Speed, acceleration, chameleons and cherry pit projectiles

This article has been downloaded from IOPscience. Please scroll down to see the full text article.

2012 Phys. Educ. 47 21

(http://iopscience.iop.org/0031-9120/47/1/21)

Download details:

IP Address: 88.200.77.175

The article was downloaded on 22/12/2011 at 16:03

Please note that terms and conditions apply.

View the table of contents for this issue, or go to the journal homepage for more

Home Search Collections Journals About Contact us My IOPscience

PAP ER S

www.iop.org/journals/physed

Speed, acceleration, chameleonsand cherry pit projectilesGorazd Planinsic and Andrej Likar

Faculty for Mathematics and Physics, University of Ljubljana, Slovenia

AbstractThe paper describes the mechanics of cherry pit projectiles and ends withshowing the similarity between cherry pit launching and chameleon tongueprojecting mechanisms. The whole story is written as an investigation,following steps that resemble those typically taken by scientists and cantherefore serve as an illustration of scientific reasoning and how scientificknowledge is built.

IntroductionIt is often argued that science subjects todayshould prepare future generations to be ableto make decisions in several situations that arerelated to science. In order to achieve this weshould in addition to teaching facts and datacreate opportunities for students to learn howscience knowledge is built, improved and appliedbut also what are its limitations. This paperdescribes a scientific explanation of a simpleactivity well known to most students that canserve for the purpose described above. Thestory is written as an investigation, followingsteps that resemble those typically done by thescientists. Such an investigative approach is foundto also be successful as a basis for teachingstrategies that engage students in learning thatmirrors scientific practice, such as the InvestigativeScience Learning Environment (ISLE) [1]1.

The ability to shoot projectiles by our ownforce has been very important for human history.From throwing stones and spears people graduallyimproved shooting techniques by using slings,catapults, bows, crossbows and other tools that

1 In ISLE, students construct new ideas themselves byfirst observing carefully selected phenomena, proposingmultiple explanations for their observations and then designingexperiments to rule out those explanations. Explanations thatthey fail to rule out are then used for further investigations andpractical applications.

help them use their own power in a more optimalway. The goal was very pragmatic: to kill animalsor enemies. Today some of these techniques arestill present, luckily as sporting activities: shotput, discus, hammer, javelin and archery, to nameonly those that are recognized as Olympic sports.Yet there is another humble and totally harmlesstechnique with as long a history as stone throwingthat remains today: cherry pit shooting. In thepresent article we will describe the simple physicsof cherry pit shooting supported by data obtainedfrom a high-speed video camera. At the end wewill explain what this technique has in commonwith the chameleon’s mechanism for projecting itstongue.

You can shoot cherry pits in two ways: byspitting the pit from the mouth or by squeezingthe pit between the fingers. Though both methodshave interesting physical backgrounds, we willfocus here on the latter one (note that in someplaces cherry pit spitting is known as an amateursport). The principle is very simple: insert a freshcherry in your mouth, eat all but the pit, spit the piton your hand, hold the pit between your thumb andforefinger, squeeze the pit as hard as you can whileslowly moving the point of compression towardsthe back of the pit—and the pit will shoot out at ahigh speed. A curious person will go beyond the‘sport level’ of this activity. She will want to knowhow the cherry pit launching mechanism works.

0031-9120/12/010021+07$33.00 © 2012 IOP Publishing Ltd P H Y S I C S E D U C A T I O N 47 (1) 21

G Planinsic and A Likar

Figure 1. Sequence of 12 video frames showing the cherry pit after launching. Time interval between successivephotos is 1/600 s. The scale on the ruler is in centimetres.

Box 1. Basic data for the cherry pit used in our experiments. The photograph belowshows the pit on the millimetre mesh as seen from two perpendicular directions.

Mass of the cherry (without the stem): 6.65 gMass of the pit (as used in experiments):mp = 0.60 g

Dimensions (see photograph on the right):a = 14 mmb = 9 mmc = 11 mm (perpendicular to a and b).

Observations and measurementsBasic data for the cherry pit used in our experi-ments are summarized in box 1. Our measure-ments are based from the analysis of high-speedvideos obtained with a Casio Exilim camera. Wefilmed videos at 600 frames s−1 and with shuttertime 1/4000 s. The shutter time was chosen afterthe approximate speed of the pit (10 m s−1) wasdetermined. At shutter time 1/4000 s the smearingof the pit due to motion (in the direction of motion)is about 3 mm, which is acceptable concerning thedimensions of the pit. A short shutter time requiresintense lighting, which was in our case obtainedusing a 1000 W studio lamp.

The speed of the cherry pit can be determinedfrom pit position measurements obtained fromsuccessive frames (figure 1). In order to getreliable measurements, the projectile must move inthe plane that is perpendicular to the camera sightand the camera should be as far away as possible(powerful zoom is of great help in this case).

The ruler has been added to set the scale of thedistances. Analysis of the photographs shows thatin the first 18 ms the pit moves nearly at a constantspeed of 13 m s−1 (47 km h−1). For repeated trialswe get similar results for the speed. Note thatduring the presented time interval the pit has fallendue to gravity only about 2 mm, meaning that forall practical cases the pit trajectory in our case canbe regarded as a straight line. Note also that themaximum speed of a javelin at the time of releaseachieved by Olympic champions (100 km h−1) isonly about twice as large as the maximum speed ofthe cherry pit launched by a middle aged physicistin average sporting condition.

Photographs in figure 1 also reveal that thepit motion is a combination of translation androtation. Closer examination of the photographsshows that the pit made one complete revolutionduring the first 10 video frames, meaning that thepit rotational speed ω was about 380 Hz.

22 P H Y S I C S E D U C A T I O N January 2012

Speed, acceleration, chameleons and cherry pit projectiles

Assumptions and simple explanationLet us first try to estimate the pit’s initial ac-celeration from the data that we obtained above.To do so, we need to make an assumption thatacceleration was constant until the pit acquired thefinal speed. Since we already know the final speedwe only need to estimate the distance at whichthe pit acquires this speed from the rest. It isreasonable to assume that this distance is equal tothe long axis of the pit a (in our case 14 mm) sincethe pit was always launched by pointing this axis inthe direction of launching. Using these data we es-

timate the acceleration to be a = v2f

2s = 6000 m s−2

or about 600 g. Since we know the mass of the pit(0.60 g) we can also estimate the force exerted onthe pit during the acceleration. Newton’s secondlaw tells us that the force is about 4 N.

Checking the resultsLet us pause for a moment and think about theresults that we have obtained so far. Comparedto g = 9.81 m s−2, 6000 m s−2 is a remarkableacceleration, but apart from this we have noparticular feeling for acceleration. We have morefeeling for force though. Anyone who has triedshooting cherry pits will agree that the force,which we have to apply with our fingers, ismuch larger than 4 N. The discrepancy betweenthe results and independent estimation (based oncommon sense in this case) indicates that thelaunching mechanism is not that simple and thatwe need to do more investigations to improve it.

Additional experimentsAgain we use high-speed video, but this time wetake a closer look at the launch itself. Figure 2shows a close view of another launch of the samepit in eight successive frames. Analysis of thephotographs shows that in this case the final speedwas about 8 m s−1.

Video analysis also reveals that during thelaunch the acceleration was not constant. Thepeak acceleration was about 1300 m s−2 and it wasachieved sometime between the sixth and seventhframes when the fingers slipped down the pointedpart of the pit. Roughly speaking this means thatthe distance on which the main acceleration occursis about half of the length of the pit. Simpleestimation as used above would give in this casean acceleration of about 2300 m s−2, showing thatthe actual peak acceleration in our previous case

was probably around 3400 m s−2 (assuming that inboth cases the launching mechanism and the waywe grabbed the pit were the same).

Improving the modelCloser inspection of the launch and additionalanalysis of the photographs gave us hints on howto improve the initial model. We decided thatour new model should take into account the shapeof the pit and the role of the fingers. Of coursewe will still try to keep the model as simple aspossible. We will assume that the pit has theshape of a double wedge and that the thumb andforefinger have equal masses and are subjected toequal muscle tensions. We will also assume thatfingers move only in a vertical direction and thatfriction and gravitational forces are negligible orirrelevant in our case. A pictorial representation ofthe model with forces acting on the three bodies isshown in figure 3.

Solution of the model and interpretation ofthe resultsThe wedge angle θ is determined by the pitdimensions and can be calculated from theequality tan θ

2 = ba . In our case we get θ ≈ 65◦.

The geometry of the model dictates that if eachfinger moves by x in a vertical direction, the pitwill move by x cot(θ/2) in a horizontal direction,which further implies that accelerations of thepit and fingers are connected by the followingexpression:

ap = af cot(θ/2).

In our case we find ap ≈ 1.6af. This equationrepresents a constraint imposed by the geometry ofthe problem. Writing Newton’s second law for thefinger and the pit we get two more equations thatdescribe the motion of the three bodies (note that inour model the same equation describes the motionof each finger). Solving these three equations givesexpressions for ap, af and F (the interested readerwill find the complete solution of the problem inbox 2). We will focus on the expression for theacceleration of the pit

ap = 2T tan(θ/2)

mp + 2M tan2(θ/2).

In our case we can simplify this expression bytaking into account that the mass of the pit mp =0.60 g is much smaller than the mass of the

January 2012 P H Y S I C S E D U C A T I O N 23

G Planinsic and A Likar

Figure 2. Sequence of eight video frames showing the closer view of the pit launch (left). Using LoggerPro software we have carried out a movie video analysis to obtain position–time and velocity-time graphs (right). The time interval between successive photos is 1/600 s. The yellow ticks on the photos indicate positions that are 2 mm apart.

s (m

m)

v (m

s–1

)

50

30

10

0.00 0.01 0.02

time (s)

time (s)

0

2

4

6

8

0.00 0.01 0.02

40

finger M . Namely, the mass of each fingeris approximately 20 g (we estimated the fingervolume and assumed the density of the finger isabout the same as the density of water). In thiscase the expression above simplifies to

ap ≈ T

Mcot(θ/2).

The result suggests that the acceleration ofthe pit is mainly determined by the tension forcein muscles, masses of fingers and the shape ofthe pit but it is independent of the mass of thepit. How can we interpret this result? Sincemp � M , the forces produced by the musclesmainly accelerate the fingers. The pit only slipsbetween the fingertips and has almost no influenceon the motion of the fingers. It is only the pitgeometry (angle θ in our case) that has some (butnot a major) effect on the pit acceleration.

Checking for consistencyThe analysis above calls for independent measure-ment of the muscle force that acts on the finger.This may look easier than it is. We need tomeasure the force with which the two fingers areacting on each other with the hand and fingers in aposition that is as close as possible to the positionduring the shooting of the pit. Conventional forcemeters are obviously not suitable for this purpose.We need a thin force meter that can be squeezedbetween the fingertips. For a rough estimation,simple plastic tubing proved to work well. Weused transparent plastic tubing with outer diameter6 mm and inner diameter 3 mm. The tubing wasstiff enough that it became flat only when it wascompressed between fingers with maximal force,similar to the force used when shooting the cherrypit (judged by personal feeling), (figure 4, left).The same tubing was then placed on the scales andpressed down until the same deformation of tubingwas achieved (figure 4, right). At that point the

24 P H Y S I C S E D U C A T I O N January 2012

Speed, acceleration, chameleons and cherry pit projectiles

Figure 3. Improved model of the cherry pit shooting mechanism (sizes are not to scale). Acceleration vectors are presented in red and force vectors are in blue. Only contact forces are shown (gravity forces are not relevant in this case): F—forces between fingers and the pit, T—muscle forces on the fingers.

M

M

F

F

T

af

af

T

θ mp ap

thumb

pit

forefinger

reading of the scales showed the mass, the weightof which is equal to the muscle force. In our casethe average force measured in this way was aboutT = 50 N.

Now we can independently estimate theacceleration of the fingers (af = T/M =2500 m s−2) and the acceleration of the pit (ap =1.6af = 4000 m s−2). This value is less than 20%off from the estimated acceleration for the first pitlaunching. Taking into account all the assumptionsthat we made in our model, the agreement is goodenough that we can accept the proposed model as

a satisfactory approximation. The model explainshow the motion of the fingers in one directionis transformed into the motion of the pit in aperpendicular direction. The model also showsthat for a typical cherry pit the acceleration ofthe pit is of the same order of magnitude as theacceleration of the fingers.

Limitations of the modelThough the proposed model proved to be usefulfor our purpose we should have in mind the maindiscrepancies between the model and the real ex-periment: in reality friction is not zero, fingers arenot equal in mass and size and they do not movealong a straight line, a real pit is not symmetric andits surface is not flat but curved. The differencesbetween the fingers and asymmetry of the pitare probably the main reasons for the torque thatcaused the observed rotation of the pit.

Different viewsLet us look at the experiment from the energy pointof view. As shown above, the main accelerationof the pit occurred around the sixth frame infigure 2, though the fingers’ muscles were undertension long before this moment. The launchingmechanism resembles in principle the one usedin catapults. First, elastic potential energy isgradually stored in the system and then it issuddenly transformed into the kinetic energy of thelaunching mechanism and the projectile. In ourcase this occurs when the fingertips slip along thepointed part of the pit.

Let us compare the kinetic energies of thefingers and the pit. Kinetic energy of the finger canbe estimated from the work done by the muscleforce T = 50 N while moving the finger over

Figure 4. Tubing force meter. Tubing is first compressed by the fingers in a similar position as when shooting thepit (left). After that the same deformation is achieved by pressing the tubing on the scales (right).

January 2012 P H Y S I C S E D U C A T I O N 25

G Planinsic and A Likar

Box 2. Mathematical treatment of the improved model.

Newton’s second law for one finger and for the pit reads as follows (see figure 3):

T − F cos(θ/2) = Maf

2F sin(θ/2) = mpap.

In addition we have a constraint equation determined by the geometry of the problem

ap = af

tan(θ/2).

Therefore we have three equations and three unknowns (ap, af and F). Solving this system ofequations we get

ap = 2T tan(θ/2)

mp + 2M tan2(θ/2),

af = 2T tan2(θ/2)

mp + 2M tan2(θ/2),

F = mpT

cos(θ/2)(mp + 2M tan2(θ/2)).

If mp � M the expression for pit acceleration takes the following form:

ap ≈ T

M tan(θ/2).

the distance equal to half of the height of the pit(b/2 = 5 mm). The calculated value for one fingeris 0.25 J. Note that we neglected friction, so thereal value for kinetic energy is somewhat lowerthan this. Translational kinetic energy calculatedfrom the maximal speed of the pit (13 m s−1) andits mass is only 0.05 J. As observed from the video,the pit also rotates. Rotational kinetic energy ofthe rigid body is equal to 1

2 Jω2, where J is therotational inertia and ω rotational speed. Sincewe are only interested in the order of magnitude,we can approximate the pit with a sphere witheffective radius r = (a + b + c)/3 = 11 mmand use the expression for J = 2

5 mr 2. Taking intoaccount the measured rotational speed of the pit(380 Hz) we calculate the rotational kinetic energyto be 0.002 J, which is only a small fraction of thetotal kinetic energy of the pit.

The values calculated above suggest that mostof the energy that was initially stored in themuscles is transformed into the kinetic energy ofthe fingers (which is sooner or later transformedinto heat) and only a small fraction is transformed

into kinetic energy of the pit. Note that this viewfrom the point of energies is consistent with theone we made when we studied forces.

Application of new knowledgeThe whole story described above can be used asan illustration of how physicists solve problemsand build new knowledge. But at the end of thestory students may ask questions like: ‘So what?Can we use this knowledge to solve some realproblems?’ The answer is YES!

A chameleon can catch prey located upto 1.5 body lengths away within a tenth of asecond by launching its tongue at an accelerationof 500 m s−2 [2]. Through history zoologistshave suggested several mechanisms to explain theimpressive performance of a chameleon’s tongue,such as that the tongue is ‘erected’ through anincrease in blood pressure or inflated by the lungs.In 1993 Zoond established the currently held viewabout how a chameleon projects its tongue [3],but the scientific debate about the role of differentparts of the tongue still goes on [2].

26 P H Y S I C S E D U C A T I O N January 2012

Speed, acceleration, chameleons and cherry pit projectiles

hyoid horn

sticky tip

accelerator musclesretractor muscles

Figure 5. Sketch of the chameleon with projected tongue (upper-left drawing), the structure of the chameleon’s tongue (lower-left drawing) and launch of the chameleon’s tongue in three successive moments (right).

A chameleon has a hollow tongue thatsheathes over a long, tapering cartilaginous spikecalled the hyoid horn. The hyoid horn is attachedto the head bones. The tongue consists of threebasic components: the sticky tip, the retractormuscles and the accelerator muscles (figure 5,left).

The accelerator muscles have a cylindricalshape and can contract radially to squeeze againstthe hyoid horn. The tongue launches when theaccelerator muscle begins to slide off the tip of thehyoid horn (figure 5, right).

The effect is similar to shooting the cherry pit,except in this case the pit (hyoid horn) remainsstationary and the squeezer (accelerator muscles)is propelled towards the prey. After hitting the preythe retractor muscles bring the tongue and the preyback to the mouth and a delicious feast can start.

Conclusions

We have followed the steps typically takenby scientists in their work. Based on initialobservations and assumptions we built a simplemodel. Since the predictions of this modelwere not consistent with measured values wedesigned additional experiments and improved theinitial theoretical model. We needed mathematicaltools to solve that model and we checked itagain against independent measurements. Weverified the consistency of different interpretationsand we became aware of the limitations of ourmodel. Finally we showed that new knowledgecan be applied in completely different situations,which can be described with an analogous model.Students should see the role each of these stepsplay in building scientific knowledge but alsorealize that science does not give absolute and

definite answers but only approximations that areconstantly improved.

Acknowledgments

The authors wish to thank Gary Williams andEugenia Etkina for valuable discussions and theanonymous referee for careful reading of the text.

Received 15 January 2011, in final form 3 April 2011doi:10.1088/0031-9120/47/1/21

References[1] Etkina E and Van Heuvelen A 2007 Investigative

Science Learning Environment—A ScienceProcess Approach to Learning Physicsed E F Redish and P Cooney, Research BasedReform of University Physics, AAPT, online athttp://per-central.org/per reviews/media/volume1/ISLE-2007.pdf

[2] Mueller U K and Krenenbarg S 2004 Power at thetip of the tongue Science 304 217–9

[3] Zoond A 1933 The mechanism of projection of thechameleon’s tongue J. Exp. Biol. 10 174–85

Gorazd Planinsic received his BSc andPhD in physics from the University ofLjubljana, Slovenia. He leadsundergraduate and postgraduate physicseducation programmes at the Departmentof Physics, University of Ljubljana and isa co-founder and collaborator in theSlovenian hands-on science centre TheHouse of Experiments.

Andrej Likar is a professor of physics atthe Faculty of Mathematics and Physics,University of Ljubljana. His research arealies in experimental nuclear physics,optimal filtering and physics education.He gives lectures on measurementsystems based on optimal feedback,measurement of ionizing radiation,development of physics and healthphysics.

January 2012 P H Y S I C S E D U C A T I O N 27