Skimming and skipping stones

Teaching Mathematics and its Applications Advance Access published November 8, 2006 TEACHING MATHEMATICS AND ITS APPLICATIONS, 2006

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doi:10.1093/teamat/hrl016

Skimming and skipping stones Steve Humble Submitted September 2006; accepted September 2006

Abstract This article presents an example of skimming and skipping stone motion in mathematical terms available to students studying A-level mathematics. The theory developed in the article postulates a possible mathematical model that is veri¢ed by experimental results.

1. Introduction Throwing a stone such that it skims across a body of water in a series of bounces, leaving a trail of expanding ripples on the water surface, is a very satisfying pastime. I have spent hours as a child and adult practising my skimming stone technique, and still fall a long way short of the Guinness World Record of 40 skips, set by Kurt Steiner (1) at the Pennsylvania Qualifying Stone Skipping Tournament on September 14, 2002. Recent mathematical developments have made it possible for this activity to be taught to students studying A-level mathematics. I took some of my A-level Mathematics students to Whitley Bay, on the North East coast of England, to experimentally test a simplified version of the new skimming stones (2,3) theory.

2. Historical background Historical records show that skimming stones was played in the 17th century, when it is said that James I of England would skim sovereigns across the River Thames in London. This is supposedly where the phrase ‘Ducks and Drakes’ came from; ‘to make ducks and drakes’, meaning to squander your money foolishly and recklessly. As with all games, there is a collection of words and phases associated with it. For example, a ‘plonk’ means a stone which sinks on the first hit with the water, or ‘pitty-pat’ means that the stone makes short skips at the end of the run. For more words and phrases associated with skimming stones, I have given some websites in the references to keep your students amused for hours learning the language of skimmers (4).

3. How to skim a stone? (i) The stone should be mostly flat, about the size of the palm of your hand and weigh around 10–15 g. (The Guinness Book of Records Skipping Stones holder recommends that you use triangular stones, stating that these tend to skip best as they are more stable). The Author 2006. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please email: [email protected]

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TEACHING MATHEMATICS AND ITS APPLICATIONS, 2006

Fig 1.

Skimming stones.

(ii) Hold the stone with your thumb and middle finger. Then put your index finger along the edge of the stone. (iii) Throw the stone as low to the water as you feel comfortable (Fig. 1.). (iv) When you throw the stone give it a flick, as it needs to have spin as well as forward motion. (v) The stone should hit the water with an angle of elevation of 10

4. Why does a stone skim? The most important aspect of skimming motion, is when the stone hits the surface of the water. The surface exerts a lift force on the stone and it is this reaction force that makes the rebound possible. The lift occurs due to the build-up of water under the front of the stone as it travels forwards, during its time in contact with the water. With each collision, energy is lost due to frictional effects and so the initial kinetic energy is dissipated after a number of rebounds. The induced spin acts mainly to maintain the orientation of the stone, and plays only a minor role in the number of possible bounces, as long as the stone is spinning at a rate of around 5 revolutions/s. To keep the mathematics at an applicable level for A-level students, I have ignored the effects of spin. For similar reasons, I have also chosen not to include in the model, irregularities in the surface of the water and air resistance.

5. Background Using discus theory from Neville De Mestre’s (5) wonderful book, and adapting ideas from Rosellini and Bocquet papers for square stones, I have been able to create theory which can be tested on a beach, in a stream or on a lake. I have kept the theory as simple as possible to make it available to students studying A-level mathematics. The topics covered are resolving forces, friction, projectiles, energy, simple harmonic motion, simple integration and algebra.

6. Mathematical theory Using discus theory we can write down the following equations of motion d2 x dvx 1 ¼ Av2 ðCD cos þ CL sin Þ ¼m dt dt2 2

ð1Þ

dvy d2 y 1 ¼ mg þ Av2 ðCL cos CD sin Þ ¼m 2 dt dt 2

ð2Þ

m m


Fig 2.

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Stone skimming model.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with m mass; v ¼ v2x þ v2y speed; density of water; CD drag coefficient; CL lift coefficient; angle of elevation between the stone and the water surface; A = ay/sin immersed area of a square-shaped stone, with side length a. As the stone hits the water (Fig. 2.) it will gradually be immersed to its greatest depth and at this point it will have the most upward lift. It will then rise to the surface and skip. This motion is modelled as simple harmonic motion (SHM). Assuming that the largest speed is in the forward motion, then v2 ¼ v2x þ v2y v2xo with vxo as the initial speed in the horizontal direction. Using this, equation [2] becomes m

d2 y 1 ay 2 ¼ mg v CL dt2 2 sin xo

With small angle , CL will dominate in the y direction. SHM theory gives the angular frequency squared as !2 ¼

CL v2xo a 2m sin

ð3Þ

Multiplying both sides of equation [1] by vx and integrating over the collision time (tc) you can obtain the loss in energy. ð tc 1 2 1 2 1 mvx mvxo ¼ vx Fx dt with Fx ¼ Av2xo ðCD cos þ CL sin Þ 2 2 2 0 ð tc 1 ¼ vx0 Fy dt given Fx ¼ Fy ¼ Av2xo CL 2 0 Assuming that Fy = mg on average over the collision period since ð1=tÞ ð tc 2 2 given tc ¼ ¼ vxo mgdt ¼ vxo mg ! ! 0

Ð tc 0

Fy dt ¼ mg then

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from equation [3]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m sin ¼ 2mg CL a

ð4Þ

Hence, the energy loss during one collision is independent of the speed. Therefore, we can write sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 2 2m sin mvx1 mvxo ¼ 2mg 2 2 CL a sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 2 2m sin mv mv ¼ 2mg 2 x2 2 x1 CL a sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 2 2m sin mv mv ¼ 2mg 2 x3 2 x2 CL a and so on until sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 2 2m sin mv mv ¼ 2mg 2 xN 2 xN1 CL a Adding all N relations gives sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 2 2m sin mv mv ¼ N2mg 2 xN 2 xo CL a

ð5Þ

The stone will stop at the final collision with the water (Nf) (Fig. 3.). This is when the total energy loss is equal to the initial kinetic energy, implying v2xNf ¼ 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2m sin ð6Þ Hence mvxo ¼ Nf 2mg 2 CL a

and

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v2xo CL a so Nf ¼ 4g 2m sin

Fig 3.

A four skip collision.

ð7Þ


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The distance between successive bounces can be calculated from standard projectile motion and energy loss equations. Equations [5] and [6] combine to give v2xo v2xN N ¼ v2xo Nf

ð8Þ

From projectile motion theory, the distance between impacts can be written as x ¼

2vy vx 2vyo vxo 2vyN vxN ; hence xo ¼ and xN ¼ g g g

ð9Þ

vyo = vyN since we have assumed that vy does not depend on the number of collisions, as it rebounds elastically. Using equations [8] and [9] with this gives rffiffiffiffiffiffiffiffiffiffiffiffiffiffi N ð10Þ xN ¼ xo 1 Nf

7. Experiments to test the theory The theory in equations [7] and [10] was tested using a digital camera with a ‘burst facility’. This facility allows 16 pictures to be taken in about 2 s at the rate of one every 1/7.5 s. To enable the analysis of the photographs, markers were placed at 50 cm intervals along the edge of the water and used as reference points. From this picture, (Fig. 4.), and other similar pictures we were able to find an approximate value for the stones speed vxo as 5 m/s. Using this value along with ¼ 1000 kgm3, m ¼ 0.1 kg, a ¼ 0.1 m and ¼ 100, it was possible to find approximate values of the parameters and CL in [7].

Fig 4.

An example of 16 shot burst photography.

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From thousands of pictures taken we selected the ones which showed the bouncing stone clearly, and from which we could analyse the validity of [10].

7.1. Experimental data Results given subsequently are from a four skip experiment (Fig. 5.). Distance between skips (cm) Collision N 0 1 2 3

Experimental 42.5 36.9 30.4 22.0

Theoretical xN ¼ xo

qffiffiffiffiffiffiffiffiffiffiffiffi 1 NNf

36.8 30.0 21.2

Results given subsequently are from a three skip experiment (Fig. 6.). Distance between skips (cm) Collision N 0 1 2

Experimental 61.9 49.0 35.0

Theoretical xN ¼ xo

qffiffiffiffiffiffiffiffiffiffiffiffi 1 NNf

50.5 35.7

Due to the camera angles the analysis of the photographs involved dealing with perspective. This gave the opportunity to discuss maths with photography and art teachers, offering opportunities for cross curricular links to develop.

Fig 5.

Example photograph of a four skip experiment. Nf ¼ 4.


Fig 6.

Fig 7.

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Example photograph of a three skip experiment. Nf ¼ 3.

Skimming stone experimenters on Whitley Bay beach.

8. Conclusion The results seem on average to validate the theory, but clearly with so many variables further experiments and refinements in the theory are necessary. I would be very interested to hear from teachers who try skimming experiments with their students. I believe, working in an experimental fashion with mathematical ideas creates not only more interest in the subject, but a greater awareness of how maths permeates all areas of the world.

Acknowledgements I wish to thank Ben Haslam for his ‘Skimming Stones’ picture and his helpful advice on photography.

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I also wish to give particular thanks to James Wong, Michelle Ngu, Jeslin Ha, Yik Shien Yee, Daniel Humble and all the other students whom I have enthused into spending time skimming stones down by the waters edge over the years (Fig. 7.)

References 1. http://pastoneskipping.com/steiner.htm. 2. Rosellini, L., Hersen, F., Clanet, C., and Bocquet, L. (2005) Skipping stones. Journal of Fluid Mechanics, 543, 137–146. 3. Bocquet, L. (2003) The Physics of stone skipping. American Journal of Physics, 71, 150–155. 4. http://www.stoneskipping.com & http://www.yeeha.net/nassa/guin/g2.html. 5. N de Mestre. The Mathematics of Projectiles in Sport. UK: Cambridge University Press.

Steve Humble (aka DR Maths) works for The National Centre for Excellence in the Teaching of Mathematics in the North East of England (http://www.ncetm.org.uk). He believes that the fundamentals of mathematics can be taught via practical experiments. He is the author of the book The Experimenter’s A to Z of Mathematics, which develops an experimenter’s investigative approach to mathematical ideas. Always having had great fun playing with maths, he enjoys teaching this to others. For more information on DRMaths go to http://www.ima.org.uk/Education/DrMaths/DrMaths.htm. Address for correspondence: Steve Humble, NCETM North East, St. Mary’s Court, 55 St. Mary’s Road, Sheffield S2 4AN, UK. E-mail: [email protected]