulae