Cha p t e r 3

HOW TO DE S C R I B E MOT IONndash K I N E M AT IC S

ldquoLa filosofia egrave scritta in questo grandissimo libroche continuamente ci sta aperto innanzi agliocchi (io dico lrsquouniverso) Egli egrave scritto inlingua matematica rdquoGalileo Galilei Il saggiatore VI

Experiments show that the properties of Galilean time and space are extracted from theenvironment by most higher animals and by young children Among others this hasbeen tested for cats dogs rats mice ants fish and many other species They all find thesame results

First of all motion is change of position with time This description is illustrated byrapidly flipping the lower left corners of this book starting at page 195 Each page sim-ulates an instant of time and the only change that takes place during motion is in theposition of the object represented by the dark spot The other variations from one pic-ture to the next which are due to the imperfections of printing techniques can be takento simulate the inevitable measurement errors

Calling lsquomotionrsquo the change of position with time is neither an explanation nor a def-inition since both the concepts of time and position are deduced from motion itself Itis only a description of motion Still the description is useful because it allows for highprecision as we will find out by exploring gravitation and electrodynamics After all pre-cision is our guiding principle during this promenadeTherefore the detailed descriptionof changes in position has a special name it is called kinematics

The idea of change of positions implies that the object can be followed during its mo-tion This is not obvious in the section on quantum theory we will find examples wherethis is impossible But in everyday life objects can always be tracked The set of all pos-itions taken by an object over time forms its path or trajectory The origin of this conceptis evident when one watches fireworksRef 49 or again the flip film in the lower left cornersstarting at page 195

In everyday life animals and humans agree on the Euclidean properties of velocityspace and time In particular this implies that a trajectory can be described by specify-ing three numbers three coordinates (x y z) ndash one for each dimension ndash as continuousfunctions of time t (Functions are defined in detail on page 201) This is usually writtenas x = x(t) = (x(t) y(t) z(t)) For example already Galileo found using stopwatch and Science is written in this huge book that is continuously open before our eyes (I mean the universe) Itis written in mathematical language

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how to describe motion ndash kinematics 69

collision

F I G U R E 43 Two ways to test that the time of free fall does not depend on horizontal velocity

ruler that the height z of any thrown or falling stone changes as

z(t) = z0 + 9841630 (t minus t0) minus 12 д (t minus t0)2 (5)

where t0 is the time the fall starts z0 is the initial height 9841630 is the initial velocity in thevertical direction and д = 98ms2 is a constant that is found to be the same within aboutone part in 300 for all falling bodies on all points of the surface of the Earth Where dothe value 98ms2 and its slight variations come fromRef 50 A preliminary answer will begiven shortly but the complete elucidation will occupy us during the larger part of thishike

The special case with no initial velocity is of great interest Like a few people beforehim Galileo made it clear that д is the same for all bodies if air resistance can be ne-glected He had many arguments for this conclusionPage 165 Can you find one And of coursehis famous experiment at the leaning tower in Pisa confirmed the statement (It is a falseurban legend that Galileo never performed the experiment)Ref 51

Equation (5) therefore allows us to determine the depth of a well given the time astone takes to reach its bottomChallenge 113 s The equation also gives the speed 984163 with which one hitsthe ground after jumping from a tree namely 984163 = 10035242дh A height of 3m yields a velocityof 27 kmh The velocity is thus proportional only to the square root of the height Doesthis mean that onersquos strong fear of falling results from an overestimation of its actualeffectsChallenge 114 s

Galileo was the first to state an important result about free fall the motions in thehorizontal and vertical directions are independent He showed that the time it takes fora cannon ball that is shot exactly horizontally to fall is independent of the strength of thegunpowder as shown in Figure 43 Many great thinkers did not agree with this statementeven after his death in 1658 the Academia del Cimento even organized an experimentRef 52

to check this assertion by comparing the flying cannon ball with one that simply fellvertically Can you imagine how they checked the simultaneityChallenge 115 s Figure 43 also showshow you can check this at home In this experiment whatever the spring load of thecannon the two bodies will always collide in mid-air (if the table is high enough) thusproving the assertion

Motion

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70 3 how to describe motion ndash kinematics

v

z

z

vx

mvz

x

mvx

hodograph phase spacegraph

x

z

configurationspace

t

z

t

x

space-timediagrams

F I G U R E 44 Various types of graphs describing the same path of a thrown stone

In other words a flying canon ball is not accelerated in the horizontal direction Itshorizontal motion is simply unchanging By extending the description of equation (5)with the two expressions for the horizontal coordinates x and y namely

x(t) = x0 + 984163x0(t minus t0)y(t) = y0 + 984163y0(t minus t0) (6)

a complete description for the path followed by thrown stones results A path of this shapeis called a parabola it is shown in Figures 18Page 38 43 and 44 (A parabolic shape is also usedfor light reflectors inside pocket lamps or car headlights Can you show why)Challenge 116 s

Physicists enjoy generalizing the idea of a pathRef 53 As Figure 44 shows a path is a traceleft in a diagram by a moving object Depending on what diagram is used these pathshave different names Hodographs are used in weather forecasting Space-time diagramsare useful to make the theory of relativity accessible The configuration space is spannedby the coordinates of all particles of a system For many particles it has a high number ofdimensions It plays an important role in self-organizationThe difference between chaosand order can be described as a difference in the properties of paths in configurationspace The phase space diagram is also called state space diagram It plays an essentialrole in thermodynamics

Throwing jumping and shooting

The kinematic description of motion is useful for answering a whole range of questions

lowastlowastWhat is the upper limit for the long jump The running peak speed world record in

Motion

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how to describe motion ndash kinematics 71

F I G U R E 45 Three superimposed images of a frass pellet shotaway by a caterpillar inside a rolled-up leaf (copy StanleyCaveney)

2008 was over 125ms asymp 45 kmh by Usain BoltRef 54 and the 1997 womenrsquos record was11ms asymp 40 kmhRef 55 However long jumpers never run much faster than about 95msHow much extra jump distance could they achieve if they could run full speed Howcould they achieve that In addition long jumpers take off at angles of about 20deg asRef 56 theyare not able to achieve a higher angle at the speed they are running How much wouldthey gain if they could achieve 45degChallenge 117 s (Is 45deg the optimal angle)

lowastlowastWhat do the athletes Usain Bolt and Michael Johnson the last two world record holderson the 200m race at time of this writinghave in common They were tall athletic andhad many fast twitch fibres in the muscles These properties made them good sprintersA last difference made them world class sprinters they had a flattened spine with almostno S-shapeThis abnormal condition saves them a little bit of time at every step becausethe spine is not as flexible as in usual people and allows them to excel at short distanceraces

lowastlowastAthletes continuously improve speed records Racing horses do not Why For racinghorses breathing rhythm is related to gait for human it is not As a result racing horsescannot change or improve their technique and the speed of racing horses is essentiallythe same since it is measured

lowastlowastHow can the speed of falling rain be measured using an umbrellaChallenge 118 s The answer is impor-tant the same method can also be used to measure the speed of light as we will find outlater (Can you guess how)Page 17

lowastlowastWhen a dancer jumps in the air how many times can it rotate around its vertical axisbefore arriving back on earthChallenge 119 ny

lowastlowastNumerous species of moth and butterfly caterpillars shoot away their frass ndash to put itmore crudely their shit ndash so thatRef 57 its smell does not help predators to locate them Stanley

Motion

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72 3 how to describe motion ndash kinematics

Length of animal [m]

running jumps

Hei

ght o

f jum

p [m

]

101010010001

101010010001

1

01

001

fleas

locusts andgrasshoppers

lesser galago

antilopeleopard

elephant

human horse

tigercat

dog

20 Wkg

standing jumps

F I G U R E 46 The height achieved by jumping land animals

Caveney and his team took photographs of this process Figure 45 shows a caterpillar(yellow) of the skipper Calpodes ethlius inside a rolled up green leaf caught in the actGiven that the record distance observed is 15m (though by another species Epargyreusclarus) what is the ejection speedChallenge 120 s How do caterpillars achieve it

lowastlowastWhat is the horizontal distance one can reach by throwing a stone given the speed andthe angle from the horizontal at which it is thrownChallenge 121 s

lowastlowastWhat is the maximum numbers of balls that could be juggled at the same timeChallenge 122 s At themoment robots can juggle three balls as shown by the Sarcoman robot on wwwphysionorthwesterneduSecondlevelMillerFirstLevelhistresearchhtml It is a challenge forrobotics to reach the maximum number of balls in the future

lowastlowastIs it true that rain drops would kill if it werenrsquot for the air resistance of the atmosphereChallenge 123 s

What about hail

lowastlowastAre bullets fired into the air from a gun dangerous when they fall back downChallenge 124 s

lowastlowastPolice finds a dead human body at the bottom of cliff with a height of 30m at a distanceof 12m from the cliff Was it suicide or murderChallenge 125 s

lowastlowast

Motion

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how to describe motion ndash kinematics 73

All land animals regardless of their size achieve jumping heights of at most 2mRef 58 asshown in Figure 46 The explanation of this fact takes only two lines Can you find itChallenge 126 s

The last two issues arise because equation (5) does not hold in all cases For exampleleaves or potato crisps do not follow it As Galileo already knew this is a consequence ofair resistance we will discuss it shortly Because of air resistance the path of a stone isnot always a parabola

In fact there are other situations where the path of a falling stone is not a parabolaCan you find oneChallenge 127 s

Enjoying vectors

Physical quantities with a defined direction such as speed are described with three num-bers or three components and are called vectors Learning to calculate with such multi-component quantities is an important ability for many sciences Here is a summary

Vectors can be pictured by small arrows Note that vectors do not have specified pointsat which they start two arrows with same direction and the same length are the samevector even if they start at different points in space Since vectors behave like arrowsthey can be added and they can be multiplied by numbers For example stretching anarrow a = (ax ay az) by a number c corresponds in component notation to the vectorca = (cax cay caz)

In precise mathematical language a vector is an element of a set called vector spacein which the following properties hold for all vectors a and b and for all numbers c and d

c(a + b) = ca + cb (c + d)a = ca + da (cd)a = c(da) and 1a = a (7)

Examples of vector spaces are the set of all positions of an object or the set of all itspossible velocities Does the set of all rotations form a vector spaceChallenge 128 s

All vector spaces allow the definition of a unique null vector and of one negative vectorfor each vectorChallenge 129 e

In most vector spaces of importance in science the concept of length (specifying thelsquomagnitudersquo) can be introduced This is done via an intermediate step namely the intro-duction of the scalar product of two vectors The product is called lsquoscalarrsquo because itsresult is a scalar a scalar is a number that is the same in for all observers for exampleit is the same for observers with different orientations The scalar product between twovectors a and b is a number that satisfies

aa ⩾ 0 ab = ba

(a + a998400)b = ab + a998400b a(b + b998400) = ab + ab998400 and

(ca)b = a(cb) = c(ab) (8)

This definition of a scalar product is not unique however there is a standard scalar prod-

Motion

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ChristophSchiller

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74 3 how to describe motion ndash kinematics

y

tΔt

Δy

secant slope ΔyΔt

derivative slope dydt

F I G U R E 47 The derivative in apoint as the limit of secants

uct In Cartesian coordinate notation the standard scalar product is given by

ab = axbx + ayby + azbz (9)

If the scalar product of two vectors vanishes the two vectors are orthogonal at a rightangle to each other (Show it)Challenge 130 e

The length or norm of a vector can then be defined as the square root of the scalarproduct of a vector with itself a = 1003522aa Often and also in this text lengths are writtenin italic letters whereas vectors are written in bold letters A vector space with a scalarproduct is called an Euclidean vector space

The scalar product is also useful for specifying directions Indeed the scalar productbetween two vectors encodes the angle between them Can you deduce this importantrelationChallenge 131 s

What is rest What is velocity

In the Galilean description of nature motion and rest are opposites In other words abody is at rest when its position ie its coordinates do not change with time In otherwords (Galilean) rest is defined as

x(t) = const (10)

We recall that x(t) is the abbreviation for the three coordinates (x(t) y(t) z(t)) Laterwe will see that this definition of rest contrary to first impressions is not much use andwill have to be expanded Nevertheless any definition of rest implies that non-restingobjects can be distinguished by comparing the rapidity of their displacement Thus wecan define the velocity 984163 of an object as the change of its position x with time t This isusually written as

984163 = dxdt

(11)

In this expression valid for each coordinate separately ddt means lsquochange with timersquoWe can thus say that velocity is the derivative of position with respect to time The speed

Motion

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how to describe motion ndash kinematics 75

F I G U R E 48 Gottfried Leibniz (1646ndash1716)

984163 is the name given to themagnitude of the velocity 984163 Derivatives are written as fractionsin order to remind the reader that they are derived from the idea of slopeThe expression

dsdt

is meant as an abbreviation of limΔtrarr0

ΔsΔt

(12)

a shorthand for saying that the derivative at a point is the limit of the secant slopes in theneighbourhood of the point as shown in Figure 47 This definition implies the workingrulesChallenge 132 e

d(s + r)dt

= dsdt

+ drdt

d(cs)dt

= cdsdt

ddt

dsdt

= d2sdt2 d(sr)

dt= dsdt

r + sdrdt

(13)

c being any numberThis is all one ever needs to know about derivatives Quantities suchas dt and ds sometimes useful by themselves are called differentials These concepts aredue to GottfriedWilhelm Leibniz Derivatives lie at the basis of all calculations based onthe continuity of space and time Leibniz was the personwhomade it possible to describeand use velocity in physical formulae and in particular to use the idea of velocity at agiven point in time or space for calculations

The definition of velocity assumes that it makes sense to take the limit Δt rarr 0 Inother words it is assumed that infinitely small time intervals do exist in nature Thedefinition of velocity with derivatives is possible only because both space and time aredescribed by sets which are continuous or in mathematical language connected and com-plete In the rest of our walk we shall not forget that from the beginning of classicalphysics infinities are present in its description of natureThe infinitely small is part of ourdefinition of velocity Indeed differential calculus can be defined as the study of infinityand its uses We thus discover that the appearance of infinity does not automatically ren-der a description impossible or imprecise In order to remain precise physicists use onlythe smallest two of the various possible types of infinities Their precise definition andan overview of other types are introducedVol III page 199 in later on

Gottfried Wilhelm Leibniz (b 1646 Leipzig d 1716 Hannover) Saxon lawyer physicist mathematicianphilosopher diplomat and historian He was one of the great minds of mankind he invented the differen-tial calculus (before Newton) and published many influential and successful books in the various fields heexplored among them De arte combinatoria Hypothesis physica nova Discours de meacutetaphysique Nouveauxessais sur lrsquoentendement humain the Theacuteodiceacutee and the Monadologia

Motion

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76 3 how to describe motion ndash kinematics

The appearance of infinity in the usual description of motion was first criticized in hisfamous ironical arguments by Zeno of Elea (around 445 bce)Ref 59 a disciple of ParmenidesIn his so-called third argument Zeno explains that since at every instant a given objectoccupies a part of space corresponding to its size the notion of velocity at a given instantmakes no sense he provokingly concludes that therefore motion does not exist Nowa-days we would not call this an argument against the existence of motion but against itsusual description in particular against the use of infinitely divisible space and time (Doyou agree)Challenge 133 e Nevertheless the description criticized by Zeno actually works quite well ineveryday life The reason is simple but deep in daily life changes are indeed continuous

Large changes in nature aremade up ofmany small changesThis property of nature isnot obvious For example we note that we have tacitly assumed that the path of an objectis not a fractal or some other badly behaved entity In everyday life this is correct in otherdomains of nature it is not The doubts of Zeno will be partly rehabilitated later in ourwalk and increasingly so the more we proceedVol VI page 56 The rehabilitation is only partial as thesolution will be different from that which he envisaged on the other hand the doubtsabout the idea of lsquovelocity at a pointrsquo will turn out to be well-founded For the momentthough we have no choice we continue with the basic assumption that in nature changeshappen smoothly

Why is velocity necessary as a concept Aiming for precision in the description ofmotion we need to find the complete list of aspects necessary to specify the state of anobject The concept of velocity is obviously on this list

Acceleration

Continuing along the same line we call acceleration a of a body the change of velocity 984163

with time or

a = d984163

dt= d2xdt2 (14)

Acceleration is what we feel when the Earth trembles an aeroplane takes off or a bicyclegoes round a corner More examples are given in Table 13 Like velocity acceleration hasboth a magnitude and a direction properties indicated by the use of bold letters for theirabbreviations In short acceleration like velocity is a vector quantity

Acceleration is felt The body is deformed and the sensors in our semicircular canalsin the ear feel it Higher accelerations can have stronger effects For example when ac-celerating a sitting person in the direction of the head at two or three times the value ofusual gravitational acceleration eyes stop working and the sight is greyed out becausethe blood cannot reach the eye any more Between 3 and 5д of continuous accelerationor 7 to 9д of short time accelerationRef 60 consciousness is lost because the brain does not re-ceive enough blood and bloodmay leak out of the feet or lower legs High acceleration inthe direction of the feet of a sitting person can lead to haemorrhagic strokes in the brainThe people most at risk are jet pilots they have special clothes that send compressed aironto the pilotrsquos bodies to avoid blood accumulating in the wrong places

In a usual car or on a motorbike we can feel being accelerated (These accelerationsare below 1д and are therefore harmless) Can you think of a situation where one is ac-celerated but does not feel itChallenge 135 s

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how to describe motion ndash kinematics 77

TA B L E 13 Some measured acceleration values

O b s e rvat i o n A c c e l e r at i o n

What is the lowest you can find Challenge 134 s

Back-acceleration of the galaxy M82 by its ejected jet 10 fms2

Acceleration of a young star by an ejected jet 10 pms2

Fathoumi Acceleration of the Sun in its orbit around the Milky Way 02 nms2

Deceleration of the Pioneer satellites due to heat radiation imbalance 08 nms2

Centrifugal acceleration at Equator due to Earthrsquos rotation 33mms2

Electron acceleration in household electricity wire due to alternatingcurrent

50mms2

Acceleration of fast underground train 13ms2

Gravitational acceleration on the Moon 16ms2

Minimum deceleration of a car by law on modern dry asfalt 55ms2

Gravitational acceleration on the Earthrsquos surface depending onlocation

98 plusmn 03ms2

Standard gravitational acceleration 9806 65ms2

Highest acceleration for a car or motorbike with engine-driven wheels 15ms2

Space rockets at take-off 20 to 90ms2

Acceleration of cheetah 32ms2

Gravitational acceleration on Jupiterrsquos surface 25ms2

Flying fly (Musca domestica) c 100ms2

Acceleration of thrown stone c 120ms2

Acceleration that triggers air bags in cars 360ms2

Fastest leg-powered acceleration (by the froghopper Philaenusspumarius an insect)

4 kms2

Tennis ball against wall 01Mms2

Bullet acceleration in rifle 2Mms2

Fastest centrifuges 01Gms2

Acceleration of protons in large accelerator 90 Tms2

Acceleration of protons inside nucleus 1031 ms2

Highest possible acceleration in nature 1052 ms2

Higher derivatives than acceleration can also be defined in the same manner Theyadd little to the description of natureChallenge 136 s because ndash as we will show shortly ndash neither thesehigher derivatives nor even acceleration itself are useful for the description of the stateof motion of a system

Motion

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78 3 how to describe motion ndash kinematics

TA B L E 14 Some acceleration sensors

Me a s u r e m e n t S e n s o r R a n g e

Direction of gravity in plants(roots trunk branches leaves)

statoliths in cells 0 to 10ms2

Direction and value ofaccelerations in mammals

the membranes in eachsemicircular canal and the utriculeand saccule in the inner ear

0 to 20ms2

Direction and value of accelerationin modern step counters for hikers

piezoelectric sensors 0 to 20ms2

Direction and value of accelerationin car crashes

airbag sensor using piezoelectricceramics

0 to 2000ms2

F I G U R E 49 Three accelerometers a one-axis piezoelectric airbag sensor a three-axis capacitiveaccelerometer and the utricule and saccule in the three semicircular canals inside the human ear(copy Bosch Rieker Electronics Northwestern University)

Objects and point particles

ldquoWenn ich den Gegenstand kenne so kenne ichauch saumlmtliche Moumlglichkeiten seinesVorkommens in Sachverhalten rdquoLudwig Wittgenstein Tractatus 20123

lsquoIf I know an object then I also know all the possibilities of its occurrence in atomic factsrsquo

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how to describe motion ndash kinematics 79

α

Betelgeuse

γ

κβ

Rigel

δ MintakaεAlnilamζ

Alnitak

Bellatrix

Saiph

F I G U R E 50 Orion in natural colours (copy Matthew Spinelli) and Betelgeuse (ESA NASA)

One aim of the study of motion is to find a complete and precise description of bothstates and objects With the help of the concept of space the description of objects canbe refined considerably In particular one knows from experience that all objects seen indaily life have an important property they can be divided into partsChallenge 137 e Often this observa-tion is expressed by saying that all objects or bodies have two properties First they aremade out ofmatter defined as that aspect of an object responsible for its impenetrabilityie the property preventing two objects from being in the same place Secondly bodieshave a certain form or shape defined as the precise way in which this impenetrability isdistributed in space

In order to describe motion as accurately as possible it is convenient to start withthose bodies that are as simple as possible In general the smaller a body the simplerit is A body that is so small that its parts no longer need to be taken into account iscalled a particle (The older term corpuscle has fallen out of fashion) Particles are thusidealized small stones The extreme case a particle whose size is negligible comparedwith the dimensions of its motion so that its position is described completely by a singletriplet of coordinates is called a point particle or a point mass In equation (5) the stonewas assumed to be such a point particle

Do point-like objects ie objects smaller than anything one can measure exist indaily life Yes and no The most notable examples are the stars At present angular sizesas small as 2 μrad can be measured a limit given by the fluctuations of the air in theatmosphere In space such as for the Hubble telescope orbiting the Earth the angularlimit is due to the diameter of the telescope and is of the order of 10 nrad Practicallyall stars seen from Earth are smaller than that and are thus effectively lsquopoint-likersquo evenwhen seen with the most powerful telescopes

As an exception to the general rule the size of a few large and nearby stars of redgiant type can bemeasured with special instruments Betelgeuse the higher of the two

Matter is a word derived from the Latin lsquomateriarsquo which originally meant lsquowoodrsquo and was derived viaRef 61intermediate steps from lsquomaterrsquo meaning lsquomotherrsquo The website wwwastrouiucedu~kalersowsowlisthtml gives an introduction to the different types ofstars The wwwastrowiscedu~dolanconstellations website provides detailed and interesting informationabout constellations

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80 3 how to describe motion ndash kinematics

shoulders of Orion shown in Figure 50 Mira in Cetus Antares in Scorpio Aldebaran inTaurus and Sirius in Canis Major are examples of stars whose size has been measuredthey are all only a few light years from EarthRef 62 Of course like the Sun all other stars havea finite size but one cannot prove this by measuring dimensions in photographs (True)Challenge 138 s

The difference between lsquopoint-likersquo and finite size sources can be seen with the nakedeye at night stars twinkle but planets do not (Check it)Challenge 139 e This effect is due to the tur-bulence of air Turbulence has an effect on the almost point-like stars because it deflectslight rays by small amounts On the other hand air turbulence is too weak to lead totwinkling of sources of larger angular size such as planets or artificial satellites becausethe deflection is averaged out in this case

An object is point-like for the naked eye if its angular size is smaller than about2 998400= 06mrad Can you estimate the size of a lsquopoint-likersquo dust particleChallenge 140 s By the way anobject is invisible to the naked eye if it is point-like and if its luminosity ie the intensityof the light from the object reaching the eye is below some critical value Can you esti-mate whether there are any man-made objects visible from the Moon or from the spaceshuttleChallenge 141 s

The above definition of lsquopoint-likersquo in everyday life is obviously misleading Do properreal point particles exist In fact is it at all possible to show that a particle has vanishingsize This question will be central in the last two parts of our walk In the same way weneed to ask and check whether points in space do exist Our walk will lead us to theastonishing result that all the answers to these questions are negative Can you imaginewhyChallenge 142 s Do not be disappointed if you find this issue difficult many brilliant minds havehad the same problem

However many particles such as electrons quarks or photons are point-like for allpractical purposes Once one knows how to describe the motion of point particles onecan also describe the motion of extended bodies rigid or deformable by assuming thatthey aremade of partsThis is the same approach as describing themotion of an animal asa whole by combining the motion of its various body partsThe simplest description thecontinuum approximation describes extended bodies as an infinite collection of pointparticles It allows us to understand and to predict the motion of milk and honey themotion of the air in hurricanes and of perfume in rooms The motion of fire and allother gaseous bodies the bending of bamboo in the wind the shape changes of chewinggum and the growth of plants and animals can also be described in this wayRef 63

A more precise description than the continuum approximation is given belowVol IV page 14 Nevertheless all observations so far have confirmed that the motion of large bodies can

be described to high precision as the result of the motion of their parts This approachwill guide us through the first five volumes of our mountain ascent Only in the finalvolume will we discover that at a fundamental scale this decomposition is impossible

For an overview of the planets see the beautiful book by K R Lang amp C A Whitney Vagabonds delrsquoespace ndash Exploration et deacutecouverte dans le systegraveme solaire Springer Verlag 1993Themost beautiful picturesof the stars can be found in D Malin A View of the Universe Sky Publishing and Cambridge UniversityPress 1993 A satellite is an object circling a planet like the Moon an artificial satellite is a system put into orbit byhumans like the Sputniks

Motion

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how to describe motion ndash kinematics 81

F I G U R E 51 How an object can rotate continuously without tangling up the connection to a secondobject

Legs and wheels

The parts of a body determine its shape Shape is an important aspect of bodies amongother things it tells us how to count them In particular living beings are always made ofa single body This is not an empty statement from this fact we can deduce that animalscannot have wheels or propellers but only legs fins or wings Why

Living beings have only one surface simply put they have only one piece of skinMathematically speaking animals are connectedVol V page 288 This is often assumed to be obviousand it is often mentioned thatRef 64 the blood supply the nerves and the lymphatic connec-tions to a rotating part would get tangled up However this argument is not so simple asFigure 51 shows It shows that it is indeed possible to rotate a body continuously against asecond one without tangling up the connections Can you find an example for this kindof motion in your own bodyChallenge 143 s Are you able to see how many cables may be attached tothe rotating body of the figure without hindering the rotationChallenge 144 s

Despite the possibility of animals having rotating parts the method of Figure 51 stillcannot be used to make a practical wheel or propeller Can you see whyChallenge 145 s Evolution hadno choice it had to avoid animals with parts rotating around axles That is the reasonthat propellers and wheels do not exist in nature Of course this limitation does not ruleout that living bodies move by rotation as a whole tumbleweedRef 65 seeds from various treessome insects several spiders certain other animals children and dancers occasionallymove by rolling or rotating as a whole

Single bodies and thus all living beings can only move through deformation of theirshape therefore they are limited to walking running rolling crawling or flapping wingsor fins Extreme examples of leg useRef 66 in nature are shown in Figure 52 The most extremeexample (not shown) are rolling spiders living in the sand inMoroccoRef 67 they use their legsto accelerate and steer the rolling direction Walking on water is shown in Figure 102 onpage 139 examples of wings are given later onVol V page 208 as are the various types of deformationsthat allow swimming in waterVol V page 210 In contrast systems of several bodies such as bicyclespedal boats or other machines can move without any change of shape of their compo-nents thus enabling the use of axles with wheels propellers or other rotating devices

Rolling is known for desert spiders of the Cebrennus and the Carparachne genus films can be found onwwwyoutubecomwatchv=5XwIXFFVOSA and wwwyoutubecomwatchv=ozn31QBOHtk Cebrennusseems even to be able to accelerate with its legs Despite the disadvantage of not being able to use rotating parts and of being restricted to one pieceonly naturersquos moving constructions usually called animals often outperform human built machines As anexample compare the size of the smallest flying systems built by evolution with those built by humans (Seeeg pixelitoreferencebe)There are two reasons for this discrepancy First naturersquos systems have integrated

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50 μm

F I G U R E 52 Legs and lsquowheelsrsquo in living beings the red millipede Aphistogoniulus erythrocephalus (15 cmbody length) a gekko on a glass pane (15 cm body length) an amoeba Amoeba proteus (1 mm size) therolling shrimp Nannosquilla decemspinosa (2 cm body length 15 rotations per second up to 2 m caneven roll slightly uphill slopes) and the rolling caterpillar Pleurotya ruralis (can only roll downhill toescape predators) (copy David Parks Marcel Berendsen Antonio Guilleacuten Robert Full John Brackenbury Science Photo Library )

In summary whenever we observe a construction in which some part is turning con-tinuously (and without the lsquowiringrsquo of the figure) we know immediately that it is an arte-fact it is a machine not a living being (but built by one) However like so many state-ments about living creatures this one also has exceptions The distinction between oneand two bodies is poorly defined if the whole system is made of only a few moleculesThis happens most clearly inside bacteria Organisms such as Escherichia coli the well-known bacterium found in the human gut or bacteria from the Salmonella family allswim using flagella Flagella are thin filaments similar to tiny hairs that stick out of thecell membrane In the 1970s it was shown that each flagellum made of one or a fewlong molecules with a diameter of a few tens of nanometres does in fact turn aboutits axisPage 210 A bacterium is able to turn its flagella in both clockwise and anticlockwise direc-tions can achieve more than 1000 turns per second and can turn all its flagella in perfectsynchronizationRef 68 (These wheels are so tiny that they do not need a mechanical connec-tion) Therefore wheels actually do exist in living beings albeit only tiny ones But let usnow continue with our study of simple objects

repair and maintenance systems Second nature can build large structures inside containers with smallopenings In fact nature is very good at what people do when they build sailing ships inside glass bottles

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how to describe motion ndash kinematics 83

F I G U R E 53 Are comets such as the beautiful comet McNaught seen in 2007 images or bodies Howcan one settle the issue (copy Robert McNaught)

Curiosities and fun challenges about kinematics

What is the biggest wheel ever madeChallenge 147 s

lowastlowastA soccer ball is shot by a goalkeeper with around 30ms Calculate the distance it shouldfly and compare it with the distances found in a soccer match Where does the differencecome fromChallenge 148 s

lowastlowastA train starts to travel at a constant speed of 10ms between two cities A and B 36 kmapart The train will take one hour for the journey At the same time as the train a fastdove starts to fly from A to B at 20ms Being faster than the train the dove arrives atB first The dove then flies back towards A when it meets the train it turns back againto city B It goes on flying back and forward until the train reaches B What distance didthe dove coverChallenge 149 e

lowastlowastBalance a pencil vertically (tip upwards) on a piece of paper near the edge of a tableHow can you pull out the paper without letting the pencil fallChallenge 150 e

The human body is full of such examples can you name a fewChallenge 146 s

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F I G U R E 54 Observation of sonoluminescence (copy Detlev Lohse)

lowastlowastIs a return flight by plane ndash from a point A to B and back to A ndash faster if the wind blowsor if it does notChallenge 151 e

lowastlowastThe level of acceleration a human can survive depends on the duration over which oneis subjected to it For a tenth of a second 30 д = 300ms2 as generated by an ejectorseat in an aeroplane is acceptable (It seems that the record acceleration a human hassurvived is about 80 д = 800ms2) But as a rule of thumb it is said that accelerations of15 д = 150ms2 or more are fatal

lowastlowastThe highest microscopic accelerations are observed in particle collisions where one getsvalues up to 1035 ms2 The highest macroscopic accelerations are probably found in thecollapsing interiors of supernovae exploding stars which can be so bright as to be visiblein the sky even during the daytime A candidate on Earth is the interior of collapsingbubbles in liquids a process called cavitation Cavitation often produces light an effectdiscovered by Frenzel and Schulte in 1934 and called sonoluminescence (See Figure 54)Ref 69

It appears most prominently when air bubbles in water are expanded and contracted byunderwater loudspeakers at around 30 kHz and allows precise measurements of bubblemotion At a certain threshold intensity the bubble radius changes at 1500ms in as littleas a few μm giving an acceleration of several 1011 ms2Ref 70

lowastlowastLegs are easy to build Nature has even produced a millipede Illacme plenipes that has750 legsThe animal is 3 to 4 cm long and about 05mmwideThis seems to be the recordso far

Summary of kinematics

The description of everyday motion of mass points with three coordinates as(x(t) y(t) z(t)) is simple precise and complete It assumes that objects can be fol-

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lowed along their paths Therefore the description does not work for an important casethe motion of images

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