Approximately certain

6505 /www/nrich/html/content/id/6505/ 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/6145">Warm-up</a></li> <li><a href="http://nrich.maths.org/7424">Try this next</a></li> <li><a href="http://nrich.maths.org/6354">Think higher</a></li> <li><a href="http://en.wikipedia.org/wiki/Order_of_magnitude">Read: mathematics</a></li> <li><a href="http://en.wikipedia.org/wiki/Age_of_the_universe">Read: science</a></li> <li><a href="http://plus.maths.org/content/cars-next-lane-really-do-go-faster">Explore further</a></li> </ul> <div> </div> 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: <ol> <li>Used to walk up the steps of the Burk Dubai skyscraper ($818$m)</li> <li>Contained in a full-sugar can of coke</li> <li>Contained in a single atom of lead (according to Einstein's equation $E=mc^2$)</li> <li>Needed to boil a kettle of tap water</li> </ol> <div>The time taken:</div> <ol> <li>For a radio wave to travel halfway around the world</li> <li>For a top sprinter to run $1$mm at top speed</li> <li>For the end of the second hand on a watch to move a distance of $1$ micron</li> <li>For a test tube of hydrogen gas to fully combust when exposed to a flame</li> </ol> <div>The distance:</div> <ol> <li>You could jump vertically up on the surface of the moon</li> <li>You could throw a tennis ball-sized lump of lead</li> <li>Between peaks of two sound waves caused by two successive hand claps in your fastest possble round of applause</li> <li>You can run in $1$ second</li> </ol> <div>The mass:</div> <ol> <li>Of the atmosphere</li> <li>Of all of the people in the world</li> <li>Of the north polar ice cap</li> <li>Of all of the living bacteria presently living on Earth</li> </ol> Can you make up other similar quantities? Be inventive! </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <h3>Why do this problem?</h3> <a href="http://nrich.maths.org/public/viewer.php?obj_id=6505&part=">This problem</a> 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. <h3>Possible approach</h3> <div>This problem could be approached in two stages. First of all the problem could be discussed as a group without any calculations being made (except in students' heads or on the back of an envelope). Once the issues are uncovered, students might wish to begin calculation or might need to turn to the internet or other resources for more information.</div> <div>It is likely that students' answers and approximations will vary. Once the task is finished, groups could feed back to the rest of the class. Is their reasoning and explanation clear? They will need to convince the rest of the class that their ordering is correct. This could be done with each group attempting each of the four different sets of data. Alternatively, different groups could attempt different parts of the task, in which case the explanations might need to take on a more detailed focus.</div> <div>There are two different levels at which the problem might be approached</div> <div>Basic: Produce a means of calculating each part with various estimated values of the data. Order the answers.</div> <div>Advanced: Produce definite upper and lower bounds for the quantities, using upper and lower bounds for the input data. The ordering is only guaranteed when these intervals do not overlap.</div> <div>Don't lose sight of the fact that the ordering is important. If a very crude approimation shows that one of the quantities is clearly largest or smallest, then that is sufficient. Of course, students might be interested in computing a more accurate answer out of general interest.</div> <div> </div> <div>This <a href="/content/id/6505/approximately%20certain.pdf">worksheet</a> has all the quantities.</div> <div> </div> <h3>Key questions</h3> <ul> <li>What is precisely stated and what is not precisely stated?</li> <li>What factors would complicate the most accurate calculation? How can we deal with these? Which factors can we neglect and which are important?</li> <li>Can you give quick, sensible lower and upper bounds on the quantities before attempting a computation?</li> <li>Is a detailed computation necessary for all of the parts of the problem?</li> </ul> <h3>Possible extension</h3> <div>The most able students should be required to approach the task with the most rigour. They might also consider the best way to represent their results. What accuracies are most relevant? Is a linear measurement scale suitable?</div> <h3>Possible support</h3> <div>Focus on the basic method of approaching the task, as mentioned in the possible approach.</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">How could you estimate each of the quantities? What extra information might you need to find out? For some questions, you may need to be more accurate in your estimations than others to arrange the list in order of magnitude. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Note the rankings, figures and methods given here are not definitive - they are only meant to provide an example of how to approach the problem. The important thing is that any estimates have been justified. Energy (Highest) 12 4 3 (Lowest), although this will depend on the mass of the person in question. 1) Mass of a typical person $ m = 60 \textrm{ kg}$ Height travelled up $h = 818 \textrm{ m}$ $\therefore$ assuming other energy uses (overcoming friction etc.) are insignificant in comparison to gravitational potential $ E_1 \approx mgh \approx 60 \times 9.8 \times 818 \approx 500 \textrm{ kJ}$ 2) Energy in can of full-sugar coke (from nutritional information) is about $180 \textrm{ kJ} $ per $100$ ml. A can contains $330$ ml of drink. Thus: $E_2 \approx 594 \textrm{ kJ} $. 3) 1 atom of lead has mass $m = \frac{m_{Pb}}{N_A} $ Einstein's equation $E = mc^2$ $E_2 = \left(\frac{m_{Pb}}{N_A}\right) c^2 = \left(\frac{207}{6.022 \times 10^{23}}\right) \times (3.00 \times 10^8)^2 \approx 3 \times 10^{-5} \textrm{ J}$ 4) Change in internal energy due to raising temperature of mass $m$ of water by $\Delta T$ $\Delta U = mc_p\Delta T$ Specific heat capacity of water (at constant pressure) $c_p \approx 4.2 \textrm{ kJ (kg K)}^{-1}$ Change in temperature from room temperature ($\approx 25\ ^\circ C$) to boiling ($= 100\ ^\circ C$) $\Delta T = 75 \textrm{ K}$ Typical volume of water to be boiled $\approx 1 \textrm{ dm}^{-3}$ which has mass $m = 1 \textrm{ kg}$ $E_4 \approx 1 \times 4200 \times 75 \approx 300 \textrm{ kJ}$ Time (Longest) 4 1 3 2 (Shortest) 1) Circumference of Earth $C \approx 40\ 000 \textrm{ km}$ Speed of light $c \approx 3 \times 10^8 \textrm{ ms}^{-1}$ $t_1 = \frac{C}{c} \approx \frac{40 \times 10^6}{2 \times 3 \times 10^8} \approx 0.07 \textrm{ s}$ 2) Top sprinter can complete 100 m in 9 s Assuming constant speed (actually varies substantially however sufficient for this calculation) $v \approx 11 \textrm{ ms}^{-1}$ To travel $1 \textrm{ mm}$ therefore would take $t_2 \approx \frac{1 \times 10^{-3}}{11} = 9 \times 10^{-5}$ 3) Angular speed of second hand $\omega = \frac{2\pi}{60} = \frac{pi}{30} \textrm{ rad s}^{-1}$ Typical length of second hand $\ell = 1 \textrm{ cm}$ Therefore speed of second hand $v = \ell \omega = \frac{2\pi}{6000} \textrm{ ms}^{-1}$ Therefore time taken to travel 1 micron $t_3 \approx \frac{1 \times 10^{-6} \times 6000}{2\pi} \approx 1 \times 10^{-3} \textrm{ s}$ 4) Flame front will travel at flame speed of hydrogen $v \approx 4 \textrm{ ms}^{-1}$ ?? Typical length of a test tube $\ell \approx 15 \textrm{ cm}$ $t_4 \approx \frac{0.15}{4} \approx 4 \times 10^{-2} \textrm{ s}$ Distance (Longest) 3 2 4 1 (Shortest) 1) Gravitational field of Moon $\approx \frac{1}{6}$ of Earth's Typical height centre of mass is raised through when jumping on Earth $h = 0.5 \textrm{ m}$ $\therefore$ height reached on Moon $d_1 \approx 6 \times 0.5 = 3 \textrm{ m}$ 2) Density of lead $\rho \approx 11 \textrm{ g cm}^{-3}$ Radius of tennis ball $r \approx 3.5 \textrm{ cm}$ Volume of tennis ball $V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \times (3.5 \times 10^{-2})^3 \approx 180 \textrm{ cm}^3$ $\therefore$ tennis ball sized lump of lead has mass $m = \rho V = \frac{11 \times 180}{100} \approx 2 \textrm{ kg}$ Mass of shot put 4-7 kg, typical range of trained atheletes being $\geq 15 \textrm{ m}$ (WR 23.12 m) $\therefore$ for an object of half the mass assume an average person could reach similar ranges $d_2 \approx 15 \textrm{ m}$ 3) From experimentation, maximum clapping frequency approximately $f \approx 10 \textrm{ Hz}$ $\therefore$ time between each clap $T \approx 0.1 \textrm{ s}$ Speed of sound $v \approx 330 \textrm{ ms}^{-1}$ $d_3 \approx 33 \textrm{ m}$ 4) Typical time to sprint 100 m for an untrained person is around 15 s Therefore in 1 s cover around $\frac{100}{15} \textrm{ m}$ $d_4 \approx 7 \textrm{ m}$ Mass (Highest) (3 1 4) 2 (Lowest) Large degree of uncertainty in top three 1) Air pressure at sea level $p \approx 10^5 \textrm{ Pa}$ Surface area of Earth $A \approx 510 \times 10^9 \textrm{ m}^2$ $\therefore$ weight of atmosphere $W \approx 5.1 \times 10^{16} \textrm{ N}$ Acceleration due to gravity $g \approx 9.8 \textrm{ ms}^{-1}$ $m_1 \approx \frac{W}{g} = 5.2 \times 10^{15} \textrm{ kg}$ 2) Average mass of one person (all ages and genders) $m \approx 60 \textrm{ kg}$ Number of people on Earth $P \approx 6 \times 10^9$ $m_2 \approx Pm = 3.6 \times 10^{11} \textrm{ kg}$ 3) Density of ice $\rho \approx 920 \textrm{ kg m}^{-3}$ Area of north polar ice cap $A \approx 10^{13} \textrm{ m}$ Typical depth of ice $d \approx 2 \textrm{ m}$ $m_3 \approx \rho Ad = 1.8 \times 10^{16} \textrm{ kg}$ 4) Number of living bacteria on Earth $N \approx 10^{30}$ Mass of single bacteria $m \approx 10^{-15} \textrm { kg}$ $m_4 \approx Nm = 10^{15} \textrm{ kg}$ </mdoxml> 2 4 0 0 0 1 1 Approximately certain Estimate these curious quantities sufficiently accurately that you can rank them in order of size Applications chemistry Applications physics Applications engineering Applications biology Calculations and Numerical Methods Estimating and approximating Measures and Mensuration Upper and lower bounds of measures Applications Maths Supporting SET Admin Individual Applications STEM - physical world Applications STEM - living world sfh10 Steve - Workshop Materials