### Far Horizon

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

### Epidemic Modelling

Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths.

### More or Less?

Are these estimates of physical quantities accurate?

# Constantly Changing

##### Stage: 4 Challenge Level:

Physical constants can only be determined by experiment and can never be known exactly, even if in principle an exact value does exists. As a result, physical quantities are given as a probable range of values with an uncertainty registered in the last two digits, as follows:

$1.234\, 5678(32) \rightarrow 1.234\, 5678 \pm 0.000\, 0032$

$245.234\, 789\, 123(45) \rightarrow 245.234\, 789\, 123\pm 0.000\, 000\, 045$

The following table contains the best currently known measurements for various physical quantities:

Name                        Value
Avogadro constant           $6.022\, 141\, 79(30) \times 10 ^{23}$ mol $^{-1}$
Atomic mass constant        $1.660 \,538\, 782(83) \times 10^{-27}$ kg
Electron mass               $9.109\, 382 \,15(45)\times 10^{-31}$ kg
Proton-electron mass ratio  $1836.152\, 672\, 4718(80)$
Proton mass                 $1.672\, 621\, 637(83) \times 10^{-27}$ kg
Neutron mass                $1.674\, 927 \,211(84) \times 10{-27}$ kg
Speed of light in vacuum    $299\, 792\, 458$ m s$^{-1}$ exactly

Consider the relationship between the error bounds for the proton-electron mass ratio and those for the electron mass and the proton mass. Are they consistent? Which appears to be known to best experimental accuracy?

Using this data, can you work out an upper limit on the mass of a mole of water? What is a lower limit?

How much uncertainty is there is the energy is contained within the mass of a mole of water, according to Einstein's energy-mass equation $E=mc^2$?

If the specific heat capacity of liquid water is about $4.1813$kJ kg$^{-1}$ K$^{-1}$ make an estimate of the number of cups of tea that you could make with this uncertain amount of energy.

NOTES AND BACKGROUND

National Institute of Standards and Technology Reference on Constants, Units and Uncertainty provides detailed information on the bounds of measurements of physical constants.

See http://physics.nist.gov/cuu/Constants/index.html for more details .

For a list of all constants, see http://physics.nist.gov/cuu/Constants/Table/allascii.txt

Interestingly, there is a strong element of statistics used to determine the probable values of constants. Key to this idea are the concepts of error and uncertainty in measurement. Cleverly designed experiments based on a strong understanding of statistics can be used to minimise this uncertainty.

To read about the essentials of expressing measurement uncertainty see http://physics.nist.gov/cuu/Uncertainty/index.html

Note that the speed of light given is a numerically exact quantity because the length of a metre has now been defined in terms of the speed of light!

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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